College Level Physics

College Level™ Physics

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TABLE OF CONTENTS Preface........................................................................................................ 1 Chapter 1: Kinematics: The Study of Motion................................................ 6 One-Dimensional Motion ................................................................................................ 6 Displacement and Measurements ................................................................................... 6 Distance ............................................................................................................................ 8 Vectors.............................................................................................................................. 8 Velocity and Acceleration ................................................................................................ 9 Falling Objects ............................................................................................................... 13 Key Takeaways ............................................................................................................... 15 Quiz ................................................................................................................................ 16 Chapter 2: Two-Dimensional Kinematics .................................................. 20 Vectors in Two Dimensions ........................................................................................... 20 Vector Addition and Subtraction via Graphical Methods ............................................. 22 Multiplication of Vectors ............................................................................................... 24 Resolving Vectors ........................................................................................................... 24 Projectile Motion ........................................................................................................... 28 Range of a Projectile ...................................................................................................... 31 Velocity in Two Dimensions .......................................................................................... 32 Key Takeaways ............................................................................................................... 35 Quiz ................................................................................................................................ 36

Chapter 3: Newton’s Laws of Motion ......................................................... 40 Dynamics ........................................................................................................................ 40 Newton’s First Law ........................................................................................................ 41 Newton’s Second Law .................................................................................................... 41 Newton’s Third Law ....................................................................................................... 43 Forces in Physics ............................................................................................................ 44 Tension ........................................................................................................................... 46 Applications of Newton’s Laws ...................................................................................... 48 Newtonian Forces .......................................................................................................... 49 Friction ........................................................................................................................... 50 Elasticity, Stress, and Strain .......................................................................................... 52 Tension and Compression ............................................................................................. 53 Key Takeaways ............................................................................................................... 55 Quiz ................................................................................................................................ 56 Chapter 4: Circular Motion and Gravitation .............................................. 60 Angular Velocity ............................................................................................................. 60 Centripetal Forces .......................................................................................................... 62 Centripetal Force ........................................................................................................... 63 The Coriolis Force .......................................................................................................... 65 Newton’s Law of Gravitation ......................................................................................... 67 Kepler’s Laws ................................................................................................................. 69 Quiz .................................................................................................................................71

Chapter 5: Work and Energy ..................................................................... 75 Work ............................................................................................................................... 75 Kinetic Energy ................................................................................................................ 77 Gravity and Potential Energy......................................................................................... 78 Conservative and Nonconservative Forces....................................................................80 Nonconservative Forces ................................................................................................. 83 Conservation of Energy.................................................................................................. 84 Transformation of Energy ............................................................................................. 86 Power.............................................................................................................................. 86 Key Takeaways ...............................................................................................................88 Quiz ................................................................................................................................ 89 Chapter 6: Momentum and Collisions ....................................................... 93 Linear Momentum ......................................................................................................... 93 Impulse .......................................................................................................................... 94 Subatomic Collisions and Momentum Conservation.................................................... 96 Elastic Collisions in One Dimension ............................................................................. 97 Inelastic Collisions in One Dimension .......................................................................... 98 Collisions in Two Dimensions ....................................................................................... 99 Key Takeaways ............................................................................................................. 102 Quiz .............................................................................................................................. 103 Chapter 7: Statics, Torque, and Rotational Motion .................................. 107 Statics ........................................................................................................................... 107 Torque .......................................................................................................................... 107 Stable Equilibrium ........................................................................................................110

Simple Machines ........................................................................................................... 111 Angular Acceleration .................................................................................................... 113 Rotational Motion ......................................................................................................... 114 Work of Rotation........................................................................................................... 117 Angular Momentum...................................................................................................... 119 Collisions of Rotating Objects...................................................................................... 120 Key Takeaways ............................................................................................................. 124 Quiz .............................................................................................................................. 125 Chapter 8: Fluid Statics and Dynamics .................................................... 129 Fluid Statics.................................................................................................................. 129 Density ......................................................................................................................... 130 Pressure ........................................................................................................................ 130 Pascal’s Principle ......................................................................................................... 132 Archimedes Principle ................................................................................................... 134 Surface Tension............................................................................................................ 136 Fluid Flow .................................................................................................................... 138 Bernoulli’s Equation .................................................................................................... 139 Flow and Turbulence ................................................................................................... 144 Viscous Fluids and Terminal Velocity ......................................................................... 144 Diffusion through a Fluid ............................................................................................ 145 Key Takeaways ............................................................................................................. 147 Quiz .............................................................................................................................. 148

Chapter 9: Temperature and Gas Laws.....................................................152 Temperature................................................................................................................. 152 Kinetic Theory .............................................................................................................. 153 Thermal Expansion of Liquids and Solids....................................................................157 Thermal Stress ............................................................................................................. 159 Ideal Gas Law ............................................................................................................... 159 Phase Changes .............................................................................................................. 161 Evaporation and Boiling .............................................................................................. 163 Key Takeaways ............................................................................................................. 164 Quiz .............................................................................................................................. 165 Chapter 10: Heat and Heat Transfer ........................................................ 169 Heat and Heat Capacity ............................................................................................... 169 Heat Transfer Methods ................................................................................................ 174 Conduction ....................................................................................................................175 Convection .................................................................................................................... 177 Radiation ...................................................................................................................... 178 Key Takeaway............................................................................................................... 180 Quiz ............................................................................................................................... 181 Chapter 11: Thermodynamics .................................................................. 185 First Law of Thermodynamics ..................................................................................... 185 Second Law of Thermodynamics ................................................................................. 189 The Four-Stroke Engine .............................................................................................. 192 Application of Thermodynamics ................................................................................. 196 Heat Pumps .................................................................................................................. 197

Refrigerators and Air Conditioners ............................................................................. 198 Entropy......................................................................................................................... 198 Key Takeaways ............................................................................................................. 201 Quiz ..............................................................................................................................202 Chapter 12: Oscillatory Motion and Waves .............................................. 206 Hooke’s Law and Oscillation .......................................................................................206 Period and Frequency ................................................................................................. 208 Simple Harmonic Motion ............................................................................................209 Pendulums ................................................................................................................... 213 Damped Harmonic Motion.......................................................................................... 215 Resonance .................................................................................................................... 216 Waves ............................................................................................................................217 Superposition and Interference ................................................................................... 219 Wave Energy ................................................................................................................220 Key Takeaways ............................................................................................................. 222 Quiz .............................................................................................................................. 223 Chapter 13: Electrical Charges and Electrical Fields ................................ 227 Static Electricity ........................................................................................................... 227 Conductors and Insulators .......................................................................................... 228 Coulomb’s Law ............................................................................................................. 230 Electrical Fields ............................................................................................................ 231 Key Takeaways ............................................................................................................. 237 Quiz .............................................................................................................................. 238

Chapter 14: Electric Potential and Electric Energy .................................. 242 Electric Potential Energy ............................................................................................. 242 Equipotential Lines ...................................................................................................... 245 Capacitors..................................................................................................................... 247 Key Takeaways ............................................................................................................. 252 Quiz .............................................................................................................................. 253 Chapter 15: Electric Current and Circuits ................................................ 257 Electric Current ............................................................................................................ 257 Ohm’s Law....................................................................................................................260 Electric Resistance and Resistivity .............................................................................. 261 Electric Power and Energy........................................................................................... 263 Alternating Current and Direct Current ...................................................................... 264 Circuits ......................................................................................................................... 265 Electromotive Force ..................................................................................................... 266 Key Takeaways ............................................................................................................. 268 Quiz .............................................................................................................................. 269 Chapter 16: Magnetism ........................................................................... 273 Magnets ........................................................................................................................ 273 Ferromagnets ............................................................................................................... 273 Electromagnets ............................................................................................................ 274 Magnetic Fields ............................................................................................................ 275 Magnetic Forces ........................................................................................................... 275 Motors ......................................................................................................................... 280 Meters .......................................................................................................................... 281

Magnetic Field and Currents ....................................................................................... 282 Electromagnetism ........................................................................................................ 284 Inductance.................................................................................................................... 286 Electromagnetic Waves................................................................................................ 287 Key Takeaways ............................................................................................................. 293 Quiz .............................................................................................................................. 294 Chapter 17: Light and Optics ................................................................... 298 Geometric Optics ......................................................................................................... 298 Reflection ..................................................................................................................... 298 Refraction ..................................................................................................................... 299 Dispersion .................................................................................................................... 302 Lens and Light Systems ............................................................................................... 303 Mirror Images ..............................................................................................................306 Wave Optics .................................................................................................................308 Key Takeaways ............................................................................................................. 310 Quiz ............................................................................................................................... 311 Chapter 18: Quantum Physics ................................................................. 314 Quantization................................................................................................................. 314 Photoelectric Effect ...................................................................................................... 316 Ionizing Radiation ....................................................................................................... 318 Ultraviolet Radiation ................................................................................................... 318 Visible Light ................................................................................................................. 319 Low-Energy Photons.................................................................................................... 319 Particle-Wave Duality .................................................................................................. 320

Heisenberg Uncertainty Principle ............................................................................... 320 Key Takeaways ............................................................................................................. 321 Quiz .............................................................................................................................. 322 Summary ................................................................................................ 326 Course Questions and Answers ............................................................... 331

PREFACE The intention of this course is to cover all of the major topics one would learn in a typical college physics course. Physics is the study of the physical attributes of matter and energy. Matter can consist of small particles and large objects—as small as subatomic particles or as large as planetary bodies and other celestial bodies. Regardless of size, everything must follow specific physical laws and principles, which will be uncovered by you as you study the material in this course. There are also mathematical aspects of the nature, applicable forces, and seemingly invisible energies involved in physical structures that will be clearly explained as part of this course. While there is mathematics involved in the study of physics, the topics we will cover can be understood in both mathematical and nonmathematical ways.

Chapter one in the course introduces the subject of kinematics, which is the study of the motion of objects without regard to the objects’ masses and without the consideration of the particular forces that may have caused the movement of the objects.

Objects are always in motion—even if they do not appear to move as there is the continuous vibrations of molecules and atoms that make up the object. In this chapter, we will look at the basics of movement, including velocity, acceleration, and the acceleration of a body that is free-falling on earth.

The focus of chapter two is the motion of an object in two dimensions. Many things do not simply go in a straight line or in an up-and- down fashion. These include celestial objects in orbit, automobiles that travel around a curve, and the arcing of a ball.

There are different equations and different vectors that apply to these types of movements, which need to be studied and memorized. Three-dimensional kinematics is very similar to two-dimensional kinematics, except that the x, y, and z axes are part of this discussion. All of the same physics principles you will learn in the course apply to motion in more than one direction, allowing for more types of situations involving the motion of objects to become solvable.

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Chapter three looks into Newton’s Laws of Motion as well as applications of these laws. It goes further in the discussion of motion to outline things like friction, drag, and elasticity when it comes to motion. Chapters one and two will involve just a discussion of motion in its purest form without an understanding of the forces behind the motion.

What necessarily follows in chapter three is a discussion of “dynamics”, which are the forces that affect the movement of the different objects and systems. It turns out that Newton’s Laws are the foundation of the study of dynamics. The laws were uncovered in 17th Century but still apply today on earth as well as in space. They apply to the study of classical mechanics, meaning that they apply to speeds less than light and sizes of objects greater than molecules.

Chapter four in the course deals with uniform circular motion, which is defined as motion in a circular path at a constant speed. It involves things like pure rotational motion, which occurs when an object is traveling a path that is centered around a single point. This is different from pure translational motion, which is motion that has no rotation associated with it. There is mixed motion as well, with circular and rotational components. In addition, related components discussed in the chapter are the Coriolis effect and Kepler’s laws of planetary motion, which also apply to circular motion.

The topics of chapter five are the concepts of work, energy, and power. Work involves the process of getting something done using forces or the transfer of energy from one state to another. Energy, as you may know, cannot be created or destroyed; it can only be transferred from one form to another. The chapter also introduces the topic of power, which is a closely related term that is the rate (energy amount per time period) at which work is done or energy converted. The relationship between energy, work, and power will be covered as part of this chapter.

Chapter six in the course deals with the subjects of linear momentum and collisions.

It involves first the topic of linear momentum, which is the velocity of something and the product of its mass. Impulse is also covered, which is the change in momentum of an object in a system. Momentum leads an object toward collisions with other objects.

There are elastic collisions and inelastic collisions that differ in their apparent conservation of momentum. Each of these topics builds upon things that will have

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already been learned in the previous chapters on force, velocity, and mass, as well as Newton’s laws.

Chapter seven opens up the topics of statics, torque, and rotational motion. While motion is primarily covered in the beginning of the course, there is an entire branch of physics associated with nonmoving forces, which are collectively referred to as statics.

Torque involves forces that act in a twisting fashion in order to cause motion or the potential for motion. This leads to the issue of rotational motion, along the lines of rotational acceleration and motion that is not necessarily uniform as will be discussed chapter four of the course.

Chapter eight in the course gets into the subjects of fluid statics and fluid dynamics.

The liquid state represents a state of matter in which there is some cohesion of the molecules that is different from that of solids and gases. There are characteristics of fluids, such as density, pressure, and other factors that will be explained as part of the chapter. In addition, there are aspects of this state of matter that specifically touch on the dynamics or flow properties of liquids in physics and biology. The flow of fluid can be relatively laminar or turbulent, depending on a variety of factors, including the viscosity of a particular liquid, which will be discussed as part of this chapter.

The focus of chapter nine is temperature and the properties of substances related to temperature, such as evaporation, humidity, and phase changes of a given substance.

This leads to a discussion of kinetics and kinetic theory as it applies to gases. The ideal gas law is covered, with some attempt to link ideal gases with real gases. As it turns out, all substances are affected by their own temperature and the temperature of their surroundings, with expansion occurring in solids, liquids, and gases to varying degrees.

This chapter combines theories and influences of both physics and physical chemistry as they apply to molecular systems and macroscopic substances.

Chapter ten in the course explores issues related to heat, which is itself a form of energy. Heat can be stored in a substance or transferred from one substance to another.

Quite often, heat is not recognized until it is in transit from one thing to another. There are different types of heat transfer methods, including convection, conduction, and

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radiation. There are fundamental issues in physics related to heat transfer and specific laws that apply to the transfer of heat energy, which are covered in this chapter.

Chapter eleven in the course is also concerned with temperature and heat transfer; however, it expands these topics further and goes into the laws of thermodynamics that apply to heat as it relates to energy and work. In earlier chapters, the topic will be on heat as a pure form of energy transfer, while this chapter is about the ability of heat transfer to perform work. Heat is like any other form of energy. In many systems you will become familiar with in this chapter, heat transfer has the ability to do things like run engines and allow machines to function. The laws of thermodynamics are not just laws of physics; they have practical applications that are seen in everyday life.

Chapter twelve delves into oscillatory motion, which is movement back and forth between two points. There are many systems that oscillate, some of which create waves.

Waves can be visual, such as the waves in a swimming pool or ocean. Other waves that aren’t commonly seen as waves include sound waves and light waves. Waves create disturbances that carry energy, from small waves that carry light energy to large waves that create tsunamis and earthquakes. Waves, as it turns out, have the energy to augment each other or to interfere with one another, which is covered in this chapter.

Chapter thirteen in the course introduces the physics of electricity by covering electric charges and electric fields. Static electricity is just one aspect of electricity that is well understood by anyone who touches an object and gets an electric shock. Also covered is the topic of electromagnetic force, which is a type of energy that applies to electrical fields. A natural part of the discussion is that of conductors of electricity and insulators of electricity, which are also a part of this chapter.

The focus of chapter fourteen is to extend the understanding of electricity to include electric potential and electrical energy. It introduces aspects of electricity such as voltage and the storage of electrical energy by capacitors. In this chapter, you will find that electrical energy and voltage are not the same thing because small batteries can have the same voltage as large batteries but will not create the same amount of electrical energy. In this chapter, the actual use of electricity in everyday electrical situations is discussed as well.

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Chapter fifteen in the course covers the topics of electric currents and circuits.

Electric current is defined as the movement of a charge from one place to in another over a period of a certain time. Such a thing has the capacity to do work. This then gets into Ohm’s law as it applies to electrical resistance and to the subject of circuits. There are AC current situations and DC current situations, which are things that many have heard about but will now understand from the perspective of physics.

Chapter sixteen focuses first on magnetism and then on the relationship between magnetism and electricity. Magnetism is common in nature and explains many things related to what we see in nature, such as the magnetic poles on Earth. Magnetism can cause electrical currents to be generated, which is also discussed as part of this chapter.

The chapter also brings into focus the topic of electromagnetism and electromagnetic waves. You will see that there is a vast range of these types of waves that go from those that heat food (which are microwaves) to waves that are much higher in frequency than those a person can see (which is the narrow spectrum of visually-seen electromagnetic waves).

Chapter seventeen in the course gets into electromagnetic waves in the spectrum of visible light and its properties. Light comes in rays from various sources and is subject to reflection, refraction, and diffraction—each being aspects of waves not unique to light waves; however, they are unique phenomena seen in everyday physics and in life. The properties of light as it relates to passing it through a lens and the properties of light as it strikes a mirror are discussed in this chapter as well as the physics of light optics.

Chapter eighteen in the course briefly introduces quantum physics, which is the physics that involves the behavior of very small things. While most properties of physics can be explained on a macroscopic scale, quantum physics describes physical principles in ways that could not have been understood in the early days of physics. As it turns out, when breaking matter down into its smallest states, the study of physics becomes very different from that which is understood on a macroscopic scale. Issues that come up when looking at matter at an atomic level are the basis of this chapter.

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CHAPTER 1: KINEMATICS: THE STUDY OF MOTION This chapter introduces the subject of kinematics, which is the study of the motion of objects without regard to the objects’ masses and without the consideration of the particular forces that may have caused the movement of the objects. Objects are always in motion—even if they do not appear to move as there is the continuous vibrations of molecules and atoms that make up the object. In this chapter, we will look at the basics of movement, including velocity, acceleration, and the acceleration of a body that is freefalling on earth.

ONE-DIMENSIONAL MOTION The study of physics naturally begins with the study of kinematics, which looks into motion without considering the cause. There is one-dimensional kinematics and twodimensional kinematics. One dimensional kinematics involves motion in a straight line, while two-dimensional kinematics looks at motion along a curved path of some sort.

DISPLACEMENT AND MEASUREMENTS The first thing that is necessary to describe the motion of an object is its position at point A at any given point in time. There needs to be some type of reference point.

Objects on earth usually use the earth as a reference point; however, for objects in an airplane, for example, the reference point is in the airplane.

Displacement is the movement relative to a reference frame or the change in position of an object. In the movement of an object in one direction, the displacement is the difference between the position of the ending point x1 and the starting point x0, which is called a delta x. Any discussion of displacement requires a knowledge of these two points. Use the Greek letter delta (∆) to describe the change in direction.

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You should know the SI units for displacement and other aspects of physics. For displacement, the SI unit is meter (m); however, kilometers, feet, miles, and inches can be used. You may need to use a calculator to transfer the units given to the SI units.

SI units in physics: •

Length—meter

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Mass—kilograms

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Time—seconds

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Electric current—Amperes

There are several prefixes used to determine things that are not SI units, such as these: •

Exameter—1018 meters (the distance light travels in a century)

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Petasecond—1015 seconds (approximately 30 million years)

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Terawatt—1012 watts (the output of a laser)

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Gigahertz—109 hertz (microwave frequency)

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Megacurie—106 curies (high degree of radioactivity)

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Kilometer—103 meters (about 0.6 miles)

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Hectoliter—102 meters (26 gallons)

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Decagram—101 grams (teaspoon of butter)

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Deciliter—10-1 liters (about half a soda can)

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Centimeter—10-2 meters (thickness of a fingertip)

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Millimeter—10-3 meters (the width of a flea)

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Micrometer—10-6 meters (microscope-sized item)

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Nanogram—10-9 grams (weight of a speck of dust)

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Picofarads—10-12 Farads (small radio capacitor)

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Femtometers—10-15 meters (size of a proton)

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Attoseconds—10-18 seconds (the time light crosses in an atom)

It is not important to know the meaning of the different measurements—only that you memorize the different prefixes, which are used in physics to remember the different sizes or measurements of things.

In displacement, there will be a direction as well as a magnitude. When it comes to distance in one direction, there will be a right-directed arrow in the positive direction and a left-directed arrow in the negative direction. The length of the arrow determines the number of units of the displacement. When doing a displacement problem, you will need to determine which is the positive direction and which is the negative direction (generally, up or to the right indicates a positive direction).

DISTANCE Distance is measured simply as the magnitude of the displacement between two positions. It is not the same as the distance traveled between them as the distance traveled may not be the shortest distance between two points. For example, distance traveled can account for overshooting the distance in one direction and then coming back to the final distance. The distance traveled will equal the distance between the two if the shortest distance between the two points was used in the traveling process. There is no sign for distance because it has no sign. Do not confuse the distance traveled by the magnitude of the displacement. The magnitude of the displacement is the change in distance from the beginning and the endpoint without a sign or direction.

VECTORS So far, you have learned the difference between distance and displacement.

Displacement has a direction and magnitude, whereas distance and distance traveled is only a magnitude. Displacement is an example of a vector and a scalar quantity.

Vectors will have a direction and magnitude. The direction of a vector in one direction is given by a plus or minus sign and will be represented by an arrow. The longer the vector length, the greater is the magnitude of the displacement.

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A scalar will be a quantity that has a magnitude but no direction. Scalars can be negative, depending on the scale. For example, it can be -20 degrees Celsius, which is because of the scale that determines temperature. There is no arrow in a scalar quantity.

Distance and speed are scalars; however, velocity and displacement are vectors.

In order to describe the direction of a specific vector, there must be a coordinate system.

In one dimension, the coordinate system is a line that starts from the left and ends on the left, usually a number that is along the x-axis. Movement to the right is positive, while movement to the left is negative. If using an up and down motion, movement upward is positive and movement downward is negative. This is done on the y-axis.

Sometimes, with falling objects, movement downward is considered positive. In reality, it doesn’t matter which direction is positive as long as you define this clearly. If not defined, use the x-axis and y-axis as is seen in mathematics to define movement in one direction.

VELOCITY AND ACCELERATION Velocity and acceleration take time into consideration. Time, in physics, represents change or an interval in which change occurs. It cannot be possible to know that time has passed unless something changes during that time. As mentioned, the SI unit for time is the second, with the change being a certain number of units per second.

As mentioned, something should change if time is a factor. There can be the length of a certain event, beginning at noon and ending at 12:30 pm. In such cases, the elapsed time is 30 minutes (note that the use of seconds here would be inappropriate). As time moves forward, this will always be a positive number and is therefore a scalar and not a vector. The delta symbol is used to represent the change in time (or delta-t). For simplicity’s sake, motion is said to start at time zero so that there is no definite need to use the delta-t designation and just “t” is used to represent time.

Velocity is the displacement of something over time. It can be listed in miles per hour or kilometers per hour. In SI units, velocity will be in meters per second. The average velocity is the displacement divided by time. This is a vector because displacement is a vector so it can be a negative number. If the object moved -4 meters relative to the initial

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placement over 5 seconds, the total velocity will be -0.8 meters per second. This does not say anything about the actual velocity at any given point in those five seconds; it only lists the average velocity over that time period.

There is such a thing as the instantaneous velocity, which is easy to represent as you look at a car’s speedometer. It is the velocity at a specific instant. The speedometer will show the magnitude of the change in distance over time but will not show the direction.

For this reason, speed is a scalar measurement, while velocity is a vector measurement.

Instantaneous velocity is the velocity of the object over an infinitesimally small period of time. In calculus, this is the limit of the change in distance as time approaches zero.

While speed and velocity are used in everyday language in the same way, remember that they are different. Instantaneous speed is the magnitude of the instantaneous velocity (or the absolute value of the velocity without a direction). It will always be a positive number. The average speed is the total distance traveled divided by an elapsed period of time. The average speed can exceed the average velocity because it represents the total distance traveled. If you start at one point, go ten miles and return in an hour, the velocity will be zero but the average speed will be 20 miles per hour.

Graphing can be used to describe motion. In such cases, the graph will be the position of the individual or object over time. Graphs can be very simple, assuming that the speed is constant during the time period. It also makes the assumption that the route taken of the person or object is a straight line, which is not always the case. Figure 1 shows the representation of a zero-velocity situation:

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Figure 1.

Acceleration is the change in velocity over time. The greater the acceleration, the greater is the change in velocity over time. The average acceleration is the rate at which velocity changes or the delta-v divided by the delta-t. The SI units for acceleration are meters per second squared. This is the velocity change every second. Remember that velocity is a vector; this means it can change in direction as well as magnitude.

Acceleration is always in the direction of the change in velocity, it is not always in the direction of motion. A slowing down of acceleration is opposite to the direction of its motion; this is referred to as deceleration.

Negative acceleration is not the same as deceleration. Negative acceleration is acceleration in a negative direction. It might or might not be deceleration because an object can be accelerating quickly but in the opposite direction from where it started. If acceleration has the same sign as the velocity, the individual or object is said to be speeding up. If acceleration has the opposite sign as the velocity, the object can be said to be slowing down. Since velocity is a vector, so is acceleration. Deceleration changes the magnitude of the velocity but does not necessarily affect the direction of the object.

There is also instantaneous acceleration, which is done similarly to instantaneous velocity and speed. It is done by calculating the acceleration over an infinitesimally

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small period of time. This can be done using calculus or by using algebra and a very small period of time. If the acceleration and the velocity have the same sign, the object is speeding up (in either direction). If the acceleration and the velocity have different signs, the object is decelerating. The average acceleration is the delta-v divided by the delta-t or the change in velocity over change in time.

In determining the average velocity, one must assume that the acceleration is going to be a constant value. With this in mind, the average velocity will be the starting velocity and the ending velocity, divided by two. If the velocity goes from 30 miles an hour to 60 miles an hour, the average velocity will be 45 miles per hour during that time. There will be a linear relationship between the displacement and the average velocity so that, if velocity 1 is twice the velocity 2, the person will go twice as far in the first instant versus the second instant.

You can also calculate the final velocity of an object by knowing the initial velocity and the rate of acceleration or deceleration (plus the time frame). If an object is traveling at 10 meters per second and decelerates at 1 meter per second squared, you can calculate the final velocity after x number of seconds, although, if it decelerates at 1 meter per second squared, you can determine that, after ten seconds, the final velocity will be zero.

Again, this relies on the fact that acceleration is a constant, which may or may not be true but must be assumed if you don’t have the advantage of complex calculus equations to determine the exact change in velocity.

When solving these types of equations, there are several things that need to be taken into account besides the fact that time always starts at zero and that acceleration is held at a constant. It helps to draw a simple sketch so you know what the vectors look like.

Remember that velocity, acceleration, and displacement are vectors and you can draw them out to identify the known quantities and what the problem is asking for.

Then find the equation that fits for the problem. If there is one equation, there will be one unknown. If there are two unknowns, there should be two equations. Keep track of the units of measurement and make sure the problem has all the same units of time, velocity, distance, etcetera. Finally, make sure the answer is reasonable and compares with what you know to be true of natural phenomena.

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FALLING OBJECTS From what you already know about the physics of motion in one direction, you can solve the problems related to falling objects. In such cases, there is a constant acceleration, independent of the mass of the object—unless you factor in things like air resistance.

Without this effect of air resistance, heavy objects will fall at the same rate as lighter objects. In the real world, air resistance does play a role in the velocity of a fall but, for short distances, it is negligible. “Free fall” is what the fall of an object is called when there is no friction or air resistance involved.

According to the force of gravity, objects will fall toward the center of the Earth. In actuality, this “force of gravity” is nothing more than “acceleration”. It turns out that acceleration is a constant and is based on the mass of the earth. While it is an acceleration number, it is given the symbol “g”. It is considered as a constant on any place on the earth. On earth, it is a constant of 9.8 meters per second squared.

In reality, gravity varies slightly depending on altitude, latitude, local topography, and the presence of underlying geological formations. The direction, however, will always be in the direction of the center of the earth. We call it a “vertical” drop even though it is actually toward the earth’s center. If the coordinate system is used, this acceleration will be a negative number as the objects are accelerating in a downward fashion. In some cases, the coordinate system is reversed so that the force of gravity and all directions are considered positive.

Velocity in these circumstances is considered vertical in nature. If an object is dropped, the initial velocity is said to be zero and the object is in free-fall. Motion will be in one direction and the acceleration will always be a constant, which is 9.8 meters per second squared.

The kinematic equations you should know for objects related to falling are described in figure 2. In such cases, acceleration is considered to be -g:

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Figure 2.

In cases where an object is thrown upward, the velocity is positive because the direction of the object is up and up is a positive direction. Because gravity will oppose the throwing upward of an object, this will be a negative value as the object’s acceleration due to gravity opposes its upward velocity. If an object is thrown downward, the velocity will be negative and the force of gravity will be negative as well but will be of the same side in that they are working in the same direction.

While it generally isn’t necessary, experiments can be done to determine the exact force of gravity at a specific point on earth. A small metal ball can be dropped and the time it takes to drop a certain distance can be measured. Using the equations in figure 2, the force of gravity can be calculated. Again, you need to choose the coordinates so that up is positive and g is negative in most situations; however, you can reverse that entirely so that the downward direction is positive and the acceleration due to gravity is positive.

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KEY TAKEAWAYS •

You will need to know the SI units for time, displacement, and distance.

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You will need to know the different prefixes that define larger or smaller units of measure.

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Displacement, velocity, and acceleration are vectors, while distance, time, and speed are scalar values.

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Free-falling objects will have the force of gravity be the number used for acceleration.

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The force of gravity is about -9.8 meters per second squared.

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QUIZ 1. What is the standard international unit for displacement? a. Kilometers b. Meters c. Feet d. Miles Answer: b. The SI or (standard international) units for displacement are meters, although any of the other units might be listed, requiring a calculation to transfer the object from its displacement unit given to the unit of meters.

2. In remembering the prefixes of SI units, what would be the greatest measurement of distance? a. Exameter b. Decameter c. Kilometer d. Centimeter Answer: a. Exameter is the largest measurement of distance, being 1018 meters, which is the distance light travels in a century.

3. Which is the smallest unit of measurement among the listed ones here? a. Exa b. Terra c. Pico d. Atto Answer: d. The term atto refers to 10-18 of something; this is an extremely small number. For example, the term “attosecond” is the time it takes light to cross through an atom.

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4. A ball is at 1 meter from a reference point and is moved 3 meters in one direction away from the reference point. What is the displacement arrow you will right? a. Two units to the right b. Two units to the left c. Three units to the right d. Three units to the left Answer: a. When determining the displacement, one uses the righthanded arrow to indicate movement away from a reference point. The units are the final position minus the initial position, leading to 2 meters to the right.

5. Which value is considered a vector? a. Displacement b. Distance traveled c. Location d. Distance Answer: a. Of these, only displacement is a vector. It is a magnitude and a direction, which can be positive or negative. The others represent the absolute value of something, which has a value but does not have a direction.

6. What is the SI unit for time in physics? a. Hour b. Minute c. Millisecond d. Second Answer: d. The SI unit for time in physics is the second. In cases where other time frames are used, you may need to use things like milliseconds or microseconds for small units of time or minute and hour for long periods of time.

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7. What is the SI unit of acceleration? a. Miles per hour squared b. Meters per hour squared c. Centimeters per millisecond cubed d. Meters per second squared Answer: d. Acceleration is the change in velocity per second so its SI units will be in meters per second per second or meters per second squared.

8. Which of the following is not considered a vector? a. Speed b. Velocity c. Displacement d. Acceleration Answer: a. Each of these is a vector having a magnitude and direction, except for speed, which does not have a direction.

9. In determining the initial and final velocities, the times to get to certain distances, and acceleration, etcetera, of an object, which value is held constant in these types of equations? a. Time b. Velocity c. Displacement d. Acceleration Answer: d. For these types of equations, the acceleration of the object is held at a constant rate. This allows for the times, distances, and velocities to be calculated.

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10. In problems related to the falling of objects, what is determined to be negligible? a. Air resistance b. Mass of the object c. Acceleration d. Change in velocity Answer: a. In problems related to falling objects, the effect of air resistance, which can be a factor in some situations, is determined to be negligible. The other factors are not negligible.

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CHAPTER 2: TWO-DIMENSIONAL KINEMATICS The focus of this chapter is the motion of an object in two dimensions. Many things do not simply go in a straight line or in an up-and-down fashion. These include celestial objects in orbit, automobiles that travel around a curve, and the arcing of a ball. There are different equations and different vectors that apply to these types of movements, which need to be studied and memorized. Three-dimensional kinematics is very similar to two-dimensional kinematics, except that the x, y, and z axis are part of this discussion. All of the same physics principles you are learning apply to motion in more than one direction, allowing for more types of situations involving the motion of objects to apply.

VECTORS IN TWO DIMENSIONS Perhaps the simplest way to understand motion in two directions is to think of walking in a city. Cities generally have uniform square blocks that have to be traversed in an xor y-axis in order to get anywhere. You will travel in a north-south or east-west direction in order to get southwest or northeast, etcetera. Figure 3 describes motion in two directions:

Figure 3.

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If you are traveling in these two directions as in figure 3, this is clearly not the route a helicopter would take, for example. What you want to know from a physics perspective is “what is the straight-line distance” between the two points A and C? From the Pythagorean theorem, you get a2 + b2 = c2, in which a equals the distance from point A to point B, b = the distance from point B to point C, and c = the distance from point A to point C. This forms a right triangle. Figure 4 demonstrates the Pythagorean theorem:

Figure 4.

In this theorem, c represents the hypotenuse of the triangle or the “straight path” from point A to point C. In the case in figure 3, you would travel 12 blocks and then 11 blocks.

According to the Pythagorean theorem, you have the square root of 12 squared and 11 squared added together or 16.3 units in total length.

Just as in chapter 1, displacement is a vector with the arrow’s length representing the length of the vector and the arrow’s direction representing a point on the coordinate. In cases as described, there can be three vectors: vector A to B, vector B to C, and vector A to C. You need to remember that the Pythagorean theorem only applies to 90-degree triangles, where two vectors are perpendicular to one another.

In the above situations, the horizontal and vertical components of two-dimensional motion are independent of each other. The horizontal direction affects only the horizontal motion and the vertical direction only affects vertical motion. Even in curved motions, there is a horizontal component (a horizontal vector) and a vertical component 21

(a vertical vector). If a ball is thrown, for example, it will have downward motion (affected by gravity) and horizontal motion (affected by the throwing horizontally).

These motions will be independent of one another.

The key to understanding the motion of a projectile is to break it into horizontal and vertical components so that its path in two dimensions can be determined. Later in this chapter, we well look at vector addition and subtraction so you can solve these types of problems using analytical and graphical methods. The important thing to remember so far is that the horizontal vector will be independent of the vertical vector—and vice versa.

VECTOR ADDITION AND SUBTRACTION VIA GRAPHICAL METHODS Remember that a vector has a magnitude and direction. The same is true of the vectors of force, displacement, velocity, and acceleration. In one direction, a vector can be positively-charged or negatively-charged. If two directions are involved, a grid system must be used so that there are coordinates along an x-axis and y-axis, which define the vectors. In such cases, you can use a graph to represent the displacement of an object or person. It is easy when using something like city blocks but coordinates can be used in any two-dimensional or three-dimensional system. In some cases, the angle will also be used to describe the vector in relationship to another vector. Figure 5 shows how to describe these vectors:

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Figure 5.

There is a head-to-tail method of adding vectors that will help you determine the displacement of the object or person. It is done by doing what was done in figure 3.

One vector is drawn horizontally, while the vertical vector is drawn at the tip of the horizontal vector. Several vectors can be “tacked” onto this with the final displacement vector being drawn from the 0,0 point on the x-y coordinate graph to the tip of the final vector.

Vectors do not necessarily have to be in a perfectly horizontal and vertical direction. If you have graph paper and a protractor that can do the angle of a specific vector and its direction, you can determine the final vector’s angle and length, along with the final displacement. Because these are just vectors, you can start with any vector and end with any vector and you will get the same answer. This is called commutative addition.

Vector subtraction is similar to vector addition. As vectors, they can be added or subtracted. If we consider B to be a vector, then -B is a negative vector. It will have the same magnitude but will not have the same direction. The direction will be opposite.

This is also referred to as “flipping the vector”. When you are “adding” this vector, you

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subtract the vector’s magnitude from the initial vector’s quantity, with the angle of the vector being the same.

MULTIPLICATION OF VECTORS While it sounds complicated, vectors can easily be multiplied. Say, for example, that you displace in a horizontal direction along the x-axis for A miles and then decide to go three times longer. This is, of course, going to be in the same direction but will be a magnitude of 3A. This is simple multiplication of a vector by a scalar (which, of course, has no specific direction, while the vector will have direction). If the scalar is a positive number, the direction of the vector will be the same; however, if the scalar is negative, the direction (angle) will be reversed. Division of a vector and a scalar will be similar to the division of anything, except that there will be a direction associated with it.

RESOLVING VECTORS In some cases, you know the final vector but will not know the vectors that went into making the final vector. In most cases, it involves finding the x-axis vector and the yaxis vector. In most cases, you will be given an angle and a magnitude that will need to be plotted on a graph using a protractor. This becomes more necessary when we discuss the physics of projectile motion and the physics of dynamics. Often, this will not be an x-axis or a y-axis, but will involve physical directions, such as north (positive Ydirection), east (positive x-direction), south (negative Y-direction), and west (negative xdirection).

There are analytical methods of vector manipulation that make use of geometry rather than rulers and protractors. Arrows are still used in order to determine things like direction but the numbers and angles are resolved with geometrical equations. You can use a protractor and rule to get a general idea; however, for practical purposes, it is more accurate to use analytics for vector manipulation.

So, how do you resolve a vector using analytics? You start with an angle and a magnitude. You can draw it out on an x and y axis. It does not have to be completely

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accurate because the goal is to determine the x-vector and y-vector that apply to it. This is shown in figure 6 :

Figure 6.

In such cases, the sum of the x-axis vector and the y-axis vector will be the final vector— vector AB. These form a right triangle, which is an angle that has 90-degrees to it on one of the angles. This brings us to the use of trigonometry, which can help resolve the x-axis and y-axis if the angle and length of the AB vector is known. For the purposes of this discussion, the x-axis vector will be called capital B and the y-axis vector will be called capital A. From trigonometry, capital A vector equals the ABvector multiplied by the cosine of the angle between AB and the capital A vector. Capital B vector equals the AB vector magnitude multiplied by the sine of the angle. This is described in figure 7:

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Figure 7.

As you can see, knowing the angle and the magnitude of the vector on the graph will resolve the A and B axis vectors. The same thing can be said, as you have determined, of knowing the resultant vector because A squared plus B squared will equal AB squared.

Another thing that might be necessary is knowing the angle. According to trigonometry, the angle theta equals the tangent to the -1 of B divided by A. These equations are shown in their entirety in figure 8 :

Figure 8.

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What this means is that you can use an scientific calculator, knowing the angle and the final vector or the two A, B vectors in order to get the values you need.

You can also add vectors once you know the perpendicular components of said vectors.

Consider two components of a walk A and B that will have a resultant displacement vector called R. This is shown in figure 9:

Figure 9.

Each vector will have an x and y component that can be determined as is seen in figure 10:

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Figure 10.

Once the components of Rx and Ry are determined, the totality of R can be determined along with the angle theta. What you need to know is that the adding and subracting of vectors is done in the same way. You can use the equations you already know to get each angle and each component of the graphed out problem so you can solve for the angle and magnitude of the resultant vector R.

PROJECTILE MOTION Projectile motion can be termed the movement of an object that is thrown or somehow projected into the air. The object thrown is known as the projectile and its path is referred to as the trajectory. After being thrown, its motion is subject to the laws of gravity. We’ve already covered the motion of falling objects. In our discussion, as in previous ones, air resistance is considered negligible. You need also to remember that you can consider the movements in perpendicular directions (as in the x and y axis) separately. Finally, you need to know that all forces, except gravity are also negligible.

Upwards direction will be positive and the force of gravity will be negative. Figure 11 describes a vector at a certain point (x, y) called S, because of a projectile’s trajectory:

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Figure 11.

So, how can the motion of a projectile be determined using physics. The first step involves graphing the projectile’s trajectory on an x and y axis. These are by nature perpendicular. You remember from what we talked about is that the length of x is equal to the displacement vector S multiplied by cosine of angle theta and the length of y is equal to the displacement vector S multiplied by the sine of angle theta.

When it comes to the velocity V, there will be a velocity in the x direction called the Vx and a velocity in the Y direction called Vy. Because these are vectors too, the Vx equals the velocity multiplied by the cosine of theta and the Vy equals the velocity multiplied by the sine of theta.

You will need to treat the motion as two independent one-dimensional movements: one being horizontal and one being vertical. There will be kinematic equations for horizontal and vertical motion, which are listed in Figure 12:

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Figure 12.

Once you know the knowns and the equations, you will be able to solve for the unknowns. These types of problem-solving abilities are the same as solving for onedimensional kinematics. The final step is solving for the x and y motions, determining the final vector or the total displacement S and velocity V using what you know about vector addition and subtraction via analytical methods.

The same equations apply to any vector (as seen in the displacement equations in two directions), including the velocity vectors. The same thing applies to the angle theta.

What you need to know is that, at any given period of time, there will be a velocity in the x direction and a velocity in the y direction. Over the length of the trajectory, the velocity in the x and y direction can be determined using equations you already know.

You need to know, though, that there are other ways to determine velocity based on calculus, which are beyond the scope of this course.

Let’s look at an example: You shoot a shell of fireworks in the air and decide that the fireworks should explode in the air at its peak above the ground. It is shot off at a 75degree angle at a speed of 70 meters per second. At what height should the shell explode and when will this happen?

The height is called the apex or the highest point Y above the ground. You know that the velocity squared is equal to the velocity initial squared minus 2 times the force of gravity times the difference between Y and the initial Y). At the apex (with regard to Y anyway), the velocity will be zero (because the fireworks are turning around). You also know that 30

the initial Y will be zero. This means that zero velocity at the apex equals the initial velocity squared minus 2 times the force of gravity times the height Y. When you shoot the fireworks at 70 meters per second, only a portion of this velocity is in the y direction. How much? Remember that the velocity in the y direction will be the total velocity multiplied by the sin of theta (which is 75 degrees). This leads to an initial velocity in the y direction of 67.6 meters per second. Taking this value, squaring it and dividing by 2 and by 9.8 meters per second squared (the force of gravity) will lead you to the height, which is 233 meters high.

You can also calculate the time it takes to do this. The equation is the height y equals the initial y plus 0.5 multiplied by the time and multiplied by the sum of the initial velocity in the y direction and the velocity at the peak (which is zero). This becomes 233 equals 0.5 times 67.6 times the time. Using a calculator, you get the time equals 6.9 seconds.

Now that you know the time spent, the angle, and the initial velocity, you can determine how far in the horizontal direction the projectile will be at its apex. The velocity in the horizontal direction will be 70 meters per second multiplied by the cosine of 75 degrees.

This will be 18.1 meters per second. The apex happens at 6.9 seconds so the distance in the x direction will be velocity multiplied by the time or will be 125 meters.

RANGE OF A PROJECTILE How does the initial velocity of a projectile affect the range of the velocity? The greater the initial velocity, the greater will be the range. The initial angle will have a big effect on the range. For a fixed initial speed (as in by a cannon), the maximum range happens at an angle of 46 degrees if no air resistance is involved. If air resistance is considered, the angle will be 38 degrees. The range is also affected by the value of the acceleration of gravity. In other words, you can drive a golf ball further on the moon than you can on Earth.

What you should know is that the range has two angles that will give the same value.

The sum of these two angles is going to be 90 degrees. This means that you’ll get the same range if you shoot the projectile at 75 degrees as you would at 15 degrees.

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The actual range of a projectile is a bit complicated. It is calculated as the initial velocity squared multiplied by the sine of 2 times the angle all divided by the force of gravity.

The assumption is that the range is small compared to the circumference of the earth. If the range is large, the earth will curve away and the acceleration of gravity changes direction. Technically, if the speed is initially great enough, the projectile will go into orbit. This is called the exit velocity.

You can calculate the time of flight of a projectile that is launched and lands on a flat horizontal surface, which works for distances that make the earth’s curvature negligible.

In such cases, the starting point with respect to y is zero and the ending point with respect to y is zero. With the displacement being zero, the time of flight is going to be calculated as 2 times the initial velocity times the sine of the initial angle divided by the force of gravity.

VELOCITY IN TWO DIMENSIONS Velocity in two dimensions can be described when someone tries to cross a stream in a boat but gets caught up in the current. The boat is then moving in a direction in which it is not pointed. The same would have to be said of an airplane stuck in a crosswind.

There is a straight direction with respect to the air but not relative to the ground. In such cases, there will be two velocity vectors that need to be added to get the actual velocity.

Remember that velocity is a vector so that the rules of vector velocity addition and subtraction still apply. Remember, too, that velocities in the x and y direction can be thought of as being separate so that the velocity in the x direction is the total velocity multiplied by the cosine of theta, while the velocity in the y direction is thought of as the total velocity multiplied by the sine of theta. Figure 13 shows the velocity equations:

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Figure 13.

Graphing of velocities is identical to displacement in two directions, with vectors on the x and y axis showing the velocities in both directions perpendicular to one another adding to make the relative velocity.

Let’s do one problem. A boat is traveling at 0.75 meters per second in a current that is going across it to the right at 1.2 meters per second. What is the relative velocity of the boat relative to the shore? Start by getting the total velocity, knowing that the Vx velocity is 1.2 meters per second and the Vy velocity is 0.75 meters per second. This is shown in figure 14:

Figure 14.

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The total velocity will be the square root of the sum of the squares of Vx and Vy or, using a calculator, you get 1.42 meters per second. The angle, according to the calculations, is 32 degrees. If you know the total width of the river, you can use the angle and the width (which will be the y direction in the figure) in order to get the x direction or the point on the river that the boat will reach at the end of the trip. This will be the displacement in the x direction after how many seconds it takes to cross the span.

For example, if the river is 100 meters in width, it will take 0.75 meters per second or 133.3 seconds to cross. In that same 133 seconds, it would traverse 1.2 meters per second multiplied by 133.3 seconds or about 160 meters downstream. This is simple math because the river crossing is the y-axis and the distance downstream is the x-axis.

When adding velocities, the relative velocities will be relative to a certain reference frame. This concept of relative velocities is one aspect of relativity, which is the study of how various observers relative to one another will measure the same phenomenon. On an airplane, the passengers have a relative velocity of zero (as they do not perceive movement relative to one another); there will also be velocity relative to the wind and velocity relative to the ground. These are examples of classical relativity, which apply when speeds are less than about 1 percent of the speed of light, which is less than 3000 kilometers per second. Modern relativity (Einstein’s relativity) applies at higher speeds, which will be discussed later.

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KEY TAKEAWAYS •

In looking at displacement or velocity in two dimensions, you need to get a coordinate system to define the x and y components of the vectors.

•

According to the Pythagorean theorem, the square of the x component plus the square of the y component will add to make the square of the hypotenuse (the actual displacement or velocity).

•

There are multiple equations you should memorize that will tell you the actual displacements and velocities in equations involving 2-dimensions.

•

The range is the total distance a projectile will go when shot from a gun, cannon, or other device when it is shot at a certain velocity and at a certain angle.

•

Relative velocity depends on the thing you are choosing the velocity to be relative to. It can be relative to the earth or to the boat or other craft.

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QUIZ 1. You are walking 9 blocks east and 5 blocks north. How many blocks do you travel if you are traveling by helicopter in a straight line? a. 14 blocks b. 17.2 blocks c. 10.3 blocks d. 15.5 blocks Answer: c. As the answer will be the square root of the squares of 9 and 5, you get 81 plus 25 or 106, of which the square root is 10.3. This is fully based on the Pythagorean theorem.

2. What must be true to use the Pythagorean theorem? a. The triangle must be on a grid. b. It must be a right triangle. c. All values for the vectors must be positive. d. Vector AB must be greater than vector BC. Answer: b. The triangle must be a right triangle, with one angle being 90 degrees. According to the theorem, AB2 + BC2 = AC2, in which each of these are vectors and the angle ABC must be 90 degrees in order to have the theorem be valid.

3. You are doing a problem in which an individual travels 9 miles east, 4 miles northwest (at 45 degrees) and 2 miles south. Which vector do you start with in drawing this? a. 9 miles east b. 4 miles northwest c. 2 miles south d. It does not matter

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Answer: d. You can start with any vector you wish because it will result in the same end-result. This problem can be solved on a graph using a head-to-tail vector analysis.

4. You have a vector A and you want to create a vector called -A. What will change in order to make this vector? a. The magnitude and direction b. The magnitude only c. The direction only d. This is the same vector Answer: c. In making the vector, you will change the direction to be the reverse of the original vector but the magnitude will be the same. This is the same thing as reversing or “subtracting” the vector.

5. You know the length of the final vector xy and the angle. How do you find the x-component of the vector? a. The vector is xy times the cosine of the angle theta b. The vector is xy times the sine of the angle theta c. The vector is xy times the tangent of the angle theta d. The vector is xy divided by the tangent of the angle theta Answer: b. The vector x will be xy times the sine of the angle theta.

This can be determined by knowing the magnitude of the final vector xy and its angle and by using a scientific calculator.

6. You know the x and y vectors on a coordinate graph and you want to know the angle of the resultant AB vector. How do you calculate this? a. The angle is the sine of x times y b. The angle is the cosine of x divided by y c. The angle is the tangent to the negative one of y divided by x d. The angle is the tangent of y divided by x Answer: c. The angle is the tangent to the negative one of y divided by x. This will require the knowledge of x and y plus a scientific calculator.

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7. In the motion of an object, what best defines the placement of the object in space at any particular point in time? a. Projectile b. Displacement vector c. Gravity d. Trajectory Answer: b. The displacement vector is the vector that will decide where the object is in space, regardless of its actual trajectory.

8. If you shoot something in the air and determine its velocity at the apex, what will this be with respect to height or the y position? a. The sine of the angle of the projectile shot b. The cosine of the angle of the projectile shot c. The force of gravity multiplied by the time it takes to reach the apex d. The velocity in the y direction at the apex will be zero Answer: d. By definition, the velocity at the apex of this kind of projectile being shot in the air will be zero because it is turning around and falling toward the earth because of gravity.

9. You shoot something from a gun to get a range of 20 feet at 80 degrees. What is the range you’ll get if you shoot it at 10 degrees? a. 200 feet b. 142 feet c. 63 feet d. 20 feet Answer: d. The same range will be gotten from 80 degrees and 10 degrees because the sum of the two angles is 90 degrees. You can shoot it out of the gun at 80 degrees or 10 degrees and you’ll get the same value for the range.

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10. In shooting a projectile, what is the exit velocity? a. The velocity at which the object goes into orbit around the earth b. The initial velocity of the object as it leaves the gun or cannon c. The velocity of the object at its apex with respect to the y axis d. The velocity of the object at its apex with respect to the x axis Answer: a. The exit velocity will be the velocity at which the object goes into orbit around the earth.

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CHAPTER 3: NEWTON’S LAWS OF MOTION This chapter looks into Newton’s Laws of Motion as well as applications of these laws. It goes further in the discussion of motion to outline things like friction, drag, and elasticity when it comes to motion. So far, there has been discussion of motion in its purest form without an understanding of the forces behind the motion. What necessarily follows is a discussion of “dynamics”, which are the forces that affect the movement of the different objects and systems. It turns out that Newton’s Laws are the foundation of the study of dynamics. The laws were uncovered in 17th Century but still apply today on earth as well as in space. They apply to the study of classical mechanics, meaning that they apply to speeds less than light and sizes of objects greater than molecules.

DYNAMICS The study of dynamics is the study of the forces that cause the movement of objects and systems. Force can be described as a push or pull, which is a vector quantity as it has a magnitude and a direction. It can be added, subtracted, and multiplied/divided like any vector. In calculations, forces are represented by arrows with a similar head to tail arrangements in graphical methods or analytical methods of determining overall forces applied to an object. You already studied how to manipulate these vectors in chapter 2 on motion in two directions.

Force, for example, can be applied to a spring by pushing or pulling on it. There will be a force applied externally as well as a force that allows the spring to return to its natural state, called a restoring force. The restoring force is said to be a standard force that applies directly to the spring. Later on, we will talk about magnetism, which is another standard force. These standard forces are reproducible in the situation they are applied to. Other forces applied in the situation can be a certain multiple of the standard force.

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NEWTON’S FIRST LAW Newton’s first law of motion is quite simple. It states first that an object at rest will remain at rest unless acted upon by an external force. Secondly, it states that a moving object remains in motion at a constant velocity unless it is acted on by an external force.

This basically means that there will be a status quo of any object unless a force is applied to it. This force, as you will discover, is a “net external force”, which will be the sum of all the vector forces applied in any situation. In objects sliding along a surface, friction becomes a force to be reckoned with.

This property of an object to remain at rest is referred to as “inertia”. This is why this law is called the law of inertia. Some objects have more inertia than others because of their mass. Heavier objects have more inertia than other objects. Mass does not vary with location or the acceleration of gravity. It depends solely on the quantity of an object or the number of molecules in the object. The mass of a gram of gold is the same as the mass of a gram of cotton balls. Mass is the same on earth as it is on the moon. Thus, it is a scalar quantity. The SI unit for mass is the kilogram.

NEWTON’S SECOND LAW Newton’s second law of motion states that there is a relationship between the force applied to an object and its motion. This law is more quantitative than the first law and is used in calculations involving force. In order to understand this law, you need to remember a few things. A change in motion is a change in velocity, which is what we already know is “acceleration”. Newton’s first law indicates that a net external force will result in acceleration.

External force implies a force outside of the system of interest. Only the external forces affect the motion of a system. For example, you can push on a wagon while sitting on the wagon all you want and it will not change the acceleration or velocity of the wagon.

You need to push on the wagon itself as this would be an external force. You need to define the boundaries of the system before you can determine which forces are internal and which forces are external.

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Acceleration is proportional to and is in the same direction as the net force on the system. This means that smaller forces cause a smaller acceleration than larger forces.

Vertical forces include the weight of an object and the support of the ground, which cancel each other out when horizontal motion is involved. Horizontal force represents the force of friction and the forces applied to the object on a horizontal surface. Friction is a force that opposes the motion past each other of objects that are touching each other. On a horizontal surface, friction is a horizontal force but you can imagine a situation where friction can be a vertical force. In a small degree, the force of wind when you are falling from an airplane is a vertical force but it is not overcome by the force of gravity.

Just as acceleration is proportional to the net forces on an object, it is inversely proportional to the mass of the object or system. The same force applied to a heavier object will not give the same acceleration as force applied to a lighter object. This leads to the mathematical equation that is Newton’s second law, or this equation: Force equals mass times acceleration. This is a cause and effect relationship among these three quantities. Because acceleration is a vector, so is force.

The SI unit of force is the Newton, abbreviated with the capital letter N. This is the force required to accelerate a 1-kilogram object or system at a rate of 1 meter per second squared. One Newton is 1-kilogram meter per second squared. In the US, the unit of force is a pound, in which 1 Newton is the equivalent of 0.225 pounds.

This leads to the issue of weight. When an object is dropped, it will accelerate toward the center of the earth because of the force of gravity. The net force on a falling object is referred to as the gravitational force, which is the object’s weight. Weight has a vector quantity because it is a force. It is essentially a downward force. The acceleration due to gravity is g or 9.8 meters per second squared. This means that the weight of a 1-kilogram object will be 9.8 Newtons. In actuality, as you already know, the weight of an object will vary according to where it is on earth. A one-kilogram mass weighs only 1.7 Newtons on earth. On earth, mass and weight are used interchangeably, even though they are not the same.

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NEWTON’S THIRD LAW This law is basically about symmetry in nature. Forces will occur in pairs so that when a force is applied to a second body, the force on the initial body will be equal in magnitude and in the opposite direction of the force initially applied. These forces do not cancel each other out because they are acting on different systems. When a swimmer pushes away from a wall and moves, there is movement because the swimmer pushes on a different system from itself (which is the wall). The force that moves the swimmer will be in the opposite direction to the push against the wall.

Rockets move forward by expelling gas in a backwards direction. There will be a large reaction force on the rocket, known as “thrust”. It is caused by the expulsion of gas in a system, such as a plane or rocket. Rockets do not push against the ground or the air behind them and will actually also work better in a complete vacuum (because of a lack of air pressure and air friction). Birds, helicopters, and airplanes also fly by exerting force on the air that is opposite to the direction they want to go.

In the case of pushing an object, there will be the forward force of pushing the object and opposing forces, such as friction and air resistance. Neutral forces, will be the downward force of gravity as this is an equal and opposite force acting on the object by the earth. Friction will be the major force involved and depends on the characteristics of the object and the substance (such as the ground) that is up against the object.

Remember that friction only applies if two objects are touching each other in some way.

You should also know that forces between components of a system will cancel each other out because they are within a system and are said to be equal and opposite in direction.

In the situation of pushing a cart, for example, if the system includes the person doing the pushing and the cart itself, there will be no net movement of these objects. There are internal forces between the cart and the pusher that do not affect the movement of the pusher and cart together. If the system does not include the pusher, the pusher can exert a certain external force on the cart, thus moving the system (which is just the cart).

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FORCES IN PHYSICS Weight or “the force of gravity” is pervasive and gets acted on at all times. If not counteracted, things will fall because of their weight. Objects placed on a surface such as a table will have a restoring force exerted upon them (similar to that of a spring) and, unless it is a heavy object or a flimsy table, you will not notice the deforming nature of the table. The greater the deformation, the greater is the restoring force. Because of the restoring force, the net external force on the load will be zero. The normal force means the force perpendicular to a surface. It is an upward force that opposes the weight of the load but is not necessarily perpendicular to the ground as it can be angulated with respect to the ground (that is, not parallel to the ground). Like other forces, normal force is a vector.

As friction opposes motion between surfaces, the acceleration is smaller when there is friction present. If friction is negligible down an incline, the acceleration down an incline will be the force of gravity multiplied by the sine of the angle of the surface, regardless of the mass of the object. This goes with the idea that all objects will fall at the same rate, regardless of mass, if other things like air resistance are not a factor.

There are steps you already have an idea how to do that involves the way to resolve weights into their respective components. The first thing you must do is to set the x and y axes so that they align with the slant of the incline upon which the weight rests. You should also recognize that some of the weight will be parallel to the slant and some of the weight will be perpendicular to the slant. Figure 15 describes this differentiation of the weights:

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Figure 15.

The weight parallel to the incline is the weight multiplied by the sine of theta, while the weight perpendicular to the incline is the weight multiplied by the cosine of theta.

Remember that weight is equal to mass times the force of gravity g, which can be substituted into the equations just mentioned.

Understand that, in this case, parallel weight is weight along the x axis and perpendicular weight is weight along the y axis. This means that the usage of sine and cosine is the reverse of what it has been before because theta is now the angle between the y axis and the downward vector, which is ninety degrees minus the theta we’ve talked about in the past. You just need to know this in order to do these equations.

The perpendicular weight is equal in magnitude and the opposite in direction to the normal force. The parallel force would be the force that pushes the object down the incline. If there was no friction, the object would slide down the incline if the force is even at a minimum. The force of friction, however, opposes the motion of the object. In such cases, friction is a force that has a magnitude and direction that goes up the incline.

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TENSION Tension is a force along the length of a medium, such as that of a rope or cable. The connector should be considered to be a flexible connector, such as a rope, string, cable, wire, or chain. It can only exert force that is parallel to its length. Tension represents a pull and cannot represent a push because the connector is flexible and has no ability to do any pushing. The actual force starts at each end of the rope or flexible connector and extends toward the middle of the connector. This is an example of Newton’s third law because it shows an equal and opposite force that doesn’t cause any movement unless one is actively pulling on the connector.

When hanging an object from a rope, the net force will be zero if the object isn’t moving.

This means that the weight is equal to the tension. This means that the force of tension and the force of the weight of the object are equal and opposite. This is the truth if one neglects the mass of the rope. This means that the tension on a 1-kilogram mass is equal to that mass multiplied by the force of gravity or 9.8 Newtons. If we cut the rope and put in a spring instead, the spring would extend out to a length that equals the downward force (weight) of the object or 9.8 Newtons.

Even if there are pullies and traction systems, the force is still just parallel to the flexible connector attached to the object. This tension is transmitted throughout the length of the connector undiminished except for the friction factors. The pullies will change the direction of the tension but will not change its magnitude as long as there is no friction.

Consider the case of a person standing on a tightrope. The tightrope will dip at an angle on both sides to a degree called theta and there will be forces of tension to the left and right of the person, called tension left and tension right. This is added to the downward force of the weight of the individual. These are the three forces at play with a tightrope situation. The net external force will be zero because the person is stationary on the tightrope.

The coordinate system in this type of situation must be chosen. The goal is to have a convenient horizontal x-axis and a perpendicular y-axis. The x-axis is chosen as the ground, while the y-axis is perpendicular to this. The rope itself cannot be an x-axis

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because it bends in the middle where the person is standing and is itself not a straight line. The tension vectors have a horizontal and vertical component (usually just a small vertical component) if the angle of the rope is minor. This leads to five forces in actuality (because the two tension forces have two components each) and these are added to the weight of the person on the tightrope.

Trigonometry will determine the x and y components of the tension on either side. The tension in the x direction or the horizontal direction will equal the total tension on one side multiplied by the cosine of theta, while the vertical component of the tension is the total tension multiplied by the sine of theta. The tension to the left and the tension to the right are equal.

Now, remember that the sum of the vertical tension to the left, vertical tension to the right, and the weight of the person equals zero because the person isn’t moving. Two times the vertical component of the right and left will equal the weight of the person.

This means that 2 times the tension times the sine of theta equals the weight of the person on the tightrope. While this sounds complicated, the important thing is to draw the vectors or forces and resolve the vertical and horizontal components. Figure 16 shows the forces added together:

Figure 16.

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APPLICATIONS OF NEWTON’S LAWS There are several situations that involve an application of Newton’s Laws. The topic that comes up in looking at these laws is called “drag force”. Consider two tugboats pushing a boat in the x and y directions at different rates of force. This is seen in figure 17:

Figure 17.

The question is, what is the drag force on the boat? This is the force of the boat that is resisting motion, which will be an opposite force to the direction of the way that the boat is being pushed. It is a frictional force exerted by fluids like air or water. The force will be the sum of the x force and the y force, with the drag force being an opposite force in magnitude and direction to the final force on the boat.

In this case, the “applied” or final force will be the square root of the x force squared plus the y force squared (from the Pythagorean theorem) and the angle theta will be the negative one tangent of the y force divided by the x force. The net force is the applied force minus the drag force. If you know the mass of the boat and its acceleration, you can determine the drag force. We will talk more about this in a minute. Suffice it to say that the drag force will not be the same as the applied force as long as the boat is moving anywhere in the direction of the applied force.

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You should also know that there will be situations where a person is standing on a tightwire or something is suspended from a wire in which there are two different angles of tension and two different tensions on either side of the weight on the wire. In such cases, the components of the tensions must cancel each other out and the sum of the vertical tensions on the wire must equal the weight of the object or person on the wire.

There will be different tensions from left to right because the object isn’t in the middle of the wire. Once you calculate the vertical components of the two tensions, you can determine the amount of tension carried by each half of the wire. The horizontal component of the tensions must be equal because the object is stationary. The trick to solving these types of drag force and tension problems is to draw a picture and identify the forces involved.

NEWTONIAN FORCES There are only four basic forces involved in Newtonian physics. These are gravitational forces, electromagnetic forces, weak nuclear forces, and strong nuclear forces. Weak and strong nuclear forces happen over very small ranges (atomic ranges) so that they are not perceived directly. This leaves only electromagnetism and gravity as the basis for all forces on a macroscopic scale. Gravitational force is the only force that is solely attractive and is not repulsive. The range of both electromagnetic forces and gravitational forces is infinity, which is different from the short range of nuclear forces.

The carrier particle for gravitational forces is the graviton, while the carrier particle for electromagnetic forces is the photon.

Gravitational force is actually quite weak. It is only because gravity is always attractive that it becomes noticeable. This is seen as applicable to earth when, in reality, there are gravitational forces regarding the sun, moon, stars, and galaxies. Gravity also affects space and time, as you’ll see when we talk about general relativity, in which space is curved near very large bodies, such as the sun and other stars. Electromagnetic forces are both electrical and magnetic. Friction, tension, and other typical forces (besides gravity) are secondary to electromagnetic interactions of atoms and molecules.

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All forces act at a distance. This is seen more obviously in the case of gravity. There is gravity affecting the attachment of the moon to the earth even though they are not connected. Friction is an electromagnetic force between atoms that do not literally touch. This brings on the concept of a force field that surrounds an object that creates the force. The field or force field becomes the thing that carries the force from one object to another. It depends on the object creating it and not on the object being acted on. This is why the force of gravity is the same for everything on the earth. A force field is also seen with electrical charges.

FRICTION Friction is a force that opposes relative motion between systems that also allow for movement as it would be difficult to walk, for example, without friction against the surface we walk on. It is always a force in parallel with the contact surface. If there are two systems in contact with one another and moving, the friction is called kinetic friction; if the object is stationary, the friction is called static friction. In addition, if an object is stationary, the static friction must be greater than kinetic friction.

There is a certain magnitude of static friction, which is equal to the coefficient of static friction multiplied by N, which is the magnitude of the normal force (the opposite of the weight of an object). In reality, the static friction can be less than this product but it can be no greater than the product of mu-s (the coefficient of static friction) and N. If the force applied exceeds the static friction, the object will move. Once moving, the magnitude of kinetic friction is mu-k multiplied by the normal force, in which mu-k is the coefficient of kinetic friction.

What are the coefficient of static friction and the coefficient of kinetic friction? These are arbitrary and unitless numbers that, of course, depend on the substance itself and the surface it resides on. There are tables that exist which will list the coefficient of friction for things like rubber on cement that can be used by engineers who study friction and the effects of friction. Friction on an incline will still be based on the coefficient of friction but, because the person is on an incline (such as a wagon on an

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incline), the weight of the wagon is not perpendicular to the slope but is perpendicular to the earth as is seen in figure 18:

Figure 18.

If the force of friction is less than the parallel weight of the wagon, then there will be acceleration down the slope. The normal weight will be mass times the force of gravity times cosine of the angle of the incline. Kinetic friction is defined as mu-k (the kinetic coefficient of friction) times the mass of the wagon times the force of gravity times the cosine of the angle of the incline. Figure 19 looks at these equations:

Figure 19.

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ELASTICITY, STRESS, AND STRAIN We have covered things like friction and drag forces that affect the motion of an object.

The topic now at hand are those things that affect an object’s shape. Deformation involves a change in an object’s shape because of the application of a force. Small deformations will “spring back” to the original shape—more so in certain situations where the object has some elasticity. The size of the deformation will be proportional to the force so that for small deformations, there is Hooke’s law that is obeyed.

So, what is Hooke’s law? In equation form, Hooke’s law states that force equals a proportionality constant multiplied by the change in deformation. The proportionality constant depends on the shape and nature of the object as well as the direction of the force. What it means is that the deformation of an object is proportional to the force.

Things like bones do not deform much and will break when a force is applied to them.

This brings up the topic of tensile strength, which is the breaking stress that will result in breakage or fracture of the material. Figure 20 graphs deformation and describes Hooke’s law and tensile strength.

Figure 20.

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TENSION AND COMPRESSION Tension and compression can be applied to a rod or wire with a resultant change in length called delta L. This delta L will be produced when a force is applied on top of the rod or wire in order to stretch it or compress it.

As it turns out, the change in length is proportional to its original length and the force applied. It is inversely proportional to the cross-sectional area of the rod or wire. This makes sense because a lesser cross-sectional area and a greater original length will make it easier to have compression or tension affect the length of the rod or wire.

There is also a constant called the Young’s modulus or the elastic modulus, which is dependent on the substance. Things with a large Young’s modulus will have a large tensile stiffness because they deform less for a given tension or compression. The units for this constant are 109 Newtons per meter squared.

You should also know the definitions for stress and strain. Stress is the ratio of force to the surface area of the object, while strain is the ratio of the change in length or delta-L and the total length or L. When you use the equation for tension and compression, you get stress equaling the Young’s modulus multiplied by the strain. These equations are listed in figure 21:

Figure 21.

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So far, we have talked about tension and compression acting on a rod or wire. What about sideways stretch? This is referred to as shearing force, in which there is deformation called delta-x. It is perpendicular to delta-L rather than parallel as is the case with tension and compression. There are similar equations, however. Figure 22 shows shear force as well as tension and compression:

Figure 22.

Shear deformation or delta-x is proportional to the force applied and the length of the object, while it is inversely proportional to the surface area of the object and another constant, referred to as the “shear modulus”, which is dependent on the substance of the object. Longer objects will have a greater shear deformation, while thicker objects will have a lesser shear deformation.

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KEY TAKEAWAYS •

Newton’s laws of motion apply to the motion of objects and the forces applied to objects.

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The force that moves something must be external to the object or system in order to have movement.

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There will be x- and y-components to the force on the object, with x and y to be determined according to the system being looked at.

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The force is described as a Newton with a person’s weight being their mass multiplied by the force of gravity.

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Normal force is an upward force that directly opposes a person’s weight with respect to the surface they are standing on.

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There is kinetic friction, which applies to movement, and static friction, which applies to something stationary.

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Deformation is the change in shape of an object because a force has been applied to it. The force may be parallel to the length of a linear object or perpendicular to a linear object, such as a nail, wire, cylinder, or even a board.

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QUIZ 1. What statement is true of the concept of force? a. It has a direction but no magnitude b. It has a magnitude but no direction c. It is a scalar quantity d. It has a magnitude and direction Answer: d. Force is a vector in that it has a magnitude and a direction.

It can be added to displacement, velocity, and acceleration as one of the vector quantities in nature.

2. What can be said of a standard force? a. It is a force that applies to different force situations. b. It is a reproducible force. c. It does not apply to things such as magnetism. d. It is an external force applied to an object.

Answer: b. A standard force is a reproducible force that applies to a specific situation. A spring or rubber band will have a certain standard force called the restoring force and the forces seen in magnetism are also standard forces. These are not applied externally to an object.

3. In describing a Newton in relationship to the pound, what is the pound equivalent of one Newton? a. 1.2 pounds b. 0.225 pounds c. 0.62 pounds d. 2.85 pounds Answer: b. While the Newton or the pound can be used to define force, you need to know that one Newton is the equivalent of 0.225 pounds.

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4. An object on earth has a mass of 2 kilograms. According to Newton’s second law, the weight of the object in Newtons is what? a. 2 Newtons b. 9.8 Newtons c. 19.6 Newtons d. 24.8 Newtons Answer: c. Weight is in Newtons and is related to the acceleration due to gravity, which is 9.8 meters per second squared. Because force equals mass times the acceleration due to gravity, the weight is 19.6 Newtons.

5. What is considered a force on earth that is acted upon at all times? a. Thrust b. Weight c. External forces d. Internal forces Answer: b. Weight is the only force of all of these that needs to be acted upon at all times. Even in orbital space, there is some gravitational force but it is far less than the weight observed on earth. Thrust and external forces do not necessarily get acted upon and internal forces, by definition, get opposed equally and oppositely within a system.

6. What is the force called that opposes the force of a load on a table? a. Restoring force b. Loading force c. Reacting force d. Elastic force Answer: a. The force on the load will be called the restoring force and is identical to the restoring force of a spring or trampoline. It isn’t noticed on most tables or other load-bearing surfaces because these are relatively firm surfaces; however, they do deform to a limited degree.

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7. In solving the weight parallel to the incline of an object that is sitting on the incline which is itself at an angle theta to the ground, what is the parallel weight? a. Mass times the force of gravity times the tangent of theta b. Mass times the cosine of angle theta c. Mass times the force of gravity times the cosine of angle theta d. Mass times the force of gravity times the sine of angle theta Answer: d. The weight that is parallel to the incline of an object that is sitting on an incline at angle theta is equal to the mass times the force of gravity times the sine of the angle theta. Once you know the angle theta and the weight of the object, you can resolve the weight of the object with respect to the x-axis or parallel to the incline.

8. What is the perpendicular weight on an incline when the incline is at the angle theta? a. Mass times the force of gravity times the tangent of theta b. Mass times the cosine of angle theta c. Mass times the force of gravity times the cosine of angle theta d. Mass times the force of gravity times the sine of angle theta Answer: c. The perpendicular weight on an incline is equal to the object’s weight (the mass times the force of gravity) multiplied by the cosine of angle theta, where theta is the angle of the incline. This perpendicular weight is the opposite force to the normal force or normal weight of an object.

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9. What is drag force? a. It is the force associated with pulling an object along an incline b. It is the force that opposes the direction of an object traveling through air or a fluid c. It is the friction of wheels on a flat surface d. It has a magnitude and direction toward the direction of movement of an object that is being pushed by another object.

Answer: b. Drag force is the force that opposes the direction of an object traveling through air or fluid. It applies to forces in water for a swimmer or boat or to a dragster traveling through the air, for example.

10. You are trying to move a box across a surface but it does not budge. What can you say about friction in such a case? a. Static friction exceeds kinetic friction b. Kinetic friction exceeds static friction c. Static friction must be infinite d. Kinetic friction must be absent Answer: a. Static friction in such cases must exceed kinetic friction because the object is not moving. Static friction is overcome and the kinetic friction becomes the operative friction force when the object becomes a moving object.

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CHAPTER 4: CIRCULAR MOTION AND GRAVITATION This chapter deals with uniform circular motion, which is defined as motion in a circular path at a constant speed. It involves things like pure rotational motion, which occurs when an object is traveling a path that is centered around a single point. This is different from pure translational motion, which is motion that has no rotation associated with it. There is mixed motion as well, with circular and rotational components. In addition, related components discussed in the chapter are the Coriolis effect and Kepler’s laws of planetary motion, which also apply to circular motion.

ANGULAR VELOCITY So far, we have concerned ourselves with motion along a straight line as well as the concepts of displacement, acceleration, and velocity. We extended the conversation to talk about motion in two dimensions. Circular motion, however, is more closely related to projectile motion, which differs primarily in that the object does not fall to the ground but instead moves throughout a curvature.

The first thing you need to understand is the concept of rotation angle. Imagine an object, such as a compact disc, rotating around the center of the CD. One can draw a line from the center to the edge that will rotate uniformly in a circle, meaning that it goes through a certain angle at a specific time. Because the CD rotates at a constant speed, the angle traversed by the line will be the same for a given period of time. So far, we’ve talked about angles being identified by the Greek letter theta, which is the same in this chapter. In this case, the rotation angle or delta-theta is equal to the delta-s divided by the radius. Delta-s is the arc length. This is shown in figure 23:

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Figure 23.

One complete revolution of the circle is the circumference of the circle, which is considered to be 2-pi-r, or two times pi times the radius of the circle. What this means is that, for 360 degrees or a complete revolution, the rotation angle is 2-pi. This leads to the concept of radians. One revolution is equal to 2-pi radians. You can determine the number of radians by taking the number of degrees, dividing by 180 and multiplying by pi. This means that 180 degrees is pi radians. One radian is about 57.3 degrees.

Angular velocity (indicated by the Greek letter omega) is measured in relation to the circle itself and isn’t in meters per second like normal velocity is listed. Instead it is the change in the angle theta divided by the change in time. The units for angular velocity are radians per second. There is a linear velocity associated with the angular velocity, which is based on the concept that, over the course of a certain number of degrees, the rotation will go a certain arc length. The linear velocity is the arc length divided by the change in time. It is defined as the radius multiplied by the angular velocity.

This means that the linear velocity is directly proportional to the distance from the center of rotation, meaning that, for a compact disc, the linear velocity is largest at the rim of the CD. This linear velocity is referred to as the tangential speed, which is the maximal linear speed on the rotating disc. In certain cases, the tangential speed of a tire is the same as the speed of the car. Larger tires with the same angular velocity will have

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a greater linear speed for the vehicle. The linear velocity is the radius of the tire multiplied by the angular speed. The angular velocity has just two directions: clockwise or counterclockwise, while linear velocity is considered tangent to the path.

CENTRIPETAL FORCES So far, we have talked about acceleration as an increase or decrease in velocity, which can change in magnitude, direction, or both. Because the direction of the linear velocity is changing all the time, so will the direction of its acceleration, even though the velocity may be the same over time. This is experienced when going around a corner; there is a sideways acceleration noted. Can this acceleration be quantified? It turns out that it can.

Acceleration in a circle points directly into the center of the circle or toward the center of the circular path. This is referred to as the centripetal acceleration. How can you show this center-seeking acceleration using vectors? Remember that velocity is a vector and acceleration is a vector. If you look at a circle with two different velocities (that have the same magnitude but different direction), the change in velocity is going to point toward the circle’s center. If acceleration is the change in velocity over the change in time, this vector will be larger if the velocity is larger. Figure 24 shows the direction and calculation of centripetal acceleration:

Figure 24.

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Because the two velocities are identical, this will lead to an isosceles triangle (which is one that has two equal sides). Based on the manipulation of knowns and equations, the centripetal acceleration is the same thing as velocity squared divided by the radius. This means that the smaller the radius, the greater is the centripetal acceleration.

You can also demonstrate a relationship between the centripetal acceleration and the angular velocity. The relationship is that the centripetal acceleration is equal to the radius multiplied by the angular velocity squared. This means the centripetal acceleration is directly proportional to the radius and the square of the angular velocity.

CENTRIPETAL FORCE There can be forces that result in centripetal acceleration. A few examples, include tension on the rope attached to a tether ball and the force of gravity affecting the moon.

There are also forces placed on the tube in a spinning centrifuge. What’s true is that any force that causes uniform circular motion is referred as a centripetal force. The direction of this force is in the direction of centripetal acceleration, even though these are not the same thing.

This brings in Newton’s second law of motion, which states that Force equals mass times acceleration. In the case of uniform circular motion, the acceleration being referred to is the centripetal acceleration. When it relates to angular velocity, force equals mass times the radius times the square of the angular velocity. Using these equations, you can do a number of problems. Let’s try one: How do you calculate the coefficient of friction car tires need on a flat curve? This basically involves calculating the centripetal force on a car that turns a certain radius at a certain linear velocity and then calculating the static friction that will keep the tires from slipping.

First, the centripetal force, which is mass times the velocity squared divided by the radius. Imagine a 900 kg car that negotiates a 500-meter radius at 25 meters per second. Doing the math, you get 900 times 25 squared divided by 500 or 1125 Newtons.

Next, you need a certain coefficient of friction to prevent the car from moving out to a greater radius (spinning out of the circle).

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Friction is the same thing as the centripetal force because it’s a flat curve. The static friction, if you can remember in the previous chapter is mu times the weight or mu times the mass times the force of gravity. You can calculate mu by taking the centripetal force and dividing by the mass and the force of gravity. This leads to a coefficient of friction of 0.13. You can also manipulate the numbers to get mu equals velocity squared divided by the radius and divided by the force of gravity.

There is the issue of banked curves to think of because banked curves allow the car to drive faster around a curve. There is the concept of an “ideally banked curve”, which is the angle at which a curve can be negotiated without the aid of friction between the tires and the road. In an ideal banking situation, the Normal Force in the horizontal and vertical directions must equal the centripetal force and the weight of the car, respectively. This would be the banking of a “frictionless” surface. Figure 25 shows this situation:

Figure 25.

In the case of the car, the external forces on the car are the weight of the car and the normal force of the road (which is perpendicular to the surface of the banked curve).

The two forces must add to give a net external force that is horizontal toward center of the curvature with a magnitude equal to the centripetal force.

This leads back to trigonometry and the study of forces that have horizonal and vertical components. The normal force multiplied by the sine of theta must equal the centripetal 64

force, which is itself mass times volume squared divided by the radius. Because the car is on the road, the net vertical force must equal zero. The vertical component is Normal force multiplied by the cosine of theta, which must also equal mass times the force of gravity. Combining these, we get these equations as seen in figure 26:

Figure 26.

THE CORIOLIS FORCE There are fictitious forces that come into play when turning a corner in a car, riding in a spinning amusement park ride, and taking off in a jet airplane that everyone experiences. You feel, for example, that you are being forced backward on an airplane when you accelerate down the runway, which is not a force at all. It is instead the force of the seat pushing on you. When you make a tight curve in one direction, you feel like you are being forced in the other direction. In reality, you are going in a straight line but the car is moving in another curved direction. This arises from the use of the car as a frame of reference.

These fictitious forces come from using something as a frame of reference that is within a specific system. Passengers will use the car or airplane as the frame of reference, while in physics, the frame of reference is the earth. Using physics, only inertial frames of reference with real forces are used. The car is considered a non-inertial frame of

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reference because it is accelerated to the side. The force that pulls you to the opposite direction of a turn is a fictitious force that has no physical origin.

Think of going around in a spinning ride at an amusement park, in which you feel pulled to the outside of the car you are riding in. You are in reality rotating together with the car but, in that non-inertial frame of reference, you feel a fictitious force called the centrifugal force that is trying to throw you off. The reason you must hang on is because you would otherwise go in a straight line.

This inertial effect, which carries you away from the center of rotation, is used in centrifuges. These will spin a sample rapidly in a circular motion, throwing the particles outward, which causes their sedimentation. The greater the angular velocity, the greater the centrifugal force. What really happens, though, is that the inertia of the particles carries them along a line tangent to the circle while the test tube in the centrifuge is being forced in a circular path by centripetal force.

This fictitious force or apparent outward force is described by Newton’s first law. This states that a body at rest will remain at rest, while a body in motion will stay in motion unless it is acted on by an external force. It is the inertia of a body with mass that will cause the object to continue in a straight line unless an outside force (centripetal force) acts upon the object spinning.

If a person throws a ball in a straight path while on a merry-go-round, it will go in a straight path and will no longer be directed along a circular path with respect to the earth. A person on the ground will see the ball going straight and will see the merry-goround circling beneath it. On the merry-go-round, on the other hand, the person will see a curvature of the ball’s path because of a fictitious force. This apparent curvature of the ball is called the Coriolis effect. It will curve in the opposite direction of the path of the circle rotating (clockwise or counterclockwise).

You should know that Earth is also an inertial frame of reference with the effect noted in things like weather systems. Because of this effect, weather systems rotate to the right in the northern hemisphere because the earth rotates counterclockwise as viewed from the north pole. It rotates in a clockwise direction as viewed from the south pole so weather rotates to the left. Wind patterns also observe the Coriolis effect. It causes

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hurricanes in the northern hemisphere to rotate in the counterclockwise direction, while in the southern hemisphere, they rotate in a clockwise direction.

NEWTON’S LAW OF GRAVITATION Gravitational force affects all things on or near the earth, including the Moon, which is a distance away from the earth. The force of gravity supplies the necessary centripetal force to allow for the rotation of the moon. These are the identical forces that cause the earth to rotate around the sun and for galaxies to rotate around a specific point. You should know that it is the weakest of the four basic forces found in nature (such as electromagnetic forces, weak nuclear forces, and strong nuclear forces). Unlike other forces, it is always attractive. It is along a line that joins the centers of mass of two objects. Figure 27 shows gravitational forces:

Figure 27.

The magnitude of force is equal in each direction, regardless of the mass of the object because of Newton’s third law. If the forces were different, the moon would leave the earth’s orbit or would crash into the earth. The center of mass is used in equations regarding this because these bodies tend to be quite large.

This brings us to Newton’s Universal Laws of Gravitation, which states that force equals the gravitational constant multiplied by the value of both masses, divided by the radius (the distance between the two masses) squared. Figure 28 shows this equation:

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Figure 28.

The gravitational constant is believed to be the same throughout the universe and has been measured to be .7 x 10-11. This is a really small force that means that gravitational forces only really apply and are only really important in larger objects like planetary objects. A person’s weight on earth is dependent upon the mass of the earth, which is 6 x 1024 kilogram. The moon’s mass is just about 1 percent of the earth’s mass.

You should remember that people do not weigh the same at different points in the earth.

The presence of mineral deposits or ice sheets beneath the earth and the presence of mountains will affect a person’s weight. The earth is also not completely round so it is thicker at the equator than it is at the poles.

Ocean tides are one observable effect of the moon’s gravity acting on the earth. Because water easily flows on the earth’s surface, a high tide is created on the side of the earth nearest to the moon, where the moon’s gravitational pull is the strongest. There are tides on the earth’s surface opposite to the moon because the earth is pulled toward the moon more than the water on its far side because earth’s center is closer to the moon than the water on the far side of the moon.

As the earth rotates, the tidal bulge keeps its orientation with the moon, leading to two tides per day (about 12 hours and 25.2 minutes every day). So, water is attracted to the moon more than earth on the near side of the moon, while the earth is attracted to the moon more than water on the far side of the moon. There is a tidal factor based on the sun but it is half the affect of the moon, depending on the angles of the earth, moon, and sun together. This leads to spring tides and neap tides with spring tides being highest and neap tides being lowest.

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KEPLER’S LAWS Kepler’s Laws relate to the orbits of different objects around the Earth or other planetary and celestial objects. Gravitational forces are forces between any two objects at a reasonable distance between one another. The laws are based on the idea that there aren’t any competing gravitational forces and that one object is big and one object is small (as is seen with satellites, including the moon).

Kepler has three laws. The first law is that planets around the sun have an elliptical orbit around the sun at one focus. This is seen in figure 29:

Figure 29.

Kepler’s second law states that each planet travels so that an imaginary line drown from the sun to the planet sweeps out equal areas in equal times. In other words, the planet moves fastest when it is closest to the sun.

Kepler’s third law is that the ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. The period of a planet is the time of the total path or orbit around the sun and the radius is the average (as it will differ in an ellipse over time). This means that T1 squared divided by T2 squared is equal to r1 cubed divided by r2 cubed. It means that if you know the time period of the moon (27.3 days) and the distance it is from the earth, you can calculate the period of any satellite at any distance from the earth.

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Key Takeaways •

Circular motion involves an angular velocity in radians per second and a linear velocity in meters per second.

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A full circle rotation is two-pi radians in total number. Two-pi equals 360 degrees in a full circle.

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Centripetal acceleration and centripetal force are vectors that act in the direction of the center of the circle.

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Centrifugal force is a fictitious force that is felt in the opposite direction of centripetal force but is not a real force but a perceived force.

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Kepler’s laws apply to small satellites in space orbiting much larger objects in space.

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Tides are the effect of the moon and less significantly the sun on the waters surrounding the earth.

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QUIZ 1. What is the relationship between the rotation angle, the arc length and the radius of a circle? a. Rotation angle is directly proportional to the arc length and the radius of a circle. b. Rotation angle is directly proportional to the arc length and inversely proportional to the radius of a circle. c. Rotation angle is inversely proportional to the arc length and directly proportional to the radius of a circle. d. Rotation angle in inversely proportional to the arc length and the radius of a circle.

Answer: b. The rotation angle is directly proportional to the arc length and inversely proportional to the radius of a circle.

2. What is the number of radians in a complete revolution around a circle? a. pi radians b. 0.5 pi radians c. 2 pi radians d. 2 pi squared radians Answer: c. The totality of a circle or a complete revolution covers an angle that is considered to be 2-pi radians.

3. The greatest possible speed of a tire happens at a certain point at the rim of the tire. What is that speed called? a. Arc speed b. Angular speed c. Tangential speed d. Linear speed

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Answer: c. The tangential speed is the maximal speed of a circular object or tire, which is at the rim of the tire or the furthest out from the center of the circle.

4. What is the direction of the linear speed of an object going in a uniform circle? a. Tangent to its path in the circle b. In a clockwise direction c. In a counterclockwise direction d. In either a clockwise or counterclockwise direction Answer: a. When it comes to linear velocity, the velocity direction is considered tangential or a tangent to its path in the circle.

5. What is the relationship between centripetal acceleration in uniform movement around a circle to the radius of the circle? a. Centripetal acceleration is proportional to the square of the radius. b. Centripetal acceleration is inversely proportional to the square of the radius. c. Centripetal acceleration is proportional to the radius. d. Centripetal acceleration is inversely proportional to the radius.

Answer: d. What this means is that, the smaller the radius or the tighter the turn around a circle, the greater is the acceleration. This is a linear inverse relationship.

6. How are the centripetal force and the mass of an object related? a. Force is inversely proportional to the mass. b. Force is inversely proportional to the square of the mass. c. Force is directly proportional to the mass. d. Force is inversely proportional to the mass.

Answer: c. This is essentially no different from Newton’s second law.

In the case of centripetal force, it is directly proportional to the mass of the object and acceleration of the object.

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7. What is the direction of the centrifugal force? a. In the direction of the path of the object in circular motion b. Toward the center of the circle of an object in circular motion c. Away from the center of the circle of an object in circular motion d. In the reverse direction of the path of an object in circular motion Answer: c. Centrifugal force is a fictitious force that is pointed away from the path of an object in circular motion. It is based solely on the frame of reference and is felt by the person in an object or vehicle that is going in a circular motion.

8. What is the Coriolis effect? a. The apparent curvature of a mass moving in a rotating system b. The actual curvature of a mass moving in a rotating system c. The outward direction of a mass leaving a rotating system d. The inward direction of a mass leaving a rotating system Answer: a. This is the apparent curvature of a mass moving in a rotating system when it leaves the path of the system. It curves in a counterclockwise direction in a circle that is going clockwise but, as is clear, this is a fictitious curvature that occurs only within the frame of reference of the observer also rotating in the circle.

9. You have the mass of the earth and the mass of the moon that are attracted to each other. What is the difference between the force of gravity between the two objects if the moon has one percent of the mass of the earth? a. The force of gravity is equal on the earth versus the moon b. There is only force from the earth c. There is 99 times more force of gravity from earth than there is from the moon d. The force of gravity on earth is 1 percent of the force of gravity on the moon

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Answer: a. The force of gravity between the two would be equal, regardless of the mass. This must be the case according to Newton’s third law about equal and opposite forces. If they were not the same, the moon would escape earth’s orbit or the moon would crash into the earth.

10. What is the proportionality of the force of gravity between two objects of different masses and the radius between them? a. The force is proportional just to the largest object and inversely proportional to the radius. b. The force is proportional to the mass of both bodies and inversely proportional to the radius. c. The force is proportional to just the masses of both bodies but not to the radius. d. The force is proportional to the masses of both bodies and inversely proportional to the square of the radius.

Answer: d. The force of gravity of two bodies is directly proportional to the masses of both bodies and inversely proportional to the square of the radius. There is a proportionality constant, which is about 6.7 x 1011

that is a part of this equation.

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CHAPTER 5: WORK AND ENERGY The focuses of this chapter are the concepts of work, energy, and power. Work involves the process of getting something done using forces or the transfer of energy from one state to another. Energy, as you may know, cannot be created or destroyed; it can only be transferred from one form to another. The chapter also introduces the topic of power, which is a closely related term that is the rate (energy amount per time period) at which work is done or energy converted. The relationship between energy, work, and power will be covered as part of this chapter.

WORK The law that states that energy cannot be created or destroyed has been proven by experiment. As more and more forms of energy were uncovered, this law has always been found to apply in experimental situations. There is no simple way to describe energy as it involves a wide variety of different types of work as well as some things that are not included in the definition of work, such as potential energy, which will be discussed.

Work is basically the exchange of one form of energy to another. For work to be done, there must be force and their must be motion and displacement in the direction of the force. Work as it applies to the use of a constant force, is the product of force and the distance through which the force acts. Work is defined through the equation: Work equals, force times cosine of theta times the displacement vector, where theta is the angle between the force vector and the actual direction of the displacement.

Remember that forces can have an x axis and y axis component. When you push on a lawnmower, you push on a handle that is not usually in a horizontal direction. This force vector must be partially in the direction of the work that needs to be done (as in across the lawn) in order for any work to be done with theta being the angle of the lawnmower handle and the direction the lawnmower is going. Bear in mind that there can be work applied to different directions and, like any vector system, these different

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aspects of work can be added together. Figure 30 shows the concept of work as it applies to the lawnmower:

Figure 30.

If someone is using a lot of energy to hold an object but the person does not move, the object itself does not move and therefore no work is done. There must be motion if work is to be done and there must be a component of the force in the direction of motion.

Carrying the object on level ground also does no work because the forces applied to the object does not move the object in the direction of movement. Carrying the object up a staircase will do work because the force of carrying it does carry the object in some direction related to the force.

The units for work and energy are the same. Because work is known to be force times the distance, the SI units for energy are Newton-meters or Joules, in which one Joule equals a kilogram meters squared per second squared. It is not a great amount of work as it is the amount of energy necessary to lift 100 grams a distance of one meter. A nonSI unit for energy is the calorie or kilocalorie. One calorie is the amount of heat required to warm 1 gram of water by one degree Celsius. One calorie is equal to 4.184 Joules.

One food calorie is actually a kilocalorie, which equals 4,184 Joules.

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KINETIC ENERGY Because work and energy have the same units, you can also say that work does not do nothing nor does it go nowhere. In the situation of a person pushing a lawnmower, the lawnmower will go at a constant speed; however, it will decrease in speed over time if the work is stopped because of friction and the transfer of energy through heat transfer.

Work to lift an object up a hill will increase the potential energy of the object that can be recovered through dropping the object off a cliff by means of the force of gravity.

We have learned that force results in acceleration. The net work in a system is the sum of work done by all external forces. This force is called the net force so that it can be said that the net work equals the net force times distance times the cosine of theta, which is the angle between the net force vector and the displacement vector. The same is true of the average work, which is the average of different forces times the distance times the cosine of theta. If force times theta is graphed on the y axis and distance is graphed on the x-axis, work is the area beneath the curve. This is described in figure 31:

Figure 31.

In typical linear work situations, where force is applied to drag something across a floor, there are no net forces involved in the vertical direction; the force in the direction of movement equals the force applied in the horizontal x-axis direction minus the force

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applied due to friction. Because force equals mass times acceleration, the net work will equal the mass times the acceleration times the distance. Another substitution that can be made involves velocity. The net work on a system can be described as half the mass times the square of the final velocity minus half the mass times the square of the initial velocity. This is explained visually in figure 32:

Figure 32.

The expression of work as it relates to mass and volume is called the work-energy theorem, which applies even if the forces vary in direction and magnitude. The quantity one-half mass times the velocity squared is referred to as the kinetic energy or KE of a mass moving at a certain speed. In actuality, this is referred to specifically as the “translational kinetic energy” because it is linear in nature and is different from the rotational kinetic energy, which will be covered later.

You should know that energy is going to be in the form of Joules, which will be kilogram-meters squared per second squared. Another way to look at it is that Kinetic Energy is force times distance so, in the case where you are given the net force and the distance, you need only multiply the two as one joule is one Newton-meter.

GRAVITY AND POTENTIAL ENERGY All work done in climbing a mountain or lifting an object against gravity does work against the force of gravity. When there is work done, energy is transformed. The work required to take an object of a mass m through a height h at constant speed will be equal to its weight or mass times the force of gravity times the height. This is called the gravitational potential energy. The energy is stored in the gravitational field of the 78

earth. Potential energy is a property of a system rather than of a single object and is basically due to its physical position. For most systems in which gravitational potential energy is used, the arbitrary setpoint for zero potential energy is the Earth’s surface.

This is an arbitrary setpoint and what’s important is the difference in gravitational potential energy.

Because all energy is energy, there must be a way to convert gravitational potential energy and kinetic energy. The best way to do this is to consider the difference in potential energy and kinetic energy without considering the step involving the work involved. The change in potential energy of a weight can be described as the mass times the force of gravity times the change in height. The change in height can be positive or negative. As something falls, it gives up its gravitational potential energy in the fall.

This potential energy is converted to kinetic energy.

It is important to note that this change in gravitational potential energy is independent of the path the object takes to get from point A to point B. It just depends on the height as it applies to a lower height on the y-axis, which might be the height above the level of the earth. The change in potential energy is equal to the kinetic energy so that the speed can be determined after a certain distance is covered. What you know about kinetic energy is that it equals one-half mass times velocity squared and that will equal mass times height times the force of gravity. Note that you don’t have to know the mass of the object because it cancels itself out in this equation. Figure 33 describes these equations:

Figure 33.

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Note that the mass cancels out in the equation. This is consistent with what you already know—that the speed of falling objects is the same thing regardless of the mass. You should also know that, when friction is negligible, the speed of a falling body depends only on its initial speed and height and not on the path the object takes. It can fall directly or take a circuitous path, such as a rollercoaster on its path. The same is true of going uphill—as long as friction is not a factor (which it normally is).

CONSERVATIVE AND NONCONSERVATIVE FORCES We have so far led up to the idea that force is what creates work. Some forces, such as gravitational force, for which work can be done against or for, depends only on the starting point and ending point of the object and not on the path. This is called a conservative force. Potential energy is also a conservative force as it does not necessarily mean the same thing as gravitational potential energy. Think of winding up a toy or an old-fashioned watch. These will also store potential energy. In both cases, stored energy is recoverable as work. In short, conservative force is one that depends only on the starting and ending point of a motion and not on the path taken.

Let’s look at the potential energy stored in a spring. This can, in some situations, obey Hooke’s law, which states that the magnitude of the force on the spring is proportional to the deformation of the spring. In such cases, the amount of deformation is equal to the amount of stretching or compression of the spring in total length. The force equals the spring’s force constant (which depends on the spring) and the change in length.

The average force of the spring will be half of its stretched length multiplied by a force constant.

The potential energy of a spring can be defined as one-half the force constant times the distance stretched squared (based on the work-energy theorem). The distance stretched is actually the difference between the length of the unstretched or uncompressed spring and the length it is stretched out. It does not depend on the path taken, just on the stretch or squeeze of the spring. Figure 34 shows the force and potential energy of a stretched spring:

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Figure 34.

Remember that, according to the work-energy theorem, the net work by all forces acting on a system is equal to its change in kinetic energy. Remember too that the work net equals one-half mass times the velocity squared (see figure 33). These two values are the same thing (change in kinetic energy and net work). If only conservative forces act on a system the work net equals the work by conservative forces).

What this means is that, if things like gravitational force or spring force does work, this will equal the change in potential energy but will be opposite in direction. Take this to mean that the total kinetic energy and total potential energy added together will equal a constant if only conservative forces are involved. This leads to the equation stating that there is conservation of mechanical energy. The total kinetic energy plus the total kinetic energy equals the system’s mechanical energy. This is seen in figure 35:

Figure 35.

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What this means is that the total mechanical energy is constant over the system. It just changes form from kinetic to potential energy and vice versa.

Let’s use an example. You have a toy car that weighs 0.1 kg and is propelled by a spring that rises 0.18 meters from the starting point. The spring is compressed 4 cm (0.04 m) with a force of 250 newtons per meter. If friction is negligible, how fast does the car go initially and how fast does it go at the top of the slope? This is described in figure 36:

Figure 36.

Using these equations in figure 36, you can solve for the initial and final velocity. In the beginning, the initial height, final height, initial speed, and final compression of the spring is zero, so the velocity is going to be related to the mass of the object, the compression of the spring, and the force constant. Putting in the numbers and solving for the velocity, you get 2 meters per second. This is the speed the car takes as it gets out

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of the gate, so to speak. At the end, the height comes into play because the car goes up a certain height against its potential energy. Solving for the velocity, you get 0.687 meters per second.

NONCONSERVATIVE FORCES Remember that forces can be conservative or nonconservative. A nonconservative force is one for which the work depends largely on the path taken. A good example of a nonconservative force is friction. Friction will depend on the path because, the longer the path, the more friction will become a force to be reckoned with. This means that there is no potential energy associated with nonconservative forces.

Work by a nonconservative force will add or take away the mechanical energy of a system. Friction is nonconservative because it creates thermal energy, which dissipates and ultimately removes energy from the system. Even if this type of energy can be retained, it cannot be completely converted back into work so it is not completely recoverable. Air resistance is also a nonconservative force. These energies and forces basically negate mechanical energy conservation idea.

So, what happens to the work-energy theorem when conservative and nonconservative forces act together. In such cases the net-work is the sum of the work by nonconservative forces plus the conservative forces. These, taken together, equal the change in kinetic energy of a system. For example, when pushing a box on an incline using an applied force, the vector goes up the incline while the force of friction goes down the incline. They oppose each other in a parallel direction. There is also the work done against gravity to consider.

Remember that the conservative work will equal the opposite of the potential energy.

The nonconservative work will equal the change in kinetic energy plus the change in potential energy. This means that the nonconservative work adds to the mechanical energy of the system. If the nonconservative work is positive, then the mechanical energy increases, such as in pushing a crate up a ramp. If the nonconservative work is negative, the mechanical energy is decreased, such as when dropping an object that deforms on the ground. These equations are explained in figure 37:

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Figure 37.

In summary, nonconservative forces depend on the path the object or person takes.

Friction is the main example of a nonconservative force that will change mechanical energy into thermal energy. This nonconservative force will change the mechanical energy of a system. When both conservative and nonconservative forces act, energy conservation can still be applied and can calculate motion as it applies to the known potential energies of the conservative forces as well as the work done by nonconservative forces.

CONSERVATION OF ENERGY Energy is always conserved. We have looked at mechanical energy, conservative forces, and nonconservative forces. Energy, however, can manifest itself in many different forms and these must be taken into account in any energy equation. So, besides mechanical energy, what other energy forms are in the equation? An example of looking at other forms of energy can be seen in the process of eating. There is the mechanical energy of digestion, the chemical energy of digestion, the oxidation of food and its use in

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other forms of chemical energy, and the thermal energy that “heats” the person to normal body temperature.

A complete list of other kinds of energy include these: •

Electrical energy

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Chemical energy

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Light energy or radiant energy

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Nuclear energy

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Thermal energy

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Sound energy

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Kinetic energy—motion energy

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Elastic energy—a form of potential energy

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Gravitational energy—a form of potential energy

So, when solving energy problems, you need to do the following in order to make sure that you have included all of the possible energy forms in the equation. First, you need to determine the entire system of interest and identify what information is given and what quantities need to be calculated. It usually helps to draw out a picture. You next define and list the magnitude of all of the forces. You need to write an equation that includes the kinetic energy, potential energy, and other energies, which are completely conserved in the beginning and end of the equation. The work done by conservative forces are included in the kinetic and potential energies. Cross out any energies that are zero or that do not apply. Solve the problem and check to see if the answer is reasonable.

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TRANSFORMATION OF ENERGY Energy is transformed all of the time. Sunlight can produce electricity in a solar cell and thermal energy in steam can be converted into mechanical energy in a turbine situation.

Food becomes chemical energy and thermal energy. Light is converted into chemical energy through the action of photosynthesis.

What’s true is that the output of useful energy or work will be less than the energy output. The efficiency of an energy conversion process is defined as the work output divided by the energy input (remember that work and energy have the same units).

Some of these efficiencies can be calculated. For example, about 40 percent of the chemical energy of coal becomes useful electrical energy. The other 60 percent gets transformed into things like thermal energy, which leaves the smokestack (or heats a room). On the other hand, an electrical motor is 98 percent efficient.

POWER Power is not the same thing as work or even energy. What you already know is that some things can be considered having a lot of power, while other things can be considered having little power. The same amount of work can be done by systems that are “powerful” as systems that are “not powerful” but there is a key difference between these two systems. Power, in a real sense, is the amount of work done over a period of time.

The SI unit for power is the Watt, in which one watt equals one joule per second. Things with high power get work done faster, while things with low power will get work done slower. Because work is the transfer of energy, power is also the same thing as the rate at which energy is expended. A sixty-watt light bulb will expend sixty joules of energy per second. Once you know the energy put into a system (kinetic and potential energy) and the time it takes to put that energy into it, you can calculate the power. A person walking a straight line will expend less energy than a person walking up a set of stairs because there is potential energy to overcome in climbing stairs that isn’t involved in walking a straight line.

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One horsepower, which is not an SI unit, is equal to 746 Watts. Horsepower or kilowatts are used to define power in larger systems because a watt is not a very large amount.

Like the transfer of energy, there is a loss of energy in systems that have power. A 60watt incandescent bulb gives off only about 5 watts of light with 55 watts dissipating into thermal energy.

The higher the power consumption rate and the longer an appliance is used, the greater is the cost of that appliance. The power consumption rate is equal to the energy divided by time. Where energy is the is what we pay for when we pay the power bill or “electric bill”. This energy is equal to the power in Watts multiplied by the time it is used. An electric bill will list the units of energy in kilowatt-hours, which can be used to define the cost of certain things electricity uses. If you keep a 60-Watt bulb going 24 hours a day for thirty days, you expend 43.2 kilowatt-hours. You can calculate the cost by looking at the cost of electricity in cents per kilowatt-hour.

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KEY TAKEAWAYS •

Work is in the same units of energy and reflect a change in energy of a system in joules.

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Work is the amount of energy that goes over a specific displacement and may not be the total amount of force applied.

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The cosine of theta is the angle between the force applied and the direction of work.

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Work will be a combination of the change in kinetic energy and potential energy if only conservative forces are involved.

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There are conservative forces and nonconservative forces; the nonconservative forces will change the mechanical work of the system but this is not the case with conservative forces.

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Power is the work done over a specific period of time; this is measured in watts (which is the SI unit) or in horsepower.

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QUIZ 1. What is work in physics essentially the same thing as? a. Power b. Change in energy c. Displacement d. Force Answer: b. Work involves any of these things but is essentially the same thing as a change in energy, which does involve forces, movement, and displacement, but is not the same thing as these things.

Power is related to work but is not the same thing.

2. How is work related to force? a. Work is equivalent to force over a period of time. b. Work is equivalent to force multiplied by a specific constant. c. Work is equivalent to force multiplied by the distance it acts on. d. Work is equivalent to potential energy multiplied by force.

Answer: c. Work is proportional to force but it is only that component of force that is acted upon over a certain distance.

3. In the equation in which work is equal to the Force times distance times the cosine of theta, what does theta represent? a. The angle between the normal weight and the horizontal direction. b. The angle between the force vector and the displacement vector. c. The angle between a pulley system vector and the applied force on the system. d. The angle between the force vector and the y-axis.

Answer: b. This is actually very simple. When doing work equations, you take the magnitude of the force vector times the magnitude of the direction/displacement vector and the cosine of the angle theta, which is the angle between them in degrees.

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4. What is the proportionality associated with kinetic energy and the velocity of a body in motion? a. Kinetic energy is directly proportional to velocity b. Kinetic energy is proportional to the square root of velocity c. Kinetic energy is proportional to the square of velocity d. Kinetic energy is inversely proportional to velocity Answer: c. Translational kinetic energy is one-half the mass times the velocity squared so it is proportional to the square of the velocity.

5. How does the velocity of something relate to its mass and height as it falls? a. The velocity is proportional to its mass and its height b. The velocity is proportional to its mass and the square root of its height c. The velocity is proportional to just the height d. The velocity is proportional to the square root of its height but not the mass Answer: d. In the equation where something falls, you need to put together the kinetic energy and potential energy equations, which cancel out the mass so this is not a factor at all. This leaves the fact that height and the force of gravity equals one-half the velocity squared or the velocity is the square root of 2 times the height.

6. You lift an object up an incline of ten meters to one-meter total height versus lifting up one meter by hand without the benefit of a ramp. What is the difference in the object’s potential energy in each situation? a. The potential energy of raising the object is a tenth the potential energy of the ramp situation. b. The potential energy of each situation will be the same. c. The potential energy of raising the object is ten times the potential energy of the ramp situation. d. The potential energy of raising the object is one-one hundredth the potential energy of the ramp situation.

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Answer: b. The potential energy of the object is directly related to its overall height and does not depend on the path that is taken.

7. When looking at the potential energy of a spring, what is it related to? a. It is proportional to the distance the spring is compressed. b. It is proportional to the square of the distance the spring is compressed. c. It is proportional to the cube of the distance the spring is compressed. d. It is the same, regardless of the spring and is proportional to the square of the distance the spring is compressed.

Answer: b. The equation for the potential energy is one-half times the force constant times the distance compressed squared. This force constant is not the same for every spring so there is a force constant involved that depends on the spring.

8. When considering the conservation of mechanical energy, what factor least destroys this concept? a. The thermal energy of the system b. Frictional forces of the system c. Air resistance forces d. Gravitational forces on a system Answer: d. Gravitational force is a conservative force so it follows the rule of conservation of mechanical energy. Things like air resistance, frictional forces, and thermal energy will take energy away from mechanical energy and cannot be turned into work, destroying the concept of conservation of conservative forces. In reality, it is difficult to find a real system that has true conservation of mechanical energy.

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9. What type of energy gets transformed into what other type of energy in the act of photosynthesis? a. Thermal energy into chemical energy b. Radiant energy into chemical energy c. Chemical energy into kinetic energy d. Radian energy into thermal energy Answer: b. Radiant energy gets turned into chemical energy in the act of photosynthesis. This is the process of sunlight creating molecules that ultimately store chemical energy through their chemical bonds.

10. The efficiency of an energetic system is defined as what? a. The percent of work output divided by energy input of the system b. The sum of all the energy inputs in a system c. The energy input divided by the work output of the system d. The sum of the energy output minus the energy input Answer: a. The efficiency of the energetic system is defined by percentages. It is the percent of the work output divided by the energy input of the system. It is a maximum of 100 percent but is actually much less than this in many energetic systems.

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CHAPTER 6: MOMENTUM AND COLLISIONS This chapter deals with the subjects of linear momentum and collisions. It involves first the topic of linear momentum, which is the velocity of something and the product of its mass. Impulse is also covered, which is the change in momentum of an object in a system. Momentum leads an object toward collisions with other objects. There are elastic collisions and inelastic collisions that differ in their apparent conservation of momentum. Each of these topics builds upon things that have already been learned in previous chapters on force, velocity, and mass, as well as Newton’s laws.

LINEAR MOMENTUM When you think of momentum, you think of something large and something fast. This intuitive understanding of momentum is actually what momentum is. In scientific circles, momentum is the mass of a system multiplied by its velocity. The SI unit for momentum is kilogram meters per second. Momentum is a vector that has a direction and magnitude because velocity is a vector.

Momentum is recognized in classical physics. It is a way to quantify motion and can also be described in terms of force as the net external force is equal to the change in momentum divided by the time over which it changes. In other words, net force equals the change in momentum over the change in time. Force acting over time will change the momentum of something and is an important thing in large-scale physics as well as the physics of subatomic particles.

Newton’s second law states that force equals mass times acceleration. From this we can determine that a change in momentum, identified by the letter p, is equal to the change in mass times velocity so, if the mass changes or the velocity changes, so will the momentum. In most cases, the mass will not change so it is more likely that a change in momentum equals a change in velocity. What do you know about change in velocity?

This is what acceleration means, so you can derive the concept that force equals mass

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times acceleration. It can apply, too, to things where the mass is changing, as in a rocket that loses mass as it expends fuel.

IMPULSE The actual effect of a specific force on an object depends on how long the force is acting on the object as well as the magnitude of the force. Small forces would have to act over a long period of time to cause the same change in momentum as a large force. The concept of this from a physics perspective is the change in momentum. This is the force multiplied by the change in time. This change in momentum is referred to as “impulse”.

It is the average net force multiplied by the time the force acts.

While you may not have heard of impulse, you intuitively understand how it works.

Things like dashboard padding in a car and airbags allow the net force to act over a much larger square area and over a longer period of time when the vehicle has a sudden stop. The force to bring the occupant to a stop will be less if it acts over a larger period of time. Longer collision times will equal decreased force applied. This is why racing cars are made to collapse versus being rigid; the impulse to the drive is less when the automobile crashes.

So far, you have determined that force, acceleration, and momentum are vectors.

Because of this, impulse is also a vector. If something strikes a wall with a certain velocity directly versus at an angle to the wall, there will be a difference in the impulse because the force on the wall will be different. This will lead to a difference in the impulse on the wall.

One issue that comes to mind is that forces are not usually constant. Forces can vary over time so that you need to get some type of average effective force that produces as close to the same result as a time-varying force as possible. Figure 38 shows the force varying over time for a ball that bounces off the floor.

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Figure 38.

The area under the curve is the momentum and, the impulse is the change in momentum between two periods of time. Of course, there can be more complex mathematical calculations that determine force over time that are beyond the scope of this course. This is why the effective force is used.

The important thing to remember about momentum is that it must be conserved. It does not seem to be conserved within a small system; however, it will be conserved if the system is sufficiently large enough. An example of this might be a football player that bounces with momentum against a goalpost. The player will have a force applied that will bounce the player backward. There will also be a minor momentum and force applied to the ground that holds the goalpost and to the earth itself, which are so minor that they are not usually put into play in equations regarding this situation.

What would happen in a system where there are two bodies of roughly the same mass that collide with one another. Say, for example, that two cars of equal mass are going in the same direction but the car in the back is faster and hits the car in the front. If friction is negligible, the car behind will slow its momentum and car in front gains momentum. The total momentum of the 2-car system will remain constant. Figure 39 shows this conservation of momentum:

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Figure 39.

In the case of the two cars, the momentum in the “before” situation will equal the momentum in the “after” situation, referring to the momentum of the entire system. An isolated system is one in which there is no external force on the system. This relies on knowing the system and making sure that the system doesn’t have any external forces that can be applied. In the case of the two cars, the velocity changes are the only things that change the momentum in reality. Friction is a force that is not included in the example; it exists, however, in the real situation.

SUBATOMIC COLLISIONS AND MOMENTUM CONSERVATION This idea of conservation of momentum applies to atomic and subatomic particles as well. Momentum is a property of all subatomic particles, including particles that have no specific mass, such as photons. What it means is that the idea of momentum may not be as closely related to mass as it seems to be. Momentum also applies to wave properties as you will see later in the course as well as to issues related to particle physics. The concept of momentum and conservation of momentum is used to identify the mass of things that are otherwise impossible to weigh.

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ELASTIC COLLISIONS IN ONE DIMENSION The conservation of momentum is useful when looking at two-object collisions.

Remember that this concept works only when there is a net zero external force on a system (only internal forces). It is most helpful to look at collisions in one dimension.

An elastic collision is a collision that also conserves internal kinetic energy, which is the sum of all the kinetic energies of a system.

What’s true is that completely elastic collisions can only happen with subatomic particles like electrons and neutrons. Macroscopic collisions can closely resemble elastic systems; however, in reality, kinetic energy is always converted into other forms of energy, such as heat or friction and sound. Things like icy surfaces, air tracks, and firm objects that do not deform are examples of kinetic energy being nearly conserved.

Go back and remember what internal kinetic energy is. This is the sum of all the kinetic energies of the objects within a given system. When two things collide on a frictionless system, both things are included as part of the system. There will be momentum before they collide and momentum after they collide. Because of conservation of momentum, there will be the same momentum before the collision as there is after the collision.

There are equations for the conservation of momentum as well as those for the conservation of internal kinetic energy that need to be included as part of the problemsolving in these situations. Figure 40 shows these Equations:

Figure 40.

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You can use these two equations in order to solve for differences in velocity in two objects undergoing a collision because there are two equations for mass and velocity that can be used to solve for two variables. Simply isolate the two equations and solve for the different velocities v1’ and v2’ if you know the initial velocities and masses of the two objects. Because solving for velocity squared involves a quadratic equation, you will need to use this equation to solve for the v1’ and v2’.

INELASTIC COLLISIONS IN ONE DIMENSION In inelastic conditions in one dimension, the internal kinetic energy is not conserved. It means that forces between two objects that are colliding may remove or add some internal kinetic energy and that internal forces may change forms of energy in the system. It can involve colliding objects sticking together and can involve heat transfer of energy. It may also convert stored energy into internal kinetic energy (as when something explodes and separate from the central mass).

An example of an inelastic collision is when two equal objects of equal speed stick together. The total kinetic energy is ½ mass times velocity squared plus ½ mass times velocity squared equals mass times velocity squared. The momentum is conserved but the internal kinetic energy is equal to zero after the collision. This is a perfect inelastic collision because it reduces the internal kinetic energy to the minimum it can have while still conserving momentum. Figure 41 shows this situation. After the collision, the kinetic energy equals zero, while before, it is mass times velocity squared.

Figure 41.

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Some objects, of course, do not stick together, so that less of the internal kinetic energy is removed. This is what happens in automobile accidents. This can also be identified by looking at a collision between two objects, in which one has a spring attached. When the two objects collide, the potential energy in the spring is “released”, resulting in both objects leaving the collision at a higher speed than before. Figure 42 shows this situation:

Figure 42.

In figure 42, the situation is frictionless so that momentum is conserved. The motion is one-dimensional. The potential energy of the compressed spring is released during the collision and is converted to internal kinetic energy. In such cases, the internal kinetic energy is increased for the system.

COLLISIONS IN TWO DIMENSIONS This involves the collision of two objects that do not directly move in a single line. This includes things like the collision of billiard balls, which go off at an angle and often scatter. This is an approach that is not much different from studying kinematics in two dimensions and studying dynamics. The goal is to establish an appropriate coordinate system and to resolve the motion into two different components along an x and y axis.

These components are solved separately. In these discussions, no rotation is

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anticipated, although this will be discussed later. In order to avoid this problem, it is important to look at the scattering of masses that cannot spin.

As usual, the net force on the system will be zero so that momentum is always going to be conserved. The most basic condition is one in which one particle is at rest with the xaxis being parallel to the velocity of the incoming particle. There will be components of the momentum that happen along the x- and y-axis. In such a system, the py or momentum in the y direction will be zero and the px is the momentum of the incoming particle only (as there will be no momentum in the other particle). This is described in figure 43:

Figure 43.

In such cases, momentum is preserved so that M1 multiplied by V1 and M2 multiplied

by V2 together will be the same value in both the before collision and after collision situations. The velocity components on the x axis will be related to the cosine of theta, where theta is the angle between the x axis and the direction of movement of the objects after the collision. The component of the velocity in the y direction is related to the sine of theta.

Remember that the total momentum is preserved, in which you need to consider the x component of the momentum and the y component of the momentum (as there will be

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both components in play). Figure 44 describes the equations involved in the previous figure:

Figure 44.

In elastic situations where there are two colliding objects having an equal mass, the situation is similar to colliding billiards balls and in the case of many subatomic particles. The internal kinetic energy will be conserved and the momentum will be conserved. In such cases, the total kinetic energy of a system will be related to the sum of half the mass times the velocity squared, with separations of the x- and y-components based on the cosine and sine of theta or the angle of each object as it relates to the x-axis and y-axis, respectively.

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KEY TAKEAWAYS •

Momentum depends on the mass and velocity of a system; it is proportional to both components.

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Impulse is the amount of force applied over a specific period of time or the change in momentum.

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Impulse and momentum are vectors with direction and magnitude.

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In collisions between two objects in one or two dimensions, there will be conservation of momentum and kinetic energy in elastic situations but just conservation of momentum in inelastic situations.

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QUIZ 1. What is the SI unit for momentum? a. Kilogram meters per second b. Kilogram meters per second squared c. Meters per kilogram per second squared d. Kilograms per second squared Answer: b. There is no easy SI unit for momentum. Instead, it is listed in the known SI units for mass and velocity, which are kilogram meters per second. It depends on the mass as well as the velocity for something.

2. If the mass of something is 2 times the mass of something else, how does this change the momentum? a. The momentum will be doubled. b. The momentum will be the square of 2 or will be quadrupled. c. The momentum will be the same. d. The momentum will be one-half the mass of the smaller object.

Answer: a. The momentum of something is directly proportional to its mass so it will double if the mass of an object doubles.

3. What are the SI units for impulse? a. Newton-seconds b. Joules per second c. Newtons d. Kilograms per square meter per second Answer: a. The SI unit will be in Newton-seconds, which means the amount of force applied over a period of time. Remember that impulse is the same as the change in momentum.

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4. Which of the following is not considered a vector? a. Impulse b. Momentum c. Speed d. Force Answer: c. Each of these is a vector quantity, having both a direction and magnitude, except for speed which, by definition, does not have a direction but only has a magnitude.

5. What is not an example of a loss of kinetic energy that can occur in macroscopic systems that aren’t completely elastic when it comes to the collision of two objects? a. Heat energy b. Friction c. Radiant energy d. Deformation Answer: c. Heat, friction, sound, and deformation of objects can all detract from the true conservation of kinetic energy that is supposed to occur in elastic systems. Radiant energy is generally not a factor in these types of systems.

6. In general, in elastic conditions in one dimension, the equations revolve around the idea that what must be conserved? a. Momentum and Internal kinetic energy b. Velocities and mass c. Total kinetic energy and velocities d. Acceleration and forces Answer: a. In elastic conditions, momentum and internal kinetic energy are two things that must be conserved in an elastic collision between two objects in one dimension.

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7. What is the kinetic energy of two identically-massed objects that approach each other with the same velocity v after they stick together in a perfectly inelastic way? a. The kinetic energy will be the mass of each multiplied by the velocity squared. b. The kinetic energy will be zero. c. The kinetic energy will be in the opposite position at one-half mass times velocity squared. d. The kinetic energy cannot be calculated in this situation.

Answer: b. If this is a perfectly inelastic collision, the equally-massed objects will have a zero velocity after the collision, which means that the kinetic energy will be zero.

8. What type of collision between two objects represents a perfectly inelastic collision? a. Two objects collide and stick together b. Two objects collide with zero friction c. Two objects collide and have springs attached that bounce off each other d. Two objects collide and neither of them are deformed Answer: a. In a perfectly inelastic condition, the two objects collide and stick together. This will take away kinetic energy from a system to the maximal degree.

9. In the situation where an object strikes another object in the x-direction and both objects travel off in opposite directions, what is the velocity component in the direction of angle theta1 of the object that goes at that angle (theta1) from the y-axis? a. Velocity after the collision multiplied by the sine of theta1. b. Velocity after the collision multiplied by the cosine of theta1. c. Velocity before the collision multiplied by the sine of theta1. d. Velocity before the collision multiplied by the cosine of theta1.

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Answer: a. The velocity will not be the same as the velocity before the collision because of the collision. As the object heads of to an angle of theta1, its velocity in the y-axis will be the velocity after the collision multiplied by the sine of theta1.

10. What will be the proportionality between the total kinetic energy of an object after colliding with another object and its velocity after the collision? a. The total kinetic energy will be proportion to the velocity squared multiplied by the cosine of theta squared where theta is the difference between the x-axis and the angle of its direction after the collision. b. The total kinetic energy will be proportional to the velocity squared multiplied by the sine of theta where theta is the difference between the xaxis and the angle of its direction after the collision. c. The total kinetic energy will be proportional to the velocity times the cosine of theta. d. The total kinetic energy will be proportional to the velocity squared.

Answer: d. The kinetic energy can be broken down into its xcomponent and its y-component; however, the total kinetic energy has both the x- and y-components so that it does not need the cosine of theta taken into account. In general, the kinetic energy is proportional to the velocity squared.

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CHAPTER 7: STATICS, TORQUE, AND ROTATIONAL MOTION This chapter opens up the topics of statics, torque, and rotational motion. While motion was covered in the beginning of the course, there is an entire branch of physics associated with nonmoving forces, which are collectively referred to as statics. Torque involves forces that act in a twisting fashion in order to cause motion or the potential for motion. This leads to the issue of rotational motion, along the lines of rotational acceleration and motion that is not necessarily uniform as was discussed chapter four of the course.

STATICS Statics involves forces that result from zero external forces in any direction, including the x-axis, y-axis, and z-axis. There is static equilibrium, in which there is no motion, and dynamic equilibrium, in which there is constant velocity. This means that there can be motion and equilibrium as long as the velocity is not changing. The concept of zero force applied must be necessary for equilibrium but there can be zero force applied and nonequilibrium if the force acts on an object that undergoes rotation because the forces cause the object to move in ways that result in rotational movement.

TORQUE While net zero force is a requirement for equilibrium, it is not the only thing necessary.

Equilibrium also involves the avoidance of accelerated rotation. A rotating body can be in equilibrium if the rate of rotation is constant. The larger the force, the greater is the ability to rotate something on its axis. The further out the applied force, the greater is the ability to turn something on its axis as well. Perpendicular force on a door at its furthest point away from the axis is what best allows the door to open.

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Torque can be defined as the turning or turning effectiveness of a force. An example of torque is what happens when you try to open a door. It depends on the radius from the axis, the magnitude of the force, and the radius of the force. In effect, torque is the rotational equivalent of a specific force. In equation form torque, as identified by the Greek letter tau, equals the radius times the Force, times sine of the angle between the force and the vector that is directed from the point of application to the pivot point.

Figure 45 describes these values:

Figure 45.

Another way of describing torque is that it is the perpendicular radius times the force applied, which can be seen in a diagram in which the force applied is not 90 degrees to the axis. Figure 46 also shows a simplified way of describing torque:

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Figure 46.

The SI unit of torque is Newton-meters in which a certain force in Newtons acts over a distance of x meters away from the hinge point or axis. From the equation, you can determine that torque is directly proportional to the radius and the force. The pivot point can be at any point on an object. In addition, torque can be clockwise or counterclockwise around the pivot point.

In order to achieve equilibrium when torque is involved, the net external torque on the system must be zero. If the net external torque on a system is zero for one choice of pivot point, it must be true additionally for any other choice of pivot point inside or outside the system of interest. Clockwise torque is considered by definition to be negative and counterclockwise is considered by definition to be positive.

A seesaw is an example of torque that can be in equilibrium. In order to balance a seesaw, the lighter person must sit further away from the center of the seesaw (the pivot point), while the heavier person can get by with sitting closer to the center of the seesaw in order to have equilibrium. Force is defined as the weight of the child or, for simplicity, their mass. The force of a child weighing 25 kilograms will be 25 times the force of gravity or about 245 Newtons. This same child sitting a meter from the pivot point will exert a force of 245 Newton-meters about the pivot point.

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STABLE EQUILIBRIUM There are three types of equilibrium: stable, unstable, and neutral. A balanced system has the center of gravity or CG directly over the pivot point. This can be seen in the case of a person standing balanced over a lever. There may be arms and legs on the person but the center of gravity is balanced over the lever.

A system is in stable equilibrium if, when there is displacement, there is a net force or torque that is opposite to the direction of the displacement. A round object in a bowl will experience a restoring force when displaced from its equilibrium position that forces the object back toward its equilibrium. Most systems are in their stable equilibrium. Unstable equilibrium is when there may be a force that is in the same direction as the displacement from the equilibrium. This would the same thing as a ball on top of a hill.

A system is in neutral equilibrium if its equilibrium is independent of displacements from its original position. A marble or pencil lying flat on a horizontal surface is an example. This is also true of a marble on a saddle, which is stable in one direction but unstable in another direction. A pencil lying horizontally on a surface is in limited neutral equilibrium because it can be displaced in one direction but not in another direction. Figure 47 shows these types of equilibrium:

Figure 47.

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Some systems are more stable than others. A pencil in stable equilibrium is less stable than a ball at the bottom of a bowl because it doesn’t take much of a displacement of the pencil to have the center of gravity outside of the stable point and to cause tipping of the ball. Stability in a person will involve lowering the center of gravity by crouching, which will be resistive of movement that tips the person over. Increasing the base of support will also increase the stability of a person or any system that needs to resist torque forces.

SIMPLE MACHINES Simple machines will multiply or augment a force that is applied to a system. Examples of simple machines include levers, pulleys, gears, screws, and wedges. Energy, as always, is conserved, and a machine cannot do more work than the energy put into it.

Machines, on the other hand, are able to reduce the input forces necessary to do the specific job it does. The mechanical advantage of a simple machine is equal to the Force output divided by the Force input. Figure 48 shows an example of a lever, which is a simple machine:

Figure 48.

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Figure 49 shows forces on a fulcrum that has a mechanical advantage on the lifting of a weight. This uses torque acting at a distance on a heavy object opposite the fulcrum:

Figure 49.

In such cases, the length input times the Force input equals the length output times the force output. What this means is that the ratio of the force output and the force input is inversely proportional to the length input divided by the length output. The further out the force on one end of a fulcrum, the less force needs to be applied to achieve a force in the opposite direction. This means is that the mechanical advantage is inversely proportional to the output length and directly proportional to the input length.

A crank is a lever that is able to be rotated 360 degrees about its pivot. While it doesn’t look like a pivot, the mathematics behind it is the same. Wheels and gears do the same thing. Figure 50 shows the mechanical advantage of a crank:

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Figure 50.

Cranks exert torque on an object using a very long radius input or length input and a very short radius or length output, so that it can exert a significant torque on the system.

Pulleys will have a mechanical advantage of one per pulley but the force changes direction. The force will not change in magnitude as long as the pulleys are friction-free.

The total force is an integral multiple of the tension on the cable. This means that two pulleys pulling a force will have two times the mechanical advantage applied to the weight.

ANGULAR ACCELERATION Thus far, we have discussed uniform circular motion and gravitation, which is motion having a constant angular velocity at a constant speed. Recall that angular velocity omega is the change in angle theta divided by the change in time and that linear velocity is equal to the radius times the angular velocity. According to sign convention, counterclockwise angular velocity is considered positive and clockwise angular velocity is considered negative.

The angular velocity is not constant when the radius is changed, such as when a skater pulls in her arms or when a turntable is turned off. This causes something we haven’t talked about yet, which is angular acceleration or alpha. This is defined as the change in angular velocity divided by the change in time, given in units of radians per second 113

squared. If the angular velocity increases, the angular acceleration is positive; if the angular velocity decreases, then the angular acceleration is negative.

The important thing to know, for example, with tires on the ground, is the linear acceleration and linear velocity. Remember that the linear velocity is tangent to the circle at the point of interest. This means that linear acceleration, is referred to as tangential acceleration. Centripetal acceleration, which is in the direction of the center of the circle, is perpendicular to the tangential acceleration. They are independent of one another. Tangential acceleration is directly related to the angular acceleration and is related to an increase or decrease in velocity but not necessarily its direction.

If you’ll remember, linear acceleration is proportional to a change in the magnitude of the velocity; with circular motion, linear velocity equals the radius times the angular velocity. What you need to know is that tangential acceleration is equal to the radius times the angular acceleration and this means that angular acceleration is the tangential acceleration divided by the radius.

ROTATIONAL MOTION Let’s look at the kinematics of rotational motion. This involves finding the relationships between the angular velocity, angular acceleration, and time. Remember that, in linear motion, the velocity is equal to the initial velocity plus the acceleration multiplied by time. If the linear velocity is equal to the angular velocity times the radius and the tangential acceleration is equal to the radius times the angular acceleration. This means that angular velocity equals the initial angular velocity plus the angular acceleration multiplied by the time. The radius cancels out of the equation. Figure 51 describes these equations:

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Figure 51.

The kinematics for rotational motion is very similar to translational kinematics we saw in chapter one and chapter 2. The difference is that, instead of x for distance, you use the angle theta, which is the number of radians traversed, and instead of average velocity, you use the average angular velocity. The angular velocity squared becomes the initial velocity squared plus two times the angular acceleration times the angle theta.

The values you are looking for are described in this picture of a fishing reel, figure 52:

Figure 52.

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With the understanding of the kinematics of rotational motion, this leads to the dynamics of angular motion, which includes the forces involved in this type of motion.

An example of this is the forces applied to a door in opening it. If the door is opened too close to the hinge, the door will not open easily and will open more slowly. The further out the force applied, the greater is the angular acceleration, which is inversely proportional to the mass. What this means is that the greater the mass of the door or the wheel being spun, the smaller is the angular acceleration.

The actual relationship is force equals mass times the radius times the angular acceleration, which is not much different from force equals mass times acceleration, which is something you already know from linear dynamic equations. Remember that torque is the turning effectiveness of a given force. The applied force will be perpendicular to the radius, in which force equals mass times the radius times the angular acceleration. Torque then equals mass times the square of the radius times the angular acceleration.

Figure 53 describes this relationship:

Figure 53.

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The term mass times radius squared is called the moment of inertia or rotational inertia.

If there is more than one particle involved, the moment of inertia is equal to the sum of the inertias of all the particles. Inertia is similar to mass in translational motion. The units for the moment of inertia are kilogram-meters squared.

There are complex formulations for inertia that depend on whether or not the object is a cylinder or a hoop. The inertia for a hoop is the total mass times the radius of the hoop squared. This means is that the net torque is equal to the moment of inertia times the angular acceleration. This means that the ability to accelerate a mass is related to its mass and the square of the distance from the center of gravity. In a solid disc, the inertia is half the mass times the radius squared.

In looking at torque and angular acceleration as it applies to inertia, you need to make the following relationship, which is “the angular acceleration is equal to the total torque divided by the moment of inertia”. This would mean that the heavier the object and the square of its distance from the center are inversely proportional to the degree of the acceleration of the object when a force is applied. In this case, the force applied would be the torque.

In such cases, linear force is related to torque and inertia is related to mass but you need to know that these things are not the same. Torque depends on three factors: the direction of the force, the magnitude of the force, and the point of application. Inertia is related to the mass and the square of the radius. In both cases, the radius plays a big role in the determination of the angular acceleration.

WORK OF ROTATION Work must be done in order to rotate different objects, such as a merry-go-round and grindstones. The net work would be the net force times the arc length traveled of the disc. Substituting the various parameters, you get the net work equaling the net torque multiplied by the angle in radians. Figure 54 shows the work of rotating a disc:

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Figure 54.

Remember that work is analogous to energy, and this is no different in rotational work.

The kinetic energy of rotation equals half of the inertia multiplied by the angular velocity squared. This means that energy can be sorted in a flywheel and used to create kinetic energy that can move things. The rotational kinetic energy of a disc or other spinning object is related to the work done by the torque applied to the object.

Another interesting question to ask yourself in understanding rotational energies is why do things roll downhill at different rates, even if they have the same mass? At the beginning of the project, the potential energy due to gravity is the starting point of the object’s energy. As it rolls downhill, all of its potential energy is converted into kinetic energy. The kinetic energy is partly translational (related to its linear motion) and rotational (or related to its rotational motion). Remember that energy is conserved.

The more energy put into rotation, the less energy goes into translation or linear motion.

This means that the greater the rotational energy, the slower the item will roll.

A can with no liquid in it will slide down a hill and won’t roll much so it slides faster. A can with thick liquid will have more rotation and will have lesser speed of translation or a lesser speed. The can with no liquid has the same mass but has less inertia than a can

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with thick liquid in it. As the rotational kinetic energy is directly related to its inertia, the can with the thick liquid in it will roll more and therefore will move slower.

What you need to take away from this is that rotational kinetic energy is essentially analogous to translational kinetic energy. Both are related to mass and velocity but, with rotational kinetic energy, the motion or velocity is rotational and with translational kinetic energy, the motion is linear.

ANGULAR MOMENTUM The topic of angular momentum goes to the issue of why the earth continues to spin and why a skater spins faster by pulling her arms in without adding extra torque. Angular momentum, which goes by the initial L, is equal to the inertia multiplied by the angular velocity, which is very similar to the linear momentum equation, which is P equals mass times the velocity, both of which have units of kilogram meters per second squared. A large moment of inertia will have a large angular momentum. Like linear momentum, inertia is the ability to carry rotation in a forward direction. As long as external torque is zero, the momentum is conserved.

The inertia of a sphere is two-fifths times the mass times the radius squared. The moment of inertia of the earth needs to be multiplied by its angular velocity, which is one revolution per day. The angular velocity in radians is 7.3 radians per second. The moment of inertia depends on the mass of the earth and its radius. Given these values, the angular momentum of the earth is roughly 7 times 1033 kilogram-meters squared per second. This is a lot of angular momentum.

When rotating a bicycle wheel, the net torque involved is the change in angular momentum over the change in time. The greater the net torque (twisting force) the more rapid is the increase in angular momentum. This is completely analogous to the relationship between force and linear momentum.

Similar to linear momentum, angular momentum is conserved as long as the net external torque equals zero. This is why the earth keeps spinning. The net torque being zero means that the change in angular momentum is zero. It would take a large torque over a long period of time to slow the earth. The only thing that affects the rotation of 119

the earth is tidal friction. This is small enough that it would take tens of millions of years before there will be a change in the Earth’s spin. This is the only external torque applied to the earth’s spin.

A skater had increased angular velocity when she pulls her arms in because she changes her moment of inertia. The moment of inertia is related to the radius squared and the angular momentum equals the inertia times the angular velocity. Because angular momentum is preserved, the decrease in moment of inertia will greatly increase the velocity. The work done in pulling her arms in will increase the rotational kinetic energy.

COLLISIONS OF ROTATING OBJECTS When certain objects collide, we have been assuming that there is no rotational spin, while in reality, things like billiard balls and bowling pins will spin. Baseball pitchers throw curve balls by putting a spin on the ball and tennis players can put a “top spin” on a tennis ball.

A simple collision involves hitting an object against a motionless stick that is attached as a lever on its far side. If the object sticks to the lever, both objects will spin about the pivot point. Angular momentum will be conserved but not kinetic energy because the two objects result in an inelastic collision. There is no torque observed at the pivot point because its anchored to a frictionless surface. If the object is seen as a point mass, the moment of inertia will be the mass times the radius squared. The angular momentum will be the inertia times the angular velocity so, if you know the value of the radius and the mass of the object striking the lever, you can determine the inertia.

Figure 55 shows the equations involved:

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Figure 55.

The example in figure 55 assumes that the mass of the lever is negligible but it can be done by adding the mass of the lever and the object that strikes it. The kinetic energy before the collision is the incoming object’s translational kinetic energy. Because this is an inelastic situation, the translational kinetic energy gets completely translated into rotational kinetic energy. The incoming kinetic energy is one-half mass times the square of the velocity. The final kinetic energy will be one-half the inertia times the square of the angular velocity.

In looking at the picture in figure 55, you can imagine that the force applied to the tip of the lever will be all translated into rotational energy and forward force on the nail itself, while force applied to the nail or pivot point will have no rotational energy and the force will be reflected back on the object. Somewhere in between is a point that results in no net force on the nail itself. This is called the percussion point.

The percussion point is the same thing as the “sweet spot” in a tennis racquet or baseball bat. This is when there is no net force on your hand when you hit the ball. Hitting the sweet spot will result in less tennis elbow or force on your batting arm. This is described in figure 56:

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Figure 56.

While rotational kinetic energy is a scalar and not a vector, rotational angular momentum is a vector. It is affected by the torque on the system. The angular momentum is defined by the right-and rule. Both the angular momentum and the angular velocity are vectors. The direction of the angular momentum will be the same as the direction of the angular velocity. Both of these will point along the axis of rotation. If you stick out your right hand and curve your fingers in the direction of rotation, the angle of the angular velocity and momentum will be in the direction of your thumb. This is seen in figure 57:

Figure 57.

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Gyroscopes will rotate around a vertical axis and the torque will be horizontal and perpendicular to its angular momentum. If the gyroscope is not spinning, it will acquire angular momentum in the direction of the torque so that it will tip over. Figure 58 shows a top that stays upright as long as it is spinning.

Figure 58.

The Earth is itself a giant gyroscope. Its angular momentum is along its axis, pointing in the direction of the North Star.

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KEY TAKEAWAYS •

Torque is a force that causes a twisting motion or that leads to the twisting of an object on an axis.

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Torque is directly proportional to the force applied and the distance from the pivot point.

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Torque can be clockwise or counterclockwise when it comes to its direction.

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Things like angular velocity and angular momentum are equivalent to the linear velocity and momentum.

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The moment of inertia varies with the object but is related to the mass and inversely proportional to the radius squared.

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Angular momentum is always conserved in any collision equations.

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Kinetic energy can be rotational or translational so that rotational kinetic energy will take away from translational kinetic energy of an object that does both rotation and linear displacement.

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The direction of the angular momentum and velocity will be upward if the object is spinning counterclockwise.

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QUIZ 1. What is one thing that defines a body that is not in equilibrium? a. A body having acceleration b. A body having velocity c. A body having mass d. A body having potential energy Answer: a. Any body having acceleration is not in equilibrium; however, a body having velocity, mass, and potential energy may still be in equilibrium.

2. What least affects the torque on an object? a. The force applied b. The direction of the force c. The starting angle of the object d. The radius from the axis at the center of the object Answer: c. Torque will depend on the radius from the axis, on the force magnitude and on the direction of the force but not necessarily on the starting angle of the object. Torque involves force that causes rotation of an object on an axis.

3. The pivot point in a diagram is to the far left and the force is pushing upward on the right. What is the direction of torque? a. It is tangential to the direction of rotation. b. It is toward the center of the axis. c. It is counterclockwise. d. It is clockwise.

Answer: c. The only two choices for torque are clockwise and counterclockwise. In the case described, the torque applied is going to be counterclockwise.

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4. How is the direction of torque defined conventionally? a. Any upward direction for torque is going to be positive. b. Any outward direction for torque is going to be positive. c. Any clockwise direction for torque is going to be positive. d. Any counterclockwise direction for torque is going to be positive.

Answer: d. By convention, torque can be clockwise or counterclockwise. By definition, counterclockwise torque is said to be positive, while clockwise direction for torque is said to be negative.

5. What is considered the advantage of a simple machine? a. It will decrease the input force necessary to do work. b. It will increase the input force necessary to do work. c. It will do more work than the energy put into it. d. It will do equalize the force output and the force input.

Answer: a. The major advantage of a simple machine is that it will decrease the necessary input force required to do the work of the machine. It cannot do more work than the energy that is put into it because energy is still conserved.

6. The mechanical advantage of a simple machine can be stated in what way? a. It is the difference between the force input and the force output. b. It is the force input divided by the force output. c. It is the force output divided by the force input. d. It is the energy input minus the work output.

Answer: c. This mechanical advantage is the ratio between the force output and the force input. Higher mechanical advantages happen when the force output is much greater than the force input because of the machine’s ability to take advantage of torque, for example.

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7. What is the relationship between the angular acceleration of a rotating system and its tangential acceleration? a. These two terms are vectors in the same direction as each other and equal in magnitude. b. The tangential acceleration is opposite in direction as angular acceleration but equal in magnitude. c. The tangential acceleration is perpendicular to angular acceleration and is independent in magnitude. d. Tangential acceleration does not have a consistent relationship to the angular acceleration.

Answer: c. The tangential acceleration is perpendicular to angular acceleration and is independent in magnitude.

8. What is the relationship between tangential acceleration, angular acceleration and the radius of the circle being rotated around? a. The tangential acceleration is directly proportional to the angular acceleration and the radius. b. The tangential acceleration is inversely proportional to the angular acceleration and the radius. c. The tangential acceleration is inversely proportional to the angular acceleration and directly proportional to the radius. d. The tangential acceleration is directly proportional to the angular acceleration and inversely proportional to the radius.

Answer: a. The tangential acceleration is directly proportional to the angular acceleration and directly proportional to the radius. The larger the radius and the larger the angular acceleration, the larger the tangential acceleration.

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9. What is true of the velocity of 2 objects of the same mass but of different inertias rolling down a hill? a. As long as they have the same mass, they will roll down the hill at the same rate. b. The object with a reduced inertia will roll slower down the hill than the object with the greater inertia. c. The object with reduced inertia will spin slower so it travels slower down the hill. d. The object with the reduced inertia will have a greater translational linear velocity down the hill.

Answer: d. High inertias mean that more of the object is involved in rotating or spinning so the energy left over for translational travel is decreased—it will roll down the hill more slowly. Rotational kinetic energy plus translational kinetic energy equals total kinetic energy.

10. If momentum in the linear sense is mass times the velocity, what is the angular momentum of a rotating object? a. Mass times its angular velocity b. Inertia times its angular velocity c. Mass times its radius times its angular velocity d. Inertia times its linear velocity Answer: b. Rather than mass, the inertia is used to calculate angular momentum and rather than linear velocity, the angular velocity is used in the equation. Otherwise, the units are the same for both and these are completely analogous to one another.

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CHAPTER 8: FLUID STATICS AND DYNAMICS This chapter gets into the topics of fluid statics and fluid dynamics. The liquid state represents a state of matter in which there is some cohesion of the molecules that is different from that of solids and gases. There are characteristics of fluids, such as density, pressure, and other factors that will be explained as part of the chapter. In addition, there are aspects of this state of matter that specifically touch on the dynamics or flow properties of liquids in physics and biology. The flow of fluid can be relatively laminar or turbulent, depending on a variety of factors, including the viscosity of a particular liquid, which will be discussed as part of this chapter.

FLUID STATICS This represents the first time we have discussed fluids or objects in a fluid state. So, what is the difference between a solid, a liquid, and a gas? In a solid, atoms are in close contact with one another and there are vibrations within the atoms but no changes with respect to the positions of the atoms with one another. Solids will resist stresses placed upon them with little deformation when pressed on. There is a lattice structure with many solids that keep a fixed distance between the different atoms.

Liquids will deform easily when stressed and cannot spring back to their original configuration when a force is placed upon them and subsequently removed.

Macroscopically, liquids will flow with molecules somewhat adherent to each other so that there is mutual attraction between the molecules. There is some resistance to compression because of the closely-packed molecules.

Gases have atoms that are separate from one another with little attraction between them. Atoms can, however, collide with one another. Gases will flow and are relatively simple to compress because of the space between the molecules. They will escape when placed in an open container. Both gases and liquids are fluids but will behave differently under certain circumstances.

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DENSITY While there can be two things that have the same mass, such as cotton balls and bricks, they do not have the same density. Density applies to solids, liquids, and gases. Density is defined as the Greek letter rho and is the mass of something divided by its volume.

The volume occupied by something is greater when there is a low density. The SI units for density are kilograms per meter cubed. Water has a density of 1000 kg per meter cubed or 1 gram per centimeter cubed.

PRESSURE Pressure is defined as the force divided by the area over which the force is applied (perpendicular to the area). It is equal to force divided by square meters. Rather than having pressure in Newtons per meter squared, the unit for pressure is the Pascal, which is equal to one Newton per meter squared. Things like tire pressure are defined as pounds per square inch or psi and, in blood pressure, mm Hg or millimeters of mercury are also used.

In working with pressure situations, you can also determine the force by looking at the pressure multiplied by the square area it acts on. With a tire or cylinder, for example, the force is always going to be perpendicular to the square area of the surface it acts on.

Pressure is not a vector but is a scalar quantity. It is exerted on all surfaces, such as all aspects of a container and on things inside the container.

There are changes in pressure related to the distance from the earth or to the pressure under water in the ocean. These changes involve things like having your ears pop in an airplane flight or having ear pain when deep diving. At the earth’s surface, the air pressure is a result of the weight of air that exists in the atmosphere above you.

Climbing a mountain will decrease the air pressure and diving deeply increases the water pressure above you. The differences in water pressure when diving will be more significant because of the greater density of water versus air. With water in a container, its weight will be mass times the force of gravity and volume will be the area at the top multiplied by its height. The bottom supports all the weight of the water. In such cases,

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the pressure equals the mass times the force of gravity times the area supporting it (which would be the bottom area of a cylinder holding the liquid). The mass would be the density times the volume. Figure 59 shows the example of water in a cylinder:

Figure 59.

The pressure due to the weight of any fluid is the average density at any depth below its surface. Rearranging the above equations, you get pressure equals the density times the force of gravity times the height.

The atmospheric pressure is another example of pressure due to the weight of a fluid, which is air in this case. This will vary because of the earth’s rotation, which creates highs and lows in weather. The average pressure at sea level is considered to be one atmosphere. Calculating the pressure in kilopascals, you get 101 kilopascals at sea level.

This means that 1.01 x 105 Newtons of pressure are exerted per square meter on the earth. Meteorologists use torr or mm Hg, in which one atmosphere is roughly equal to 760 mm Hg or 760 torr.

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PASCAL’S PRINCIPLE Pressure can be defined as force per unit area so pushing on the fluid will increase the pressure as is done when the heart pushes on blood in the heart during contraction.

This doesn’t work as well in an open system because fluid will flow away from the area of pressure.

When pushing on a fluid that is enclosed, the pressure is transmitted without diminishment. This is referred to as Pascal’s principle. What it means is that the pressure is not diminished in the entirety of the fluid’s volume. What it also means is that total pressures from different sources of fluid in the same container are additive.

This can be applied in hydraulic systems, such as those that operate automobile brakes.

A simple hydraulic system involves a piston in a cylinder. Force is applied to a small area that gets transmitted to a larger surface area with the pressure unchanged. This is described in figure 60:

Figure 60.

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There is a relationship between forces in a hydraulic system. In the figure, the two pistons are at the same height. The pressure times the area on the left can be described as force divided by surface area 1 (A1). The pressure is the same throughout so that, on the right side, the same pressure is equal to force divided by the surface area on the left or A2. If the force on the left is 1 Newton and the right side has a surface area of 5 times that, the force on the right will be 5 Newtons. What this means is that the ratio of force to surface area will be the same on the left as on the right.

With automobile brakes, a lever system is used to press brakes and exert a force on a small piston. The pressure is sent to brakes with a larger surface area, resulting in a much larger force applied to brake pads, which stop the vehicle. Because energy cannot be added or destroyed, only so much work can go into this type of system so, with power brakes, pumps add energy to the system in order to help them work better.

Gauges that measure the pressure of the blood and tire pressure are set not to an actual zero pressure but to read zero when the pressure is equal to atmospheric pressure. This brings up the issue of “absolute pressure”, which is the atmospheric pressure plus the pressure reading on the gauge. It is impossible to have a truly negative absolute pressure.

There are various ways to measure pressure. An aneroid gauge uses a flexible bellows connected to a mechanical indicator that measures the pressure. A manometer is a simple U-shaped tube, in which one end is open to the air and the other connected to a sealed container. Air pressure pushes down on each side equally so the effect cancels out. If both sides of the U-shape are open, the level will be equal on both sides. If there is one side with greater pressure when compared to the open side, there will be a difference in the height of the columns, which reflects the pressure differences between the two sides.

A barometer is a device that measures atmospheric pressure. This involves a nearly pure vacuum above a column of mercury. Remember that the atmospheric pressure will equal the height, density, and gravity force multiplied together. It measures the atmospheric pressure as well as altitude because it will decrease with a higher altitude.

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For these devices and for blood pressure machines, the units are often millimeters of mercury rather than Pascals.

ARCHIMEDES PRINCIPLE This topic gets into the idea that some things are buoyant and others are not. Buoyancy involves things being able to float in water as well as things floating in air, such as helium balloons. If the buoyant force is greater than the object’s weight, the object will rise; likewise, if the weight is greater than its buoyancy, the object will sink. The buoyant force is the net upward force on any object within a fluid. This will be the difference between the upward force and the downward force.

According to Archimedes principle, the buoyant force on an object equals the weight of the fluid it displaces. This is valid, whether the object is partially or completely submerged. Boats will float because of their shape. They will displace more water, experiencing a greater buoyant force. It turns out that you can calculate the buoyant force on an object. Figure 61 shows the buoyant force of an object according to Archimedes principle:

Figure 61.

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The density plays a great role in Archimedes’ principle. It is the average density of an object that will determine whether or not it floats. If the average density is less than that of the surrounding fluid, it will float. This is because fluid will contain more mass and more weight in that same volume. The buoyant force will be greater than the weight of the object. Dense objects—denser than the fluid—will sink. Unloaded ships are less dense so they are less submerged. The fraction submerged is the ratio of the volume of the subject divided by the volume of the total object. This is equivalent to the ratio between the volume of the fluid divided by the volume of the object.

What this means is that the volume submerged (which is the same thing as the volume of fluid displaced) is related to the density. From what we know about density, volume, and mass, the average density of the object divided by the density of the fluid will equal the fraction submerged. So, obviously, a packed ship with a greater density will have a greater submersion ratio and will be submerged in water to a greater degree.

This leads to a discussion of what’s known as specific gravity. This is the ratio of the density of an object compared to water. It has no units because it is a ratio. The calculation for specific gravity is that it equals the average density divided by the specific gravity of water. Figure 62 is an example of a hydrometer, which is a tube that has a dense substance in the bottom dropped in a fluid. It sinks to a calibrated level, depending on the density of the fluid:

Figure 62.

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According to Archimedes principle, the buoyant force impacts the object’s apparent weight. The apparent weight loss one experiences in water is equal to the weight of fluid displaced. This is why a person feels like they weigh less in water than they do in air.

SURFACE TENSION Surface tension is related to cohesive forces, which are the attractive forces between molecules of the same type. Liquid can be held within open containers because of the cohesive forces between the molecules. Adhesive forces are the forces that hold molecules of different types together. Liquid water adheres to panes of glass because of adhesive forces. Both of these are considered attractive forces.

It is the cohesive force between molecules that causes the surface of a liquid to contract to the smallest possible surface area. This is called the liquid’s surface tension. A needle can rest on the surface of water—not because it is less dense and floats—but because it is supported by the surface tension of the water. If the needle were placed on its end, it would sink because the force of its weight would be spread over a smaller surface area of the water. The restoring force is related to the surface tension of the water and results in a net upward restoring force that exceeds the weight of the object. In fact, the restoring force is equal to the surface tension, noted by the Greek letter, gamma.

The greater the cohesive force of the liquid, the greater the surface tension. The surface tension is the force per unit length of a stretched liquid membrane. The SI units for surface tension is units per meter. Soapy water can form bubbles because it has a lesser surface tension than plain water. Water forms droplets that are roughly spherical because of their inward surface tension.

The pressure inside a bubble is four times the surface tension divided by the radius of the bubble. Bigger bubbles have smaller interior pressures. This is why a soap bubble ultimately explodes. The pressure becomes less inside the bubble as it gets bigger. If a hole is placed in the bubble, the bubble would decrease in radius and the pressure will increase within the bubble.

Inside the alveoli of the lungs, the air is naturally forced out because of the shrinkage of the diameter of the alveoli in the exhalation process. Surfactant is a substance in the 136

alveoli that allows them to expand and prevents total collapse of the alveoli during exhalation. This makes surfactant in the lungs different from that of detergent, which simply lowers the surface tension of a bubble but will not prevent its total collapse.

Adhesive forces are between two molecules of different substances, such as between water and wax, which have little adhesive forces between them. The contact angle is the angle between the tangent to the liquid surface and another surface. The greater the contact angle, the less is the adhesive force between a droplet of water and the surface it’s resting on. Figure 63 shows the contact angle of a droplet with a great adhesive force and a droplet with a lesser adhesive force:

Figure 63.

This brings up the idea of capillary action. This is the tendency of a fluid to be raised inside a narrow tube, called a capillary tube. The actual effect depends on the relative strength of adhesive and cohesive forces. If the contact angle is greater than 90 percent, the liquid will be depressed; if the contact angle is less than 90 percent, the liquid will be 137

raised. The Liquid will have a meniscus, which is the curved area of the liquid in a capillary tube because of its adhesive forces and surface tension. The meniscus will be downward in mercury and upward in water. Figure 64shows the meniscus of water:

Figure 64.

The smaller the radius of the capillary tube, the greater is the height of the liquid because a smaller tube will hold less mass. In addition, the greater the fluid density, the less the liquid will rise in the capillary tube. This means that the height is inversely proportional to the density and inversely proportional to radius.

FLUID FLOW The flow rate, as defined by the letter Q, is the volume of a fluid passing through an area divided by the time. The SI units for flow rate of a fluid is meters cubed per second.

There are many other units used for flow rate used in other circumstances, such as liters per minute or liters per second. In a tube, it is the area of the tube times the average velocity of the liquid.

Flow rate is different from velocity. The greater the velocity of the water in a river, the greater the flow rate; however, the flow rate depends on the size of the river. The actual relationship between the average velocity and the flow rate is that the flow rate is the cross-sectional area times the velocity. The larger the conduit, the greater the flow rate.

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As the cross-sectional area of a conduit decreases, the speed must increase in order to have the same flow rate. The ratio of the area and velocity must be the same in order to have the flow rate be the same through a conduit. This is referred to as the equation of continuity, which applies to any incompressible fluid.

BERNOULLI’S EQUATION So, when fluid flows into a channel that is narrower, its speed increases. This means that its kinetic energy increases. One has to wonder where this kinetic energy comes from as energy is neither created nor destroyed. This goes back to the idea that net work equals one-half mass times velocity squared minus one-half mass times the initial velocity squared. This is shown in figure 65:

Figure 65.

Net work done will increase the fluid’s kinetic energy. Remember, too, that the pressure equals area times force so that the pressure will drop in a rapidly-moving fluid. This explains why shower curtains bulge inward when the shower is on and running. In the same way, cars will experience a force toward a parallel-moving truck passing beside a car. This is because the high velocity of the truck will create negative pressure in the air (which is a fluid) around it.

This leads to Bernoulli’s equation, which is the relationship between pressure and velocity in fluids. This is described in figure 66:

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Figure 66.

This basically means that the absolute pressure plus one-half times the density times the pressure times the velocity squared plus the density times the force of gravity times the height equals a constant. Figure 67 shows the Bernoulli’s equation:

Figure 67.

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This basically means that there is conservation of energy for an incompressible fluid in the absence of any friction. You need to know that there are other forms of energy that come into play including thermal energy because of fluid viscosity. It applies to static fluids and moving fluids.

In the absence of movement, there are static fluid forces indicating that pressure = the pressure at the surface of the fluid plus density times gravity times height. We already know this because we know that pressure is greater with greater height below the surface. This is the change in potential energy of a fluid.

At a constant height the kinetic energy will be conserved, so that the pressure drops at increasing speed. The difference in pressure is related to one-half times the density times the change in velocity squared. The change in pressure is proportional to the change in velocity squared.

There are many processes that pertain to Bernoulli’s principle, which uses reduced pressure in high-velocity situations in order to move things about. This is known as “entrainment”. It allows for pumps to raise water over small heights. Examples of using entrainment include atomizers that squeeze a jet of air that result in droplets of perfume being entrained. The lift of an airplane wing is an example of Bernoulli’s principle.

Wings are tilted to allow air to flow faster over the top. This reduces the pressure on top of the wing, leading to “lift” or a net upward force. Sails of ships will also apply Bernoulli’s principle. Height too is proportional to the square root of the velocity.

Pouring a fluid such as water is not difficult, and yet, pouring fluid like pancake syrup is more difficult. This is because of “fluid friction”, which is its viscosity. So far, we have been dealing with fluid that has no viscosity and therefore has “laminar flow” or nonturbulent flow. Turbulence will happen because of velocity, resulting in characteristic swirls and eddies that mix layers of fluid together.

When fluid becomes turbulent, the layers will mix and there will be velocities in different directions. Consider the movement of smoke rising. It will first rise faster in laminar flow but will rise more slowly and will swirl more easily in turbulent flow because the turbulence will increase the friction between the fluid molecules, restricting its flow. Figure 68 shows laminar flow and turbulent flow:

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Figure 68.

The lines of flow are referred to as streamlines. These will be smooth and continuous in a forward direction in laminar flow but will break up and be in all directions in turbulent flow. There will be friction between the layers of laminar flow that will restrict its flow.

Force is directly proportional to velocity but, at a certain speed, the velocity will be slowed down because of an increase in turbulence. Force is also directly proportional to the coefficient of viscosity, identified by the Greek letter eta (η).

Consider flow between two plates of a certain area. The greater the viscosity coefficient, the greater the force required. Force equals the coefficient of viscosity multiplied by the velocity and the cross-sectional area, divided by the length between two plates. The SI units for the coefficient of viscosity is Newton-seconds-meters squared. The viscosity of a fluid varies from fluid to fluid.

Fluid flow happens because of a difference in pressure. The flow will be from high pressure to low pressure. The greater the difference between two points, the greater the flow rate. It is related to the difference in pressure between two ends of a tube divided by the resistance to flow. Resistance to flow is related to many things that affect flow rate. It is greater in longer tubes and is greater with high viscosity. Turbulence is going to increase resistance, while increased diameter of a tube will decrease resistance to flow.

If viscosity is zero, the fluid is frictionless and resistance to flow will be zero. The speed of a viscous fluid will be greatest at the midstream of the flow because of the drag effects at the boundaries. Poiseuille’s law is that for resistance of a fluid. It results in the following equation for resistance of a fluid in a horizontal tube of uniform radius and length. This is outlined in figure 69:

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Figure 69.

What this means is that just by doubling the radius, the resistance will go down by a factor of 16. This has a much greater impact on resistance than on any other factor. This law has been specifically applied to blood because it explains why there needs to be a pressure difference put upon by the heart in order to overcome changes in resistance by the changes in radius put upon by things like plaques. Flow will be equal to the difference in pressure times pi times radius to the fourth divided by eight times the coefficient of friction times the length. This looks like what’s seen in figure 70:

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Figure 70.

FLOW AND TURBULENCE Sometimes turbulence can be predicted. If there is a smooth tube and low flow, the flow will be laminar. If the velocity increases, even in a smooth tube, the flow will be turbulent. At intermediate flows, the flow can change from turbulent to laminar.

Turbulent flow can be heard in checking blood pressure as the Korotkoff sounds, which are knocking sounds that affect the flow of blood at certain radii of the brachial artery.

These sounds are also seen when listening for aneurysms of the aorta or other main arteries. Heart murmurs represent turbulent flow around damaged valves.

As it turns out, the Reynolds Number is a number that can predict whether flow is going to be laminar or turbulent. For flow of a uniform diameter, the Reynolds number is the product of density, speed, and radius, and divided by the coefficient of viscosity. This means that the higher the coefficient of velocity, the lower is the Reynolds number. The Reynolds number has no units. The lower the number, the more laminar is the flow.

Numbers less than 2000 indicate that the flow is laminar. Flow will be unstable for Reynolds numbers between 2000 and 3000. Flow will be turbulent for Reynolds numbers above 3000.

VISCOUS FLUIDS AND TERMINAL VELOCITY Flow can be laminar, turbulent, or a combination of the two. There is another form of the Reynolds number defined for an object moving in a fluid, such as a person riding a bike through air. This is described as the fluid density multiplied by the velocity of the

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object and its length, divided by the fluid’s coefficient of viscosity. If this number is about one, the flow will be laminar, especially if a smooth shape exists for the object. If this number is between one and ten, there is a transition to turbulent flow. If this number is greater than ten, the flow will transition to turbulent around the object. For very large numbers, greater than 106, the flow will be turbulent.

One of the consequences of viscosity is a resistance force called the viscous drag force exerted on a moving object. This force depends on the object’s speed and will be proportional to speed at low speeds and proportional to the speed squared at high speeds (high Reynolds numbers). This explains why cyclists travel in packs because there will be decreased drag force on the secondary riders and the rider in the front will take turns with the others so that the headwind is traded off between cyclists.

A consequence of this is that the drag force will mean that, with an object falling, its acceleration will cause it to have a greater drag force as it falls, which slows its acceleration until it reaches a critical speed, referred to as its terminal velocity or terminal speed, with zero acceleration. The object will fall at the same terminal velocity until it reaches the bottom. This is the case for particles of sand falling in the ocean and for sky divers. Terminal speed will be the greatest for low-viscosity fluids and with objects that have high densities and small sizes. This means that sky divers will fall faster with hands near their side than they will with their arms spread out.

DIFFUSION THROUGH A FLUID Atoms and molecules will have a “random walk” through a fluid, resulting in diffusion due to random thermal molecular motion. Odors can even diffuse through solids like the ice in your freezer. It is a slow process that eventually has macroscopic influences.

The average distance of random molecular motion is proportional to the square root of time. In fact, the root-mean-square distance is equal to the square root of 2 times the diffusion constant for a molecule in a medium and time.

The units for the diffusion constant are meters squared per second. The larger the molecule, the smaller is the diffusion constant because molecular speed is inversely proportional to molecular mass. Diffusion constants will increase with temperature,

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decrease with molecular mass, and will be related to the kinetic energy of the molecule in the liquid. Diffusion proceeds from an area of higher concentration to an area of lower concentration. Cohesive and adhesive forces will all affect the values of the diffusion constant.

Osmosis involves the transport of water through a semipermeable membrane from a region of high concentration to an area of low concentration. It is driven by the imbalance in water concentration from one side to another. Water is more concentrated in dilute solutions and is less concentrated in concentrated solutions (the opposite of the concentration of the solute in solution).

Osmosis can create a substantial difference in pressure across a membrane. The pressure will rise until there is a back pressure that stops the osmosis, which is also called the “relative osmotic pressure”. This is the pressure reached when a solution is equal in concentration of water on both sides of the membrane and the net transfer of water is zero. This osmotic pressure is described in figure 71.

Figure 71.

On the other hand, dialysis is the transport of any other molecule through a semipermeable membrane due to differences in their relative concentration difference.

Both osmosis and dialysis can be reversed by the back pressure on one side of the membrane. Active transport involves the transport across a membrane using energy from the cell in biological systems.

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KEY TAKEAWAYS •

Density of a liquid or fluid is its mass divided by its volume.

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The concepts of pressure mainly apply to fluids that are gaseous in nature.

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Liquids will maintain their shape in an open container.

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Pistons will exert forces that act on other areas because the pressure across the fluid system will be the same throughout the fluid.

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Buoyant force equals the weight of the water it displaces, called Archimedes principle.

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Specific gravity is equal to the density of a fluid as it relates to the density of water.

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There are cohesive forces and adhesive forces that apply to a liquid and itself or another substance.

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Flow rate is the volume of a fluid traveling per a unit of time.

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Flow can be laminar or turbulent or a combination of the two.

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There is the Reynolds number, which predicts the turbulence of a fluid or an object traveling through fluid.

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Osmosis relates to the flow of water through a semi-permeable membrane.

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QUIZ 1. What are the SI units for density? a. Grams per meter cubed b. Kilograms per meter cubed c. Milliliters d. Grams per milliliter Answer: b. The actual SI units for density are kilograms per meter cubed. This is relatively impractical so often grams per milliliter or grams per centimeter cubed are often used because water has a density of one gram per cubic centimeter or one gram per milliliter.

2. What is the relationship between density, mass, and volume of a fluid? a. Density is proportional to the mass and volume. b. Density is inversely proportional to mass and inversely proportional to volume. c. Density is proportional to mass and inversely proportional to volume. d. Density is inversely proportional to mass and proportional to volume.

Answer: c. The density of something is its mass divided by its volume so it is directly proportional to the mass and inversely proportional to its volume.

3. What is the way to measure the pressure using a barometer? a. Mm water b. Kilopascals c. Atmospheres d. Mm mercury Answer: d. Both blood pressure and barometric pressure are measured in millimeters of mercury. Barometric pressure equals the atmospheric pressure in the environment; it can also measure the pressure at certain altitudes.

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4. According to Archimedes’ principle, the buoyant force on an object equals what? a. The volume of fluid it displaces b. The weight of the fluid it displaces c. The density of the object d. The square area of the object Answer: b. The buoyant force on an object equals the weight of the fluid it displaces. If the amount of displacement can be determined in water, for example, the weight of this water will be equal to the buoyant force.

5. In determining the capillary action of a liquid, what does it most depend on? a. The liquid’s adhesive forces with the capillary tube b. The liquid’s cohesive forces with itself c. The difference between the adhesive forces and the cohesive forces d. The ratio of the adhesive forces and the cohesive forces Answer: c. The difference between the adhesive forces and the cohesive forces determines the capillary action of the liquid.

6. A liquid is raised to a height because of capillary action. What is the proportionality of this height with the density of the liquid and the radius of the tube? a. The height will be inversely proportional to the density and inversely proportional to the radius. b. The height will be directly proportional to the density and directly proportional to the radius. c. The height will be inversely proportional to the density and directly proportional to the radius. d. The height will be directly proportional to the density and inversely proportional to the radius.

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Answer: a. The height depends on the surface tension but is inversely proportional to both the density of the liquid and the radius of the tube.

7. According to Bernoulli’s principle, in looking at the lift height of a static wing and its velocity, this velocity is related in what way to its height? a. Height is proportional to its velocity. b. Height is inversely proportional to its velocity. c. As height rises, this is proportional to the square of its velocity d. As height rises, this is proportional to the square root of its velocity.

Answer: c. The lift height will be proportional to the square of its velocity. This means that velocity will exert an exponential effect on the lift height of a static wing in fluid such as air.

8. What will not increase turbulence in a fluid? a. High viscosity b. Decreased flow of fluid c. Obstruction to flow d. Change in flow direction Answer: b. In fact, increased flow of fluid will increase the turbulence because it increases the friction between layers of fluid flowing and increases the friction between the fluid and its environment.

9. In an object falling through a viscous fluid, what is the terminal velocity or terminal speed? a. Its final speed before it reaches the bottom b. Its speed when the acceleration is zero c. Its speed when its mass is zero d. Its speed when the force of gravity is maximal Answer: b. This terminal speed will be when its velocity is so great that there are drag forces that reduce the acceleration to zero. It will be in effect when substances fall in water as well as when sky divers fall from a plane through the air.

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10. What can be said about the diffusion constant for a particular molecule in water? a. It will increase with temperature and increase with molecular size. b. It will decrease with temperature and decrease with molecular size. c. It will decrease with temperature and increase with molecular size. d. It will increase with temperature and decrease with molecular size.

Answer: d. The diffusion constant of a given particle will increase with temperature and decrease with molecular size so that the greater the temperature, the greater the diffusion and the lower the molecular size, the greater the diffusion.

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CHAPTER 9: TEMPERATURE AND GAS LAWS The focus of this chapter is temperature and properties of substances related to temperature, such as evaporation, humidity, and phase changes of a given substance.

This leads to a discussion of kinetics and kinetic theory as it applies to gases. The ideal gas law is covered, with some attempt to link ideal gases with real gases. As it turns out, all substances are affected by their own temperature and the temperature of their surroundings, with expansion occurring in solids, liquids, and gases to varying degrees.

This chapter combines theories and influences of both physics and physical chemistry as they apply to molecular systems and macroscopic substances.

TEMPERATURE Temperature is a measurement of the heat of something relative to another based on a scale that has been arbitrarily set. While the absolute value of temperature is arbitrary, it is certain that different substances behave according to their temperature. Physical properties of substances are reproducible based on their temperature, including their volume, electrical resistance, color, and the ability to emit infrared radiation.

The three most commonly used temperature scales are Fahrenheit, Kelvin, and Celsius.

Any temperature scale can be used based on differences between a substance’s properties and two different points along a scale. With the Celsius scale, the freezing point of water is arbitrarily set to zero degrees and the boiling point for water is arbitrarily set to 100 degrees Celsius at standard atmospheric pressure. The centigrade scale is similar to the Celsius scale but is not used as such anymore. On the Fahrenheit scale, the freezing point of water is 32 degrees and the boiling point of water is 212 degrees. One degree Celsius is 1.8 times larger or 9/5 larger than a degree Fahrenheit.

The Kelvin scale is quite similar to the Celsius scale in that one degree in Celsius is the same span of a degree in Kelvin. The big difference is that, with Kelvin, the freezing point of water is 273.15 degrees Kelvin and the boiling point of water is 373.15 degrees Kelvin. The Kelvin scale is an absolute scale, in which zero is absolute zero, in which all

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molecular motion would theoretically stop. In physics and many scientific circles, it’s the Kelvin scale that is used.

For the calculation from Fahrenheit to Celsius, you take nine-fifths of the difference between the Fahrenheit degrees and 32. The difference between Kelvin and Celsius is 273.15 degrees, with the Celsius scale just being a version of the Kelvin scale. Standard temperature is said to be 25 degrees Celsius, which is approximately “room temperature”.

You should know that absolute zero on the Kelvin scale isn’t the lowest possible temperature recorded or possible. The coldest temperature on Earth recorded has been 183 degrees Kelvin or -89 degrees Celsius. The lowest temperature recorded has been 1 x 10-10 degrees Kelvin, which is far below the level of absolute zero. The coldest place outside Earth in the known universe outside of a laboratory is 1-degree Kelvin in the Boomerang Nebula. The highest known temperature in experimental science is in the range of 1012 degrees Kelvin. In the known universe, the temperature of the interior of a neutron star is 109 degrees Kelvin.

You need to recognize that thermometers take their own temperature and not necessarily the temperature of their environment. There needs to be thermal contact between the thing being measured and the thermometer so that there is thermal equilibrium. Time must pass before this can truly happen as there needs to be transfer of heat. If two systems are in thermal equilibrium and a third system is in equilibrium with a third system, all three systems are in equilibrium. The Zeroth law of thermodynamics indicates that this is true.

KINETIC THEORY Kinetic theory is related to all phases, gaseous, liquid, and solid; however, it is most applied to gases. This is because the pressure of gases is highly related to the kinetics of different gas molecules at various temperatures. The assumption with gases is that atoms and molecules are in continuous random motion dependent on temperature.

Pressure of gases can be explained by kinetic theory. In a chamber of N numbers of gas molecules with a single molecule mass of m in a certain volume V will collide with one 153

another and with the walls of the container at a certain speed. Force can be described as occurring in the x and y and z directions. The force in one direction x will equal the change in pressure over the change in time or the Mass times the velocity in the x direction squared divided by the length L. There are three assumptions when describing kinetic theory: 1. The gas has particles separated by large spaces and move in random directions.

2. The molecules undergo perfectly elastic conditions without loss of energy.

3. The transfer of kinetic energy between molecules is “heat”.

In such a case, force is exerted on a square area L-squared. Force is described in figure 72:

Figure 72.

Gas molecules will have specific molecular speeds, referred to as the MaxwellBoltzmann distribution. What this means is that there will be a variation in speeds, in which some speeds are fast and some are slow. There is a peak probability of speed of

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particles that is related to the average molecular speed. Figure 73 describes the Maxwell-Boltzmann distribution of molecular speed at different temperatures.

Figure 73.

What this means is that the velocity of particles of gases will shift to higher speeds at higher temperatures and will be broadened by higher temperatures. As you can see, this is a probability distribution for ideal gases close to thermodynamic equilibrium. Heavier molecules will have a wider distribution (greater range of molecular speeds) but will be slower than lighter molecules. It applies to ideal gases.

Temperature is directly proportional to the average translational kinetic energy of molecules in an ideal gas. Heat applied to the system will indicate more rapid temperature of a gas molecule. This fact is crucial to the development of kinetic theory of gases. The gas in a container will exert an outward pressure on the walls of the container in elastic collisions with the sides of the container and with each other.

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With these assumptions, you can determine that pressure times the volume equals the number of molecules multiplied by the temperature and by K, which is the Boltzmann constant. Using the equation as described in figure 72 and this equation, you get the equations described in figure 74:

Figure 74.

The average kinetic energy of a molecule is going to be one-half times the mass times the velocity squared. This relates then to the equation that the kinetic energy is equal to 1.5 times Boltzmann constant times the Temperature. This average kinetic energy of a molecule is referred to as the thermal energy. There are two components to the kinetic energy of a thermodynamic system, which are the kinetic energy plus the potential kinetic energy.

The kinetic energy is due to the motion of particles, such as rotation, vibration, and translational movements. In ideal gases, there is no inter-particle interaction. This means that there is no potential energy and only kinetic energy exists. With single-atom gases, such as the noble gases, there is just one atom in the gas. This is referred to as a monatomic gas. The kinetic energy is just the translational energy as there is no real rotational energy and no vibrational energy. This involves just three degrees of freedom—x axis, y axis, and z axis translation only.

Diatomic gases have five degrees of freedom: the three axes and two rotational degrees of freedom. With this, the internal energy U is equal to five-halves times N (the number of molecules) times Boltzmann constant times the temperature. With monatomic gases, 156

the internal energy is just three-halves times N times Boltzmann constant times the temperature. What this means is that the degrees of freedom matter when it comes to the internal energy of a gas. Each degree of freedom contributes one-half times the Boltzmann constant times Temperature.

So, what is Boltzmann constant? Without getting into the specifics of how this is calculated, you need to know that this is the gas constant R divided by Avogadro’s number (which is the number of atoms in a mole of a substance). Boltzmann constant equals 1.38 × 10-23 kilogram-meters squared per second squared per degree Kelvin or Joules per degree Kelvin.

The average kinetic energy of a molecule is independent of the type of molecule. The average translational kinetic energy depends only on the absolute temperature (multiplied by a constant). This kinetic energy is very small when you think of its macroscopic energies so that one does not feel the collision of air molecules on the skin, just the total macroscopic pressure.

The root-mean-square velocity of a molecule is the square root of the sums of the velocities squared of the velocities in all three axes. The mean free path, which is the distance a molecule can move on average between collisions of molecules, is very small.

The faster the root mean velocity of air molecules, the faster that sound vibrations can be transferred through the air. The speed of sound increases with the temperature and is greater in gases that have smaller molecules. This is why people sound differently when inhaling helium.

THERMAL EXPANSION OF LIQUIDS AND SOLIDS Thermal expansion of gases is obvious but you need to know that solids and liquids expand with greater temperature. Raising the temperature of the gas in a balloon will increase its buoyant force or upward force. Alcohol will expand in an alcohol thermometer. Bridges and railroad tracks will have expansion joints that allow them to expand freely in higher temperatures and contract freely in lower temperatures.

Thermal expansion depends on the substance as well as the actual temperature.

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An increase in temperature means there is an increase in kinetic energy of the atoms.

Solids are packed together but their molecules will push against one another, resulting in a slightly greater distance between neighboring molecules, adding to the size of the whole body. It will increase the solid’s size by a certain fraction in each dimension.

One can conceive of linear thermal expansion, which is thermal expansion in one direction or dimension. In such cases, the change in linear length is equal to the length multiplied by the change in temperature multiplied by the coefficient of linear expansion, which will itself vary with temperature somewhat and will vary with the substance. Nevertheless, the coefficient of linear expansion is accurate for small changes in temperature and, over large changes in temperature, an average value of this coefficient can be used.

While it is nice to look at the linear expansion of a bridge, for example, you need to know that solids and liquids expand in all dimensions. The areas and volumes will increase with temperature and holes in a solid will get larger with temperature because molecules around the hole will get further apart. In two dimensions, the change in area will be the original area, twice the coefficient of linear expansion, and the change in temperature.

As you can imagine, there is thermal expansion in all three dimensions. In such cases, the change in volume equals the coefficient of volume expansion times the original volume times the change in temperature. This introduces the coefficient of volume expansion, which is equal to approximately three times the coefficient of linear expansion. This value will vary with the type of substance.

Objects will contract with decreasing temperature but you need to know that this is not completely the case with water. Water actually expands with increasing temperature as long as the temperature is greater than 4 degrees Celsius. Between 0 and 4 degrees Celsius, water expands with decreasing temperature so that is why ice floats on water.

Ice is less dense than liquid water. Water is densest at 4 degrees Celsius. As a pond cools, the colder water will drop and warmer water on top will cool until there is a uniform temperature of 4 degrees. This leads to freezing of the top layers of water that stays at the top.

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THERMAL STRESS Thermal stress is created on an object due to thermal expansion or contraction.

Thermal stress can be destructive, such as when expanding gasoline ruptures a gas tank.

This can be helpful in manufacturing, where a cylinder can be placed over another cylinder in hot temperatures. As they cool, these two cylinders will become tighter together as the “holes” in the cylinder contract and get smaller at lower temperatures.

Thermal stress explains the weathering of pavement with changes in temperatures as well as the expansion of ice when it freezes.

IDEAL GAS LAW Gases are compressible fluids, having the largest coefficients of volume expansion.

These large coefficients mean that they expand the most with regard to temperature changes. Most gases have the same coefficient of volume expansion. There is a wide separation of atoms and molecules in gases so that the forces between molecules can be negligible. Motion is considered to be quite fast so no two molecules are in contact with one another for very long. What this means is that the properties of a gas depend more on the number of atoms per unit volume and temperature than on the actual molecular type.

According to the ideal gas law, Pressure times Volume equals the number of molecules times the temperature times the Boltzmann constant. If you’ll remember, the Boltzmann constant is 1.38 x 10-23 Joules per degree Kelvin. Boltzmann constant in chemistry is referred to as the ideal gas constant. Figure 75 shows the ideal gas law:

Figure 75.

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The ideal gas law is based on two other laws: Charles’ law and Boyle’s law. According to Charles’ law, the volume of a gas is proportional to the temperature at a fixed pressure.

According to Boyle’s law, pressure times volume is a constant at fixed temperature. With Boyle’s law, as the volume contracts, the pressure will increase. With Charles’ law, it states that with increasing temperature of a gas, the volume will increase as long as the pressure stays the same.

In chemistry, there is the concept of a “mole”, which is defined as the number of atoms or molecules found in 12 grams of carbon-12. The number of atoms in a mole is known as Avogadro’s number. A mole of gas, at the same temperature and pressure, will contain the same number of molecules, independent of the molecular content of the gas or the type of gas. One mole of any atom or molecule equals 6.02 x 1023 particles. This is true for non-gases as well. For gases, this value is approximately 22.4 liters per mole for any gas at standard temperature and pressure.

It is sometimes more common to use the ideal gas law as it applies to the number of moles of a substance rather than on the number of atoms and molecules. To do this, the equation needs to integrate Avogadro’s number. This leads to the rearrangement of the equation to pressure times volume equals the number of moles of the gas, the gas constant, and the temperature multiplied together. The universal gas constant R is defined as 8.31 Joules per mole per degree Kelvin, 1.99 calories per mole per degree Kelvin, or 0.082 liter-atmospheres per mole per degree Kelvin.

The ideal gas law is closely related to energy. The numbers on the right side of the ideal gas law equation, which is number of atoms multiplied by Boltzmann constant, and multiplied by the temperature, translates into the amount of translational kinetic energy of a certain number of atoms at an absolute temperature. This number is in Joules as is the pressure times the volume of a gas. In dealing with these types of equations, you need to keep track of the SI units. Pressure is in Pascals, meters-cubed is volume, and degrees Kelvin is the temperature.

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PHASE CHANGES Within the range of temperatures, a substance can be in gaseous, liquid, or solid form.

Gases act very much like ideal gases at high temperatures but fail at lower temperatures.

Molecules will interact with one another and condensation occurs. There will be a significant decrease in volume as the substance becomes liquid. At even lower temperatures, the substance becomes solid. The volume becomes smaller but does not reach zero due to the mass of the molecules. Gases will become liquid at high pressures as well.

Pressure-volume diagrams will plot the behavior of a substance based on these parameters. When a substance is an ideal gas, there is a particular relationship between pressure and volume. The volume of a gas will decrease as the pressure increases. At fixed temperatures, this leads to a series of hyperbolas called isotherms. At some point, the temperature becomes too low and the gas no longer behaves ideally. There is a critical point above which a liquid cannot exist. The critical pressure is the minimum pressure needed for liquid to exist at a critical temperature. Figure 76 shows these values:

Figure 76.

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In the figure, each line represents an isotherm of equal temperature along the line. At lower temperatures, a critical point is reached, where liquid first becomes possible.

With regard to the terminology referring to a vapor, a vapor is a gaseous phase that exists at a temperature below the boiling temperature.

A more easily understood graph on phase changes is the phase diagram. This plots the pressure of a substance versus its temperature. If you know the pressure and the temperature, you can determine what phase the substance is in. The boiling point of water is, for example, 100 degrees at one atmosphere of pressure. However, as the pressure rises (as in a pressure cooker situation), water can exist as a liquid in order to cook food at much higher temperatures. The curve for water shown in figure 77 , reaches a critical point above which liquid cannot possibly exist because the temperature is too high and the kinetic energy of the system is too great.

Figure 77.

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At sufficiently low pressures, there is no liquid phase but only a gaseous and solid phase.

This phase change from solid to gas is referred to as sublimation. Note that, at a single point, called the “triple point”, all three phases will exist in equilibrium. This occurs at 0.01 degree Celsius.

Vapor pressure is the pressure at which a gas coexists with its solid or liquid phase. This depends on the substance and its temperature so that an increase in temperature increases the vapor pressure. Partial pressure is the pressure that a gas would create if it occupied the total available volume. According to Dalton’s law of partial pressures, the total pressure is the sum of partial pressures of the different gases in a specific volume. Different gases have pressures that add up to the total pressure of a gaseous system.

EVAPORATION AND BOILING This topic first gets into the idea of relative humidity. This is the amount of water vapor in the air compared to the maximum possible. At maximum saturation, the relative humidity is 100 percent and there is no possibility of evaporation. Temperature will affect the relative humidity on any given day. At the dew point, the relative humidity is 100 percent, resulting in fog due to condensation of water that stays in suspension. As the temperature raises, there is greater evaporation and things will dry out.

The vapor pressures will increase with temperature because of higher molecular speeds at higher temperatures. At 100 percent humidity, the partial pressure of water is equal to the vapor pressure so no more water can become gaseous. Evaporation will take place at any humidity less than 100 percent. If the partial pressure is less than the vapor pressure, evaporation will take place and if the partial pressure is greater than the vapor pressure, condensation will take place.

Decreased atmospheric pressure will result in a decreased partial pressure of water and lower humidity. This involves a greater evaporation of water from food as would be the case with freeze-drying. Food is subjected to lower pressure in a vacuum as well as lower temperature, which dries out and freezes the food in ways that cannot be done if there wasn’t a vacuum associated with the cooling process.

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KEY TAKEAWAYS •

The temperature of something is based on an arbitrary scale; Kelvin is the scale most commonly used in science, although Celsius has an equivalence when it comes to what a degree means.

•

Gases behave in ideal ways when the temperature is increased to a certain level.

•

The ideal gas law states that pressure multiplied by volume equals the molar quantity of a molecule multiplied by the ideal gas constant and the temperature.

•

Gases do not behave ideally at lower temperatures.

•

Phase diagrams involve pressure versus temperature and the different phases of a substance at different pressures and temperatures.

•

The triple point in a phase diagram is an equilibrium point between solid, liquid, and gaseous phases of a substance.

•

Relative humidity is relative to 100 percent humidity, at which the vapor pressure of water in air reaches its partial pressure. This humidity level is called the dew point.

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QUIZ 1. Which scale system is not traditionally used in measurement in physics? a. Fahrenheit b. Centigrade c. Celsius d. Kelvin Answer: b. The centigrade scale is similar to the Celsius scale but is no longer in use in science. The Celsius scale is named after the scientist who invented the scale.

2. According to the Celsius scale, what is the freezing point of water at standard pressure? a. 273 degrees b. 32 degrees c. -32 degrees d. 0 degrees Answer: d. The arbitrary setting for the freezing point of water at standard pressure using the Celsius scale is zero degrees. This goes along with the boiling point of water in the Celsius scale, which is 100 degrees Celsius.

3. What is the approximate value of absolute zero in terms of degrees Celsius? a. 0 degrees b. -100 degrees c. -732 degrees d. -273 degrees Answer: d. Absolute zero in Kelvin is actually -273.15 degrees Celsius, which is the time at which all molecular motion in gases would theoretically cease. It does not indicate the lowest temperature possible.

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4. What is the zeroth law of thermodynamics about? a. Thermal equilibrium b. Absolute temperature c. Rate of thermal transfer d. Kinetics of gases Answer: a. The zeroth law of thermodynamics indicates that when two objects are in equilibrium and one is in equilibrium with a third object, all of the systems are in equilibrium with one another.

5. What does the internal kinetic energy of a gas molecule least related to? a. The degrees of freedom of the molecule b. The temperature c. Boltzmann constant d. The potential energy Answer: d. Because the assumption is that inter-molecular collisions are negligible, potential energy is not considered and only the kinetic energy is considered. This internal kinetic energy is equal to one-half the Boltzmann constant times the temperature per degree of freedom afforded by the molecule.

6. How many degrees of freedom are seen in a diatomic molecule? a. 1 b. 3 c. 5 d. 7 Answer: c. A diatomic molecule will have five degrees of freedom. This will equal three degrees of freedom for each of the three axes plus two degrees of rotational freedom that aren’t seen in a monoatomic molecule.

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7. What can be said about the coefficient of volume expansion versus the coefficient of linear expansion when it comes to substance exposed to different temperatures? a. The coefficient of volume expansion is roughly twice the coefficient of linear expansion b. The coefficient of volume expansion is the square of the coefficient of linear expansion c. The coefficient of volume expansion is the cube of the coefficient of linear expansion d. The coefficient of volume expansion is roughly three times the coefficient of linear expansion Answer: d. Because volume occurs in three linear components, the coefficient of volume expansion is roughly three times the coefficient of linear expansion.

8. According to Boyle’s law of gases, what is constant when the temperature is held constant? a. Pressure multiplied by number of molecules b. Pressure multiplied by volume c. Number of molecules divided by volume d. Volume of a gas Answer: b. Boyle’s law is a restatement of the ideal gas law which indicates that pressure multiplied by volume is equal to the number of molecules multiplied by a constant multiplied by the temperature in Kelvin. Part of this makes the assumption that, at a given temperature, the pressure multiplied by the volume will be a constant.

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9. The traditional phase diagram for a substance involves what two things plotted together? a. Pressure versus volume b. Temperature versus volume c. Pressure versus temperature d. Volume versus temperature Answer: c. The phase diagram for a substance involves the plotting of pressure versus temperature.

10. In a phase diagram, what happens above the critical point? a. A solid becomes a liquid b. A liquid cannot exist at any pressure. c. A solid can become gaseous. d. Gases condense into liquid.

Answer: b. Above the critical point on the pressure/temperature phase diagram, a liquid cannot exist at any pressure because the temperature is too high and there is too much kinetic energy in the system.

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CHAPTER 10: HEAT AND HEAT TRANSFER This chapter explores issues related to heat, which is itself a form of energy. Heat can be stored in a substance or transferred from one substance to another. Quite often, heat is not recognized until it is in transit from one thing to another. There are different types of heat transfer methods, including convection, conduction, and radiation. There are fundamental issues in physics related to heat transfer and specific laws that apply to the transfer of heat energy, which are covered in this chapter.

HEAT AND HEAT CAPACITY Heat is a form of energy, which is something we have already determined also involves things like force multiplied by distance. Temperature is related to heat; in particular, it is proportional to the average kinetic energy of molecules and atoms. Systems that contain heat will have a specific internal energy that will be higher with greater temperature.

Anytime two objects at different temperatures become in contact with one another, there is a transfer of energy. In such cases, the heat energy gets transferred from one object to another until thermal equilibrium is reached. This does not result in work being done because there is no force acting through a distance. This leads to the definition of heat being the spontaneous transfer of internal energy because of a temperature difference. Heat is energy, while temperature is not.

The SI unit for heat is the Joule. The calorie is commonly used to describe heat, which is a non-SI unit for heat energy. The calorie is the energy necessary to raise a gram of water by one degree Celsius (from 14.5 to 15.5 degrees). Another common unit for heat is the kilocalorie, which is also called the food calorie. One kilocalorie is 1000 calories.

While in general heat is not considered to be work, there is such a thing as the “mechanical equivalent of heat”. This is the work that can be done in order to transfer energy into or out of a given system. As calories are often considered to be related to

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heat and Joules are not, it is important to recognize that one calorie is the same thing as 4.186 Joules.

Heat that is added or taken away from a system will not only change its temperature; it will change its internal energy. While heat and internal energy can be considered equivalent, one cannot say that a certain thing has a certain “heat content”. There is also no such thing as “work content”. Instead, what best defines heat is its transfer from one system to another rather than an absolute value of heat. We say that a certain food has a certain number of “calories” but this is not related to the heat of the food but to the energy that can be gotten out of the consumption of the food.

Heat transfer results in a temperature change. Heating something will raise an object’s temperature, while cooling something will reduce its temperature. The assumption is that there is no phase change involved and that there is no work done on a system or by the system. There are three factors that determine the amount of heat transferred: 1) the phase of the substance, 2) the mass of the system, and 3) the change in temperature.

In defining heat, Q is the initial used to describe heat.

The absolute temperature is proportional to the average kinetic energy of an atom or molecule in a system. Greater masses will have greater internal energy in its totality.

The transfer of heat is also going to be proportional to the temperature difference between two objects. Finally, the substance is important as well as its phase. The heat necessary to raise a gram of water will be greater for water than it will be for alcohol, for example. The phase of the substance (gas, liquid, or solid) also affects the ability to give off or accept heat in a heat transfer situation.

Quantitatively, what this means is that a relationship can be made between heat transfer and the issues just discussed. Basically, it means that heat transfer or Q is equal to the mass times the specific heat times the change in temperature. The specific heat, which goes by the symbol c, is dependent on the material and the phase it is in. This is the amount of heat necessary to change a kilogram of mass by 1 degree Celsius. The units for specific heat are Joules per kilogram per degree.

The values of specific heat must be looked up in tables that apply to the different molecules and different phases. Specific heat is somewhat temperature-dependent,

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particularly for gases. Specific heat for water is much greater than that for glass and iron, so it takes more to raise the temperature of water than it does these items. Water has one of the greatest specific heats of any material on Earth.

During a phase change, such as when ice melts to become liquid water, there is no temperature change. In this way, energy is necessary to melt a solid because the cohesive bonds between molecules in a solid need to be broken apart. This energy will change the phase but will not change the temperature. This is also true of vaporizing water into a liquid. The temperature change happens after the phase change is complete. This means that, when melting ice in a soda, the temperature will stay the same until all the ice has melted.

On the other hand, energy, usually thermal energy, is released when condensing water vapor into liquid water and when freezing water. This is because work is done by the cohesive forces when molecules are brought together. This energy must be given off or dissipated in order to allow the molecules to remain together. The energy involved is based on the number and strength of the bonds in the molecule. As it turns out, there are equations that help to determine the heat required to change the phase of a mass of a substance, as is seen in figure 78:

Figure 78.

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The latent heat is measured in units of Joules per kilogram. The latent heat of fusion and the latent heat of vaporization depends on the substance, particularly on the strength of its molecular forces. These two values are called latent heat coefficients.

These are called latent because, in phase changes, energy enters or leaves a system without causing a temperature change in the system. This means that the energy is hidden.

With ice into water, it takes 334 kilojoules of energy to melt a kilogram of water and 2256 kilojoules to change a kilogram of liquid water into water vapor. Both of these are far greater than it takes to raise the temperature of liquid water or gaseous water up a degree or so after they change into their respective forms. These phase changes can have a great stabilizing effect even on temperatures not near the melting and boiling point.

An example is the fact that, in humid climates, the temperature cannot go much above 35 degrees. In the same way, temperatures in humid weather rarely fall below the dew point because a great deal of heat is released in the condensation of water vapor.

The temperature will raise in a linear fashion at about 0.5 calories per gram per degree when ice is at -20 degrees Celsius until it reaches 0 degrees Celsius. Then the temperature is held steady until all the ice is melted at about 79.8 calories per gram before again rising to 100 degrees in a linear fashion (at about 1 calorie per gram per degree). Then the temperature is steady until water absorbs 539 calories per gram. After all the liquid is vaporized, the temperature rises again at a rate of 0.482 calories per gram per degree. This means that the rise in vapor temperature takes the least amount of energy per kilogram per degree Celsius. Figure 79 shows this phase change energy pattern when it comes to water:

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Figure 79.

Water can, of course evaporate at temperatures below the boiling point but it takes more energy. At body temperature, perspiration from the skin requires 2428 kilojoules per kilogram, which is higher than the latent heat of vaporization of water at 100 degrees.

This is the heat that comes from the skin in order to cool the body. This is inhibited by high humidity situations.

This leads to a real-world application of these principles. When fruit growers face low temperatures, they spray their fruit with water. When the water freezes, it releases heat to the fruit, paradoxically preventing the freezing of fruit below freezing temperature, which would otherwise damage the fruit.

We’ve talked a little bit about sublimation, which is the direct transition from a solid to a vapor phase, which can happen with frozen water in ice cubes and with snow. The reverse can happen. Think about when frost forms on windows without going through the liquid stage. Smoke can leave dry ice without having the solid carbon dioxide enter

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the liquid phase. Other substances, like the phenol in moth balls, will go through sublimation.

All phase changes will involve heat transfer. In the case of a solid to vapor transition, the energy required is mass times the latent heat of sublimation, which is the energy required to change a kilogram of a substance to vapor via sublimation. This value is a different value than the latent heats of fusion and the latent heats of vaporization and entirely depend on the substance. As in all phase changes, there is heat necessary to have sublimation occur.

HEAT TRANSFER METHODS In this section, the different methods of heat transfer are introduced. Whenever there is a temperature difference, heat transfer will occur. It can be rapid or slow and can occur through different routes. There are things that will control heat transfer, such as wearing layers to control heat loss, using insulation, and having white roofs to reflect the summer sunlight. So many processes in real-life circumstances will involve heat transfer so that it is difficult to imagine a situation where there is absolutely no heat transfer.

The three main mechanisms of heat transfer to be discussed include these mechanisms: •

Conduction—this is heat transfer in stationary matter that occurs by physical contact. The term “stationary” refers to the macroscopic object. Molecularly, there is continual thermal motion of atoms and molecules at any temperature above absolute zero. An example of this is the heat transfer from an electric stove burner and the bottom of a pan.

•

Convection—this is heat transfer by the macroscopic movement of a fluid and is the type of transfer that takes place in forced-air furnaces and in the weather itself.

•

Radiation—this is the type of heat transfer that happens when the sun warms the earth, when microwaves heat food, and even when there is thermal radiation emanating from the human body.

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CONDUCTION Conduction can be experienced in several everyday ways. The cold in a house is felt more acutely on bare floors than it is on carpeting, even though the temperature of both are the same. This is because some materials will conduct thermal energy faster than others. In general, things that conduct electricity well are also good heat conductors, while things that don’t conduct electricity are poor heat conductors.

The average kinetic energy of any molecule in a hot object will be higher than the average kinetic energy of a cold object. If two molecules collide, the energy from the molecule with a greater kinetic energy will be transferred to the molecule with a lesser kinetic energy. This will result in a net flux of heat from the hot object to a cold object.

The heat flux will depend on the temperature difference of the two objects. Eventually the heat transfer decreases to zero and equilibrium is achieved.

What is obvious is that the conduction of heat depends on the cross-sectional area between the two objects. The greater the cross-sectional area, the faster the rate of heat transfer. Another factor in conduction is the thickness of the material through which the heat transfers. The thicker the material, the longer it takes to transfer the energy. It explains why thick clothing is more protective against the cold than thin clothing.

The final thing that affects the rate of heat transfer is a coefficient of thermal conductivity, which is dependent upon the substance. This leads to the equation for the rate of heat transfer, which is shown in figure 80:

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Figure 80.

In the situation in figure 80, the distance d refers not to the distance between the two objects because they are in contact with one another but to the thickness of the slabs that are conducting heat between the two objects. Small conductivity coefficients (which are dependent on the substance) will mean that the substance is a good insulator and won’t conduct heat as quickly. It is safe to say that the conductivity coefficient of carpeting is less than that of bare floors as the rate of conduction is less with carpeting than with bare floors.

The ratio of d (thickness) to k is called the R factor with the rate of conductive heat transfer being inversely proportional to R. The larger the R factor, the better is the insulation of the substance. This R factor is used all the time when looking at things like refrigerators and insulation. It is, however, done in non-metric units of feet squared times degree Fahrenheit times hours divided by British thermal units.

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CONVECTION Convection involves the large-scale flow of matter. It is the kind of temperature change that affects the weather. There is a large amount of flow of air from the hot tropics on Earth to the poles with the flow being easterly in the northern hemisphere. In addition, the flow of water in the cooling system of a car will cool the pistons of the car. Blood vessels will dilate near the surface of the body to allow heat to dissipate by the flow of the blood so that sweating can occur. Cold air results in shrinkage of these vessels in order to reduce loss of heat from the body. Convection also helps in the loss of heat through breathing.

Convection is more complex than conduction. Hot air rises because its density decreases with increased temperature. Natural convection happens with the ocean currents and with large-scale atmospheric circulation with a transfer of energy from one part of the globe to another. While heat is transferred via conduction or radiation on a stove, the pot of water it heats will heat via convection and the rising of hot water from the bottom surface of the pot.

Cold wind is more chilling to the body than still air that is cold because convection combines with conduction in the body in order to increase the rate at which energy is transferred from the body to the surrounding air. This is what the wind-chill factor is all about as it gives an idea of what the air temperature would feel like if the wind were not blowing. The wind chill depends on the wind speed and on the actual temperature outside.

Interestingly, although air can transfer heat rapidly via convection, it is a poor conductor of heat and is actually a good insulator. The addition of wall insulation will prevent airflow so that the loss or gain of heat is decreased. The same is true of using double-paned glass with an airspace between them. In that case, the air between the windows acts as an insulator. The air spaces in animal fur has multiple convection currents that insulate an animal from temperature changes.

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RADIATION Radiation happens whenever heat comes from a fire or from an energy source like a microwave or the Sun. Similarly, you can feel the heat from an oven even without touching it. No conduction and no convection happen when the sun shines on the earth because there is no contact and no air movement. There is no medium required for the propagation of electromagnetic waves. These waves can come from x-rays, gamma rays, microwaves, infrared radiation, radio waves, visible light, and ultraviolet radiation.

With the heat transfer associated with fires, it isn’t the visible light that causes heat transfer but the infrared radiation. Visible light actually transmits little thermal energy.

Convection is in play as it transfers energy away from the observers because of wind and rising air secondary to decreased density. As you can imagine, conduction plays little role in heat transfer in the case of fires.

What’s true is that all objects absorb and give off electromagnetic radiation in some form or another. The rate of heat transfer depends on the color of the object. Black will absorb heat more readily and white will reflect it to a greater degree. Black also radiates heat better than lighter objects. The ideal radiator is therefore the ideal absorber, while poor absorbers will radiate heat more poorly. Colored objects will behave differently, depending on where they are in the visible range. Skin, for example, absorbs infrared radiation more readily and makes people more sensitive to this type of radiation.

The rate of heat transfer by means of emitted radiation can be determined by what’s called the Stefan-Boltzmann law of radiation. This relates to the rate of heat transfer rather than the actual heat transferred. Figure 81 shows the equation:

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Figure 81.

In the image, the symbol e stands for the emissivity of the object, which is a measure of how well an object radiates. A jet-black object has an e equal to 1, while a perfect reflector has e equal to zero. Most object will have an e value between these two levels.

Light bulb filaments have an e value of 0.5, while carbon black (which makes up printer toner ink) has an emissivity of 0.99. Skin, regardless of color, has an emissivity of 0.97.

Note that the rate of heat transfer is related to the fourth power of the absolute temperature in degrees Kelvin as well as the square area of the object emitting heat.

This can be seen in the breaking up of coals on a fire. When broken up, the square area of the broken coals is greater so they emit heat at a greater rate. Remember that it is the rate of heat transfer that is affected and not the total amount of heat transferred.

The net rate of heat transfer is related to the temperature of the object and the temperature of the surroundings. According to the Stefan-Boltzmann law of radiation, the net Q is seen (as in figure 81) as proportional to the difference between the fourth power of the temperature of one item and the fourth power of the temperature of the surroundings. It is also directly proportional, as the law states, to the emissivity of the object but not its surroundings.

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KEY TAKEAWAY •

Heat is the transfer of energy of a substance to another substance and is not the same thing as the temperature of a substance.

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Heat is energy and, rather than work, which is the force acting over a distance, it involves the spontaneous transfer of energy from one source to another because of a temperature difference between the two.

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There are several types of heat transfer, such as conduction, convection, and radiation.

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The rate of heat transfer in conduction can be established between two objects.

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The rate of heat transfer in convection is difficult to calculate because there are many factors involved, such as the rising of air as it heats and factors like wind.

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Convection plays a big role in determining the earth’s weather patterns.

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All things will emit electromagnetic radiation. Things that easily absorb electromagnetic radiation are also things that emit electromagnetic radiation more.

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The wavelength of the electromagnetic radiation determines its ability to radiate heat.

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QUIZ 1. What is the best physics definition of heat? a. It is the internal energy of a system. b. It is the work done between two systems of different energy levels. c. It is the potential of the system to do work because of its internal energy. d. It is the spontaneous transfer of internal energy from one object to another.

Answer: d. Heat is not the same thing as temperature. It involves the spontaneous transfer of stored internal energy from one object to another, which is transferred because of differences in temperature.

2. What is the SI unit for heat? a. Joule b. Degree c. Kilocalorie d. Calorie Answer: a. Remember that heat is energy and, in all cases, SI units for energy are joules, regardless of the type of energy being talked about.

Degrees have nothing to do with actual energy and both the calorie and kilocalorie are non-SI units for heat energy.

3. On what condition or situation is heat transfer least dependent upon? a. Mass of the system b. Temperature change in the system c. Phase of the objects d. Potential energy of the system Answer: d. The mass of the system, the phase of the objects, and the change in temperature of the system are highly linked to heat transfer; however, the potential energy of the system is not usually related to heat transfer.

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4. What is the best definition of specific heat of a substance? a. The mass of a substance that will raise its temperature by one degree Celsius if one Joule of energy is applied to it. b. The joules of energy held in a gram of a substance. c. The amount of energy necessary to raise a kilogram of a substance by one degree Celsius. d. The number of kilograms of a substance that will transfer one degree of heat to a kilogram of water.

Answer: c. The specific heat depends on the substance and on its phase.

It is the amount of energy necessary to raise a kilogram of a substance by one degree Celsius.

5. Which substance has the greatest specific heat? a. Copper b. Iron c. Alcohol d. Water Answer: d. Water has the highest specific gravity of almost any type of substance. Liquid water has double the specific heat when compared to ice.

6. What is true of the heat involved to change a phase? a. Heat is necessary to both freeze and condense a substance. b. Heat is released to both freeze and condense a substance. c. Heat is released to freeze but is necessary to condense a substance. d. Heat is necessary to freeze but is released in condensing a substance.

Answer: b. Heat is released when freezing and condensing a substance.

It is necessary to melt and vaporize a substance, depending on the latent heat of vaporization and on the heat of fusion.

7. What form of heat transfer occurs during forced heating from a furnace that heats a house?

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a. Radiation b. Convection c. Conduction d. Evaporation Answer: b. Convection heating involves the movement of warm air through ducts that heat a room or house. This is an example of the heat transfer method of convection.

8. What form of heat transfer occurs when the sun warms the planet earth? a. Radiation b. Convection c. Conduction d. Condensation Answer: a. This is a form of radiance or radiant heat, in which actual electromagnetic waves of energy cause a rise in temperature of whatever place the electromagnetic waves reach. This is radiation.

9. When looking at conduction and insulation, what is the R factor? a. A measure of the degree to which a substance conducts heat. b. A measure that is based on the square area of the substance. c. A measure of the energy required to heat a substance through conduction. d. A measure of the insulating ability of a substance.

Answer: d. The R factor is a measure of the insulating ability of a substance so that the higher the R factor, the less the heat can conduct through it. The rate of conduction is inversely proportional to the R factor.

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10. What type of heat transfer is involved in the weather? a. Convection b. Conduction c. Radiation d. Evaporation Answer: a. Heat transfer from the hot equators to the cold poles, along with the spin of the earth, involves convection and the large transfer of heat from one place to another via air currents.

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CHAPTER 11: THERMODYNAMICS This chapter is also concerned with temperature and heat transfer; however, it expands these topics further and goes into the laws of thermodynamics that apply to heat as it relates to energy and work. In previous chapters, the topic has been on heat as a pure form of energy transfer, while this chapter is about the ability of heat transfer to perform work. Heat is like any other form of energy. In many systems you will become familiar with in this chapter, heat transfer has the ability to do things like run engines and allow machines to function. The laws of thermodynamics are not just laws of physics; they have practical applications that are seen in everyday life.

FIRST LAW OF THERMODYNAMICS In earlier chapters, we discussed conservation of energy and it was determined that energy is conserved. What the first law of thermodynamics does is to apply this conservation of energy to systems that involve the transfer of heat and the work done by heat energy. The first law of thermodynamics indicates that the change in internal energy of a system is equal to the net heat put into the system minus the net work done by the system. Internal energy is described with the letter U so you can say that delta U equals Q minus W, where work is identified by the letter W and Q is the net heat put into the system.

If Q is positive, heat is put into the system. If W is positive, work is done by the system.

Any difference in the two is stored in the system as internal energy. Heat engines apply the first law of thermodynamics by transferring heat into them so that they can do work.

This first law is basically using the conservation of energy principles and includes heat transfer as part of the equation.

Remember from the previous chapter that heat transfer is driven entirely by the temperature differences between two systems. Work, on the other hand, is the organized process of applying a force through a given distance. The transfer of heat and the application of work are both considered energy in transit as both can be stored in a

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system or given off by a system. Both of these types of energy transfers will change the internal energy or “U” of the system. Internal energy is not the same thing as heat and is not the same thing as work.

The internal energy of a system can be looked at from a molecular standpoint. In such cases, the internal energy will be the sum of the potential and kinetic energy of the molecules and atoms in a system. Kinetic energy plus potential energy is the same thing as mechanical energy. Because there are many atoms and molecules in a system, one looks at the averages of all of the molecules in the system. The internal energy of the system is independent of how it got to that level.

For the purposes of understanding internal energy of a system, you need to imagine the system going from state A to state B. Regardless of how it gets to these states, the delta U or change in internal energy is the difference between the internal energies of these two states. The change in U is independent of the path. On the other hand, the heat energy and the work energy do depend on the path. The Q (heat energy) will be positive if heat is put into the system and the W (work energy) will be positive if work is done by the system or out of the system.

A system that most people are familiar with is that of human metabolism. This is a system that follows the first law of thermodynamics. Body heat will leave the system and will be negative. The body will do work and this will be positive. Adding food to the system will increase the potential energy of the system that is released into the body by means of metabolism. If one eats just the right amount of food and does work, the internal energy of the system will be constant. Eating too much will cause storage of calories as fat and the delta U will be positive. Exercising too much or eating too little will lead to a delta U that is negative.

The basal metabolic rate is the rate at which food is converted into heat energy and when the body is at rest. This adjusts when the body eats too little; it is a way of compensating for undereating. Eventually, the basal metabolic rate will get so low that weight loss will not happen as quickly as it once was. Only exercise helps overcome this decrease in BMR. Exercise will increase the rate of heat lost and will even increase the BMR at rest.

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It should be noted that some thermodynamic processes are irreversible. One cannot “unburn” a fire once it has burned. In the same way, you cannot add useful energy to the body by sitting out in the sun to heat the body. You need to eat to gain energy and metabolism is a one-way process. Photosynthesis is also an irreversible form of energy input. Light energy goes into chemical potential energy, which can only go in one way.

The first law of thermodynamics applies to certain devices, such as the heat engine.

Examples of heat engines include automobile engines and steam-driven turbines. These engines involve an input of heat in such a way that there is an output of work. No system like this is perfect so there will be an amount of heat put out that is lost and makes the engine inefficient. What this means is that the Q out or heat output will never be zero.

A piston system involves work done at a constant pressure. This is referred to as an isobaric process. As the pressure is constant, the exerted force will be constant and the work done will be the force of the piston times the distance it travels. The pressure multiplied by the change in volume displaced by the piston will equal the work done. As long as there is a positive change in the volume, the work will be positive and will be done by the gas that pushes on the piston.

Figure 82 shows the pressure/volume curve of an isobaric system. Because pressure times the change in volume is work, the area under the curve going from point A to point B is the work done by the system. We’ve talked about pressure and volume in the past and the same principles apply to these types of simple systems:

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Figure 82.

In situations where pressure is not the same throughout, the area under the pressure/volume curve will still be the work done. In such cases, the average pressure is used to determine the work done by the system. The work that will be done depends on the path taken and not just the endpoints. Anytime the volume change is zero, this is referred to as an isochoric process and no work will be done. Figure 83 shows the work done in a system where pressure and volume are changing:

Figure 83.

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Note that figure 83 is a perfect system in which every process is reversible when, in fact, many thermodynamic processes are not reversible and involve a loss of heat from the system. Later on, we will see a more realistic picture of what this looks like in a fourstroke engine.

In some work situations involving pressure and volume, there can be an adiabatic process, which is defined as a process that has no transfer of heat. This means that Q equals zero. These can only be done if there is a great deal of insulation in the system or if the process is so fast that heat transfer does not occur.

In fact, in an adiabatic process, the change in temperature must be negative because work will be done at the expense of internal energy. An example of an adiabatic process is the release of gas into the air by a pressurized cylinder. The gas will be colder than the gas in the cylinder and there will be less work done. Figure 84 describes these simple engines and the processes involved:

Figure 84.

SECOND LAW OF THERMODYNAMICS While, in theory, work can be done that is reversible, this doesn’t take place in most thermodynamic processes. In theory, an irreversible process is one that will depend on the path taken. A cold object next to a hot object will never get colder because the direction in heat only goes in one direction—from hot to cold. Friction is irreversible as well because, when it stops an object, it does not cool the object so it moves again. Gases

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expand in an area with the molecules evenly distributed. According to the first law of thermodynamics, reversible processes are not forbidden. This is the case, however, with the second law of thermodynamics.

According to the second law of thermodynamics, heat spontaneously moves from high temperatures to low temperatures but the reverse is never possible. Heat goes in a oneway direction from hot to cold and not the other way around. This is just the first expression of the second law of thermodynamics.

This takes us back to heat engines, which include jet engines, gasoline and diesel engines, and steam turbines, which use heat transfer to power them. In a heat engine, there will be heat transfer from a hot reservoir to a cold reservoir with work done as part of this. Figure 85 describes a heat engine and the work done in the system:

Figure 85.

What figure 85 shows is that heat transfer drives the system, with the work done being the heat transferred into the system subtracted by the heat leaving the system. The heat leaving the system in a heat engine will be positive because no heat engine is perfect and will have no heat output as part of their activity. This leads to the second expression of

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thermodynamics: it is impossible in any system for heat transfer to work in a cyclic process whereby the system returns to its initial state.

What this means is that, in a piston system, where the piston goes up and down, or in a rotating turbine system, where there is a cyclic process, there cannot be a perfect conversion between heat transfer into a system and work done. It means that the process will never be 100 percent efficient. The efficiency of a system is the work output divided by the heat input or heat transfer into the system. Because work equals heat input minus heat output, the efficiency can be the described as 1 minus the heat output divided by the heat input. What this looks like is seen in figure 86:

Figure 86.

An efficiency of one or 100 percent is possible only if there is no heat transfer into the environment. All Q will be positive because heat goes in one direction. Ideally the efficiency will be 100 percent; however, this is not the case so that it will be less than 100 percent because there will be some heat output.

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THE FOUR-STROKE ENGINE This represents a simple piston engine system, which involves the use of an ignitable gas that does work on the system and drives the engine. Figure 87 is a four-stroke internal combustion engine described visually:

Figure 87.

There are four cycles: intake-exhaust, compression, ignition, and power. The compression stroke involves work being done on the system, increasing the pressure and temperature. There is heat transfer in the next part, causing a pressure increase and temperature increase. This happens so quickly in a combustion engine that volume is nearly (but not completely) constant. The final part, power is is (nearly) adiabatic process so that work is done to a greater degree because the pressure is greater. Exhaust leaves the system and the volume is returned to its initial state.

During the first part, air is mixed with fuel during intake. In the second part, compression of the air-fuel mixture occurs in a process that is nearly adiabatic. Work is done on the compressed gas. In the power stroke, there is ignition of the air-fuel mixture, creating thermal energy that increases the pressure, pushing on the piston.

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This also does work, and is nearly adiabatic. In the exhaust stroke, hot gas is expelled in order to prepare the engine for another cycle.

What you should note is that it is necessary for heat to transfer out of the system in order for there to be a net amount of work done. The pressure volume work done in this cycle, called the Otto cycle, is described in figure 88, which is defined as the work done in the cycle. If no heat transfer is done, then no work is done. The lower the temperature on the path A to B, the less work needs to be done to compress the gas. In addition, the higher the temperature along the path from C to D, the greater the work done.

Figure 88.

In figure 88, paths A to B and C to D are adiabatic and are the compression strokes and power strokes, respectively, of the internal combustion engine. Paths BC and DA are isochoric and are the ignition and intake-exhaust aspects of a four-stroke engine (note that intake and exhaust happen at the same time). Work is done on the AB path but more work is done on the CD path, which allows a net amount of work output done. This increase in work occurs along the CD path because the temperature is increased along 193

this path versus the AB path. The isochoric processes are done so quickly that the volume of the piston does not have a chance to catch up.

In summary, these are the parts of the four-stroke engine: •

Exhaust-Intake D to A: The decrease in pressure when the gas-air mixture is added and exhaust takes place (isochoric).

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Compression A to B: The increase in pressure when the piston is pressed on (adiabatic).

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Ignition B to C: The ignition of the spark plugs and further increase in pressure (isochoric).

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Power C to D: The decrease in pressure and increase in volume (adiabatic).

This example of the Otto cycle and the four-cycle engine reflects what happens in normal fuel-powered engines that turn fuel into heat and finally into power that drives the engine. During the compression phase, the gases are compressed so they can be ignited, furthering the change in pressure that drives the power in the system, which is what does the work.

Heat engines cannot be 100 percent efficient. Why is this the case? Because, heat must be transferred to the environment. This heat transfer is referred to as waste heat. The Carnot engine and the Carnot cycle reflect the most efficient cyclical cycle process possible. This would technically be a never-ending engine that requires just a small amount of input to get the engine going. What defines the Carnot cycle versus the Otto cycle is that only reversible processes are at work.

This leads to a third aspect of the second law of thermodynamics. This is that a Carnot engine between two temperatures will have the greatest possible efficiency of any type of heat engine operating between the two different temperatures. All engines using just reversible processes will have the same maximum efficiency when operating between the same given temperatures. Figure 89 shows the Pressure-Volume diagram for a Carnot cycle engine. There are two isothermal and two adiabatic processes, both of which are reversible in theory:

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Figure 89.

Remember that the efficiency of a perfect heat engine is one minus the ratio of the heat output and the heat input. For a perfect heat engine, the ratio is equal to the ratio of the absolute temperatures of the heat reservoirs. It leads to the definition of the Carnot efficiency or maximum efficiency is related to the actual temperature differences between both aspects of the system. In reality, the actual efficiency is about 0.7 or 70 percent of the Carnot maximum. The greatest efficiency would happen when the temperature ratios are as small as possible.

The problem with the Carnot engine is that it has no ability to do any power. It would only be possible if the cold reservoir were at absolute zero, which is impossible. Because all of the engines operate at levels greater than absolute zero, the ability to be efficient is limited. The maximum efficiency could happen when one reservoir is at the highest possible temperature and the other reservoir were at the lowest possible temperature.

This would increase the area under the Pressure-Volume curve.

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In summary, the Carnot cycle is a theoretical cycle that is not possible in reality. It represents the most efficient cyclical process possible and only involves reversible processes that are adiabatic and isothermal. Any engine that could do this would be called the Carnot engine. Carnot engines, should they exist, have the maximal possible efficiency. Friction and other dissipative processes naturally reduce the efficiency of an engine.

APPLICATION OF THERMODYNAMICS There are so-called engines, like refrigerators, air conditioners, and heat pumps that are basically heat engines run backwards but not in reverse. Heat transfer runs from a cold reservoir into a hot one, requiring the input of work, which is also going to be converted into heat transfer. This means that the heat transfer cooling the refrigerator will be equal to the heat output plus the work done. The goal is to have the heat transfer to occur from a cool environment into a warm environment, such as a refrigerator in a warm room. Figure 90 shows what this looks like schematically:

Figure 90.

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HEAT PUMPS A heat pump involves heating a room without burning fuel. The heat comes from the outside air, even if that air is cold. The work of the pump is the only thing that is paid for; the downside is that it takes twice as much energy to heat the space than would be the case in a fuel-powered system. It requires electrical energy instead of fuel energy, which is often more expensive to do.

A heat pump involves a refrigerant that gets turned into a gas by the cold outdoor air.

This gas drives a compressor that raises the temperature, forcing it into the heated space. There is a transfer of heat into a space before the gas condenses back into a liquid, where it goes back outside to be expanded again. This is described in figure 91:

Figure 91.

As you can imagine, heat pumps don’t work very efficiently unless the temperature differences are small. They work much better in moderate climates than they do in cold climates. Friction, in this case, transfers some heat out into the cold reservoir before it gets back into the pump. The work is done by the electricity used to drive the pump.

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REFRIGERATORS AND AIR CONDITIONERS Air conditioners and refrigerators will cool something down in a warm environment. As with heat pumps, they require heat transfer from cold to hot, which is expensive. Air conditioner quality is judged by how much heat transfer occurs from the cold environment compared to how much work is done. This leads to the definition of coefficient of performance, which is the waste heat divided by the work put into the system. The coefficient of performance of a refrigerator is equal to the coefficient of performance of a heat pump minus 1.

Actual air conditioners have a coefficient of performance ranging from 2 to 6, which is better than those of a heat pump, which isn’t very efficient. There is an energy efficiency rating system (or EER) that has been developed to identify the efficiency of a refrigerator, which is given a 1 to 5-star rating. The higher the EER, the cheaper it will be to operate the air conditioner but the more expensive it will be at the outset.

ENTROPY There is still another way of expressing the second law of thermodynamics, which is related to the issue of entropy. Entropy is the tendency of systems in nature to become more disordered so that less energy is available to do work. Entropy is a measure of how much energy is not available for work. This takes us back to the Carnot cycle, which involves theoretically reversible processes of thermodynamics and theoretically involves the ratio of the temperature output and the temperature input rather than the heat output and the heat input. Figure 92 shows these equations:

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Figure 92.

As in the rest of this chapter, Q is the heat transfer into or out of a system and T is the absolute temperature at which the reversible reaction takes place. The SI units for entropy are Joules per degree Kelvin. So far, this applies to irreversible engines. On the other hand, it is possible to determine the entropy for real situations.

Why is it possible to determine the change in entropy of a system? It’s because entropy is a state system that is independent of the path involved. This is similar to the internal energy of a system. What needs to be done is to imagine or find a reversible path between the two states and to calculate the change in entropy from that equation. When two reservoirs get together, there will be a loss of entropy from the hot reservoir and a gain of entropy from the cold reservoir. The total change in entropy will be the total change in entropy from both the hot and cold reservoir together. For a Carnot engine, the total change in entropy will be zero. To restate this: reversible processes do not affect the total entropy of the universe (which is the system plus its surroundings).

Because real systems are not reversible, there is another way to think of the second law of thermodynamics as it relates to entropy: The total entropy of a system will either increase or remain constant in any process. Entropy never decreases. This is why heat transfer cannot occur spontaneously from hot to cold. Entropy is not conserved but

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increases in all real systems. In Carnot engines, which are just theoretical, the entropy remains at a constant.

When entropy increases, a certain amount of energy becomes permanently unavailable to do work. The energy is not lost but its character is changed so that some of it can never become work. The unavailable work is in the SI units of Joules, which means that it equals the change in entropy times the lowest temperature utilized.

How is it possible for a system to decrease entropy? Of course, it requires a transfer of energy. Think of it as taking bricks to build a house. This involves using the energy of the workers and equipment in order to take the disorder and turn it into a more ordered system. The sun’s energy will decrease the entropy of the earth but it does not decrease the entropy of the entire universe. This means that the second law of thermodynamics is never truly violated.

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KEY TAKEAWAYS •

The first law of thermodynamics states that the change in internal energy of a system is the difference between the heat put in and the work output of the system.

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Heat energy into the system is positive and work energy out of the system is positive.

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There are adiabatic, isothermal, isochoric, and isobaric processes involved in energy systems.

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The work done is the area bound by a pressure-volume curve in an engine.

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Heat always goes from hot to cold areas in a system.

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A Carnot engine involves a theoretical engine in which the processes are always reversible.

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The second law of thermodynamics applies also to the fact that entropy is always increasing or held constant.

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In order to decrease the entropy of a system, energy must be put into it from an external source.

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QUIZ 1. According to the first law of thermodynamics, what is the change of internal energy of a system determined by? a. The net work done by the system b. The net heat output of the system c. The change in its temperature before and after work is done d. The difference between the heat put in and the work output of the system Answer: d. The change in internal energy of a system is the difference between the heat put in and the work output of the system.

2. What is the internal energy of a given system? a. The potential energy of the system b. The difference between the potential energy and the kinetic energy of a system c. The sum of the potential energy and the kinetic energy of a system d. The kinetic energy of the system Answer: c. The internal energy of a given system is the sum of the potential energy and kinetic energy of a system. It is independent of how the energy gets into the system.

3. Which of these biological processes is not considered a one-way energy process in that it is reversible rather than being irreversible? a. Photosynthesis b. Glucose metabolism c. Fat storage d. Exercise Answer: c. Fat storage and the loss of fat are reversible processes that change the potential energy of the system. The other biological processes are considered irreversible processes.

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4. Under what circumstances in a traditional heat engine will the heat output be zero? a. When the work output is at its maximum potential b. When the system has low friction c. When the heat input is less than the work output d. There is no situation in a heat engine where the heat output is zero Answer: d. There is no situation in a heat engine where the heat output is zero. Anytime the heat engine is running there will be some inefficiency and there will be some heat output.

5. According to the second law of thermodynamics, what is the case? a. Heat always goes from hot to cold areas in a system. b. Heat output will always exceed work output. c. Heat engines are imperfect systems. d. Work output will always exceed heat output.

Answer: a. According to the second law of thermodynamics, heat always goes from hot to cold areas in a system. Heat and work output will be different with different systems with more efficient systems putting out more work than heat but this is not always the case.

6. In a heat engine, the efficiency will be related to what? a. The work put in minus the work put out. b. The work put out minus the work put in. c. The heat put out minus the heat put in divided by work. d. The work output divided by the heat put in.

Answer: d. Ideally, the work output should be equal to the heat put in so that the efficiency will be 100 percent. The efficiency, however, will be the work output divided by the heat input, which will be less than 100 percent.

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7. In a four-stroke engine, what does the intake involve? a. The increase in pressure when the gas-air mixture is added. b. The decrease in volume when the piston is pressed on. c. The ignition of the spark plugs and increase in pressure. d. The release of gases to the outside.

Answer: a. The intake involves the increase in pressure when the gasair mixture is added to the system. This mixture gets ignited under great pressure after compression, which leads to great power exerted.

8. What happens during the exhaust phase of the four-stroke engine? a. Q intake is increased. b. Q is held the same. c. Q intake is decreased. d. Q output is increased.

Answer: d. The exhaust portion of the four-stroke engine happens at the same time as the intake phase but actually involves the output of heat in the release of exhaust from the engine. The intake of gas is at a colder temperature but, because this is not ignited, the intake part adds to the potential energy of the system and not to the Q intake.

9. What example represents a Carnot engine? a. A nuclear reactor b. A propane-powered engine c. A diesel-powered engine d. None of these is a Carnot engine Answer: d. A Carnot engine is just a theoretical engine that has only reversible processes. These types of processes cannot be reversible and heat will be lost so that heat energy must continually be added in order to drive the engine.

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10. Under what conditions does a heat pump work better to heat indoor air when compared to the outdoor air? a. When the temperature difference isn’t very great. b. When the fuel system involved is propane versus gasoline. c. When the outdoor air temperature is at its lowest. d. When there is a great difference between the outdoor temperature and the indoor temperature.

Answer: a. These heat pumps work better to heat the indoor air when there is a limited difference between the indoor temperature and the outdoor temperature.

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CHAPTER 12: OSCILLATORY MOTION AND WAVES This chapter delves into oscillatory motion, which is movement back and forth between two points. There are many systems that oscillate, some of which create waves. Waves can be visual, such as the waves in a swimming pool or ocean. Other waves that aren’t commonly seen as waves include sound waves and light waves. Waves create disturbances that carry energy, from small waves that carry light energy to large waves that create tsunamis and earthquakes. Waves, as it turns out, have the energy to augment each other or to interfere with one another, which is covered in this chapter.

HOOKE’S LAW AND OSCILLATION According to Newton’s first law of motion, an object that is oscillating will have forces moving it back and forth. Deforming something that is plastic or elastic ultimately involves a restoring force that gets the object back into its normal shape. As it does that, there will be a gain in momentum and movement to the opposite direction for a little bit until the oscillatory motion is dampened so that the motion dissipates. Ultimately, the motion comes to a rest.

In simple oscillations, there will be oscillation as long as the restoring force is directly proportional to the level of displacement. We have otherwise discussed Hooke’s law as it describes deformation, in which force equals a constant multiplied by its displacement from equilibrium. The force is actually the negative of the constant times displacement because the restoring force will be opposite that of the displacement. The force constant is related to the rigidity of the system. The greater the restoring force, the stiffer is the system. Units for the constant are in Newton’s per meter.

According to Hooke’s law, there will be a restoring force that is proportional to the amount of displacement and will be based on a constant. This means that a graph of force to length displaced will be linear based on the force constant. This type of force diagram can be applied to a spring as well, even though the direction of the spring would

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be from compression to extension rather than from left to right (in most cases). Figure 93 shows Hooke’s law as it applies to a spring:

Figure 93.

In order to produce a deformation, work needs to be done. Force must be exerted through a distance. If the only thing that happens is deformation, meaning that no work goes into sound energy, thermal energy, or kinetic energy, it gets stored within the deformed object as potential energy. Potential energy in a spring is equal to one-half multiplied by a constant multiplied by the square of the distance deformed. This energy will be the area under the curve of a force-distance graph as is shown in figure 94:

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Figure 94.

PERIOD AND FREQUENCY In looking at the displacement of a guitar string, there will be a steady sound for a period of time. The vibration of the string takes the same time for each vibration over a period of time. Periodic motion is motion that repeats itself at regular time intervals.

The time it takes for a single oscillation is constant and is called the “period”, called by the letter T. The SI units are that of seconds but it could represent any period that makes sense. Periodic motion is, by definition, repetitive motion.

The frequency is not the same thing as the period. If something has a period of 0.5 seconds, its frequency will be 2 cycles per second so that the frequency is the inverse of the period. The SI unit for the frequency is in Hertz, which is the same thing as a cycle per second. Vibrations can be singular or multiple but oscillations are defined as being repetitive over several cycles.

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SIMPLE HARMONIC MOTION There are certain oscillations that can be described as simple harmonic motion or SHM. This is the name given to oscillatory motion for a system whenever the net force can be described by Hooke’s law. This is referred to as a simple harmonic oscillator. As long as there is no damping of the motion by friction or other forces, this type of oscillator will oscillate with equal displacement on either side of the equilibrium position. The maximal displacement is referred to as the amplitude, identified by the letter X. The units of amplitude are the same as they are for displacement, which are meters for objects like a spring but, for sound oscillations, the units will be in pressure units. The amplitude is related to the energy stored in the oscillation.

An object that is attached to a spring which is sliding along a frictionless surface is a simple harmonic oscillator. The amplitude will be X and the period will be T. The maximum speed occurs as it passes through its equilibrium point. The stiffer the spring, the smaller the period will be. The greater the mass of the object, the greater is the period.

The period and the frequency are, as mentioned, related to one another but neither are related to the amplitude of the spring. A guitar string will make the same sound regardless of how hard it is struck. Because the period will be constant, a simple harmonic oscillator can be used to run a clock. The period will be related to the stiffness of the system because of its force constant. High stiffnesses have high force constants and a smaller period.

In addition, more massive systems will increase the period (think of a heavier person bouncing more slowly on a diving board). Mass and the force constant are the only factors that affect the period and the frequency of things in simple harmonic motion.

This is defined according to figure 95:

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Figure 95.

Simple harmonic motion can be described in terms of waves. In fact, all simple harmonic motion is intimately related to the sine and cosine waves. For instance, if the restoring force of a spring or an oscillating object can be described exclusively by Hooke’s law, then the wave will be a sine function. In such cases, the displacement over time in any simple harmonic motion is related to the equation described in figure 96:

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Figure 96.

The equations seen in figure 96 show the differences in the velocity, displacement, and acceleration over time. At the start of motion, the velocity is negative because the system is moving back toward the equilibrium point. The equations for velocity and acceleration are no different from Newton’s second law, which is Force = mass times acceleration or the force constant multiplied by distance and divided by mass. Figure 97 shows the acceleration and velocity of wave forms:

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Figure 97.

What this means is that the velocity will be maximum at the point near the zero point on the wave or when the object is crossing over the equilibrium point. The acceleration will be opposite to the displacement and will be directly proportional to the displacement.

The maximal V or Vmax will be the square root of the ratio of the force constant multiplied by the displacement divided by the mass. This is seen in figure 98:

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Figure 98.

PENDULUMS A simple pendulum is defined as an object swinging or suspended from a light wire or string. Pendulums have the ability to perform simple harmonic motion. Displacement in a pendulum is related to the arc length, defined by the letter s. The net force on the object will be the tangent to the arc, which will be equal to mass times the force of gravity times the sine of theta. The tension on the string will exactly cancel the component mass times the force of gravity times the cosine of theta, which is parallel to the string at all times. Figure 99 shows the pendulum swinging:

Figure 99.

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As long as the restoring force is directly proportional to the displacement, this would be a simple harmonic oscillator. For small angles less than 15 degrees, the sine of theta will be roughly equal to theta, which means that the restoring force will be mass times the force of gravity times theta. Theta will be expressed in radians so that the displacement will equal the length of the cord on a pendulum times the angle theta in radians. This means is that the force will be roughly equal to the mass times the force of gravity times the length of the cord times the arc length. These equations are seen in figure 100:

Figure 100.

What figure 100 ties into is the force constant k, which is mass times the force of gravity divided by the length. For angles less than 15 degrees, this will behave according to Hooke’s law and the restoring force will be directly proportional to the displacement.

This will make the pendulum a simple harmonic oscillator when the angle of displacement is less than 15 degrees.

As for the energy of a simple harmonic oscillator, the energy will be shared between the elastic potential energy and the kinetic energy. What we already know is that this will equal one-half multiplied by the mass times the square of the velocity (kinetic energy) and one-half times the force constant times the displacement squared (the potential energy). The sum of these two things will be constant and will be shown in figure 101:

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Figure 101.

DAMPED HARMONIC MOTION In reality, simple harmonic motion does not go on forever and guitar strings stop making sound; in the same way, a swing will stop swinging unless the child is pushed.

Because of nonconservative forces and friction, the presence of completely undamped motion is rare.

With a small amount of damping, the period and frequency are nearly the same as for simple harmonic motion; however, the amplitude gradually decreases. This is because the force will remove energy from the system, usually in the form of thermal energy.

The energy will cause a change in kinetic and potential energy of the system, so that the system eventually stops oscillating. Figure 102 shows damped oscillation.

Figure 102.

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As the dampening increases, the period and frequency will begin to be affected so that the object moves toward equilibrium. In some cases, such as with a car suspension, the goal is to dampen them as quickly as possible. This leads to a critical dampening, which is defined as the condition in which there is dampening to the maximal degree as quickly as possible. The goal is not to underdamp the system or overdamp the system.

An underdamped system will oscillate through the equilibrium position and an overdamped system will move more slowly toward equilibrium than one that is critically damped. Critical damping will return the system to equilibrium as quickly as possible without overshooting.

RESONANCE All objects will have a resonance frequency. This is the natural frequency of the object when a periodic force is applied to it. The driving force will put energy into a system but will not put energy into a system necessarily at the resonance frequency of the object.

The natural frequency of an object is the frequency it would oscillate if there were no driving force and no damping force. If you drive or force a system at its resonance frequency, the amplitude will naturally increase. This phenomenon is called resonance.

This is why it takes a certain frequency to break a glass. The frequency of a certain sound will reach the resonant frequency of the glass so that the glass vibrates and increases in amplitude, with the amplitude being so great eventually that it breaks the glass. Heavy damping will reduce the amplitude; the less damping at the resonant frequency will narrow the resonance frequency. Little damping is necessary for musical instruments like pianos, while automobile suspension systems will require heavy damping, which will reduce the amplitude.

When one tunes a radio, the resonant frequency is being adjusted so that the radio oscillates only at the station’s broadcast frequency. Magnetic resonance imaging used in medicine involves resonating hydrogen nuclei using incoming radio waves. Swinging a child on a swing involves pushing at the natural resonant frequency of the swing to achieve maximum amplitude. The efficiency of energy transfer from the driving force into the oscillator is best at the resonant frequency.

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WAVES So far, oscillations have been clearly discussed but we have not talked about waves.

What is a wave? This is a disturbance that propagates from the place it was created. For water, the wave can be seen; for sound waves, the wave can be seen as air pressure. For earthquakes, it can be felt as earth moving. A wave in water is an up and down disturbance of the water surface. There will be peaks and troughs of the wave. The time for a complete peak to peak motion is the wave’s period T with the frequency being 1 divided by the period. The wave’s velocity v is the speed at which the wave moves across a surface or in air. This is also referred to as the propagation velocity. The water itself in a wave does not move forward, it just goes up and down with a certain energy being propagated. The wavelength, identified by the Greek letter lambda is the peak to peak distance.

The wavelength, which can also be described as the distance between adjacent identical parts of the wave, will be a distance that is parallel to the direction of the propagation.

The speed will be the time of one period or lambda divided by T. It can also be described as the frequency multiplied by the wavelength. Figure 103 shows these relationships:

Figure 103.

Velocity can be speed of water waves, speed of light, or speed of sound. You can get the velocity by knowing, for example the time between wave crests (which is the period) and the wavelength.

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Waves that travel like ocean waves will propagate in the horizontal direction, while the surface is disturbed in the vertical direction. These ocean waves are called transverse waves or shear waves. The disturbance is perpendicular to the direction of propagation.

In contrast, there can be a longitudinal wave or compression wave, in which the disturbance is parallel to the direction of propagation. In a longitudinal wave, the size of the disturbance is an amplitude of X, which is independent of the speed of propagation.

A spring that is set in a wave fashion is a longitudinal or compression wave as is shown in figure 104:

Figure 104.

Waves may be transverse, longitudinal, or a combination of these. Water waves in the ocean are a combination of transverse and longitudinal waves. Electromagnetic waves and light waves are all transverse or shear waves, while sound waves in air and water are considered longitudinal waves. Sound waves in solids are both longitudinal and transverse. Earthquake waves under the Earth’s surface will have both longitudinal and transverse components. They will propagate at different speeds.

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SUPERPOSITION AND INTERFERENCE Most waves are not simple waves. Most waves are complex because they are added together and subtracted from one another. This can be seen when looking at the waves on the ocean, which will have different heights at different times. Waves will superimpose upon themselves onto one another. This is referred to as superposition.

The disturbances in these cases will be referred to as forces that will either add or subtract from one another. If two waves come together with the same peaks and same troughs, this will be called constructive interference. The wave will have twice the amplitude of individual waves but the wavelength will be the same.

On the other hand, if two waves are identical and have the trough of one happen at the peak of another, this is referred to as destructive interference. Because the disturbances are in direct opposition to one another, the end result is a zero amplitude and no wavelength as the waves will cancel one another completely. In order for this to occur, the waves must be precisely identical and directly opposite to one another. This can happen to sound waves when stereo sound is used.

Most wave additions and subtractions are not so simple. A large wave with a long wavelength and a small wave with a short wavelength will add together and will create an odd, complicated wave that has some periodicity that can be described. This will be a combination of constructive and destructive interference.

Some waves will vibrate and will seem not to move directly. These types of waves can be seen in wave addition and subtraction of waves traveling in opposite directions. The resultant wave is described as a standing wave. Standing waves can be seen in organ pipes and with guitar strings; they look like quivering or vibration and will not appear to travel in any direction whatsoever. Nodes are the points where the string does not move and the wave disturbance is zero, while an antinode is the location of maximum wave amplitude. The frequency is related to the velocity of the wave.

The lowest frequency of this type of wave addition and subtraction is called the fundamental frequency. This type of thing can be seen in sound waves, in which the

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harmonics will be multiples of the fundamental frequency. The fundamental frequency is seen in figure 105:

Figure 105.

Taking this to another level, you can see that hitting two adjacent keys on a piano is an unpleasant experience because there will be the superposition of two waves of similar but not the same frequencies. There will be two waves that go in and out of phase. This will lead to a “beat” heard when the waves are the loudest. At any point in time, the wave amplitude x will be equal to the total amplitude multiplied by the cosine of the multiplication of 2 times pi times the frequency times the time. The amplitude at any point in time will be the amplitude of one wave added to the amplitude of another. The beat frequency will be the average of the frequencies of two separate waves. Piano tuners will listen until the beat goes away to zero frequency.

WAVE ENERGY All waves will have energy. Think just of earthquakes, which have energy that can shake entire villages and cities until they are destroyed. Ultrasounds have energy that treats muscle strains and lasers have energy that can destroy tissue. The energy of a wave is related to its amplitude. Waves are displacements that are resisted by a restoring force.

The larger the displacement, the larger the force needed to create the wave. A waves energy is directly proportional to the square of its amplitude so that work is equal to a force constant multiplied by the displacement squared.

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Another thing that affects the energy of a wave is the time. More energy is applied over time with waves being concentrated or spread out. This leads to the definition of the intensity of a wave, identified by the letter I, which is power per unit area. The SI unit for intensity is watts per meter squared. Other units commonly used include the decibel, which is 10-3 Watts per meter squared.

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KEY TAKEAWAYS •

Oscillation is the back and forth movement of an object that has elasticity.

•

There is a restoring force that tends to draw an object from a point of displacement to equilibrium.

•

Hertz is a term that describes the cycles per second of an oscillating object.

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Pendulums can behave like simple harmonic oscillators when operating at low arc lengths.

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Everything has a resonant frequency, which is based on its natural frequency.

This is how a certain frequency of sound can break a glass.

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Waves will have certain periods, frequencies, and amplitudes.

•

Waves can add or subtract from one another in constructive or destructive interference.

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The beat frequency is the average of two frequencies of waves heard together.

•

There are longitudinal and transverse waves, depending on the direction of their amplitude.

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QUIZ 1. What is the direction of a restoring force when a pipe made of plastic is deformed when held at a fixed point? a. Toward the fixed point b. At a tangent to the direction of the deformation c. In the direction of the displacement d. In the opposite direction of the displacement Answer: d. The direction of the restoring force will be in the opposite direction of the displacement. If the displacement of the pipe is toward the left, the restoring force will be directly opposite and directed toward the right.

2. What is the restoring constant for the deformation of an object based on when the object is deformed or forced to oscillate? a. The distance it is deformed. b. The substance it is made from. c. The distance from the fixed point. d. The force applied to the object.

Answer: b. The force constant for an oscillating object that is deformed is completely related to the substance it is made from. Some things will be very rigid and will have a great restoring force, while some things will be floppy and will have a small restoring force.

3. In simple harmonic motion, what is the relationship between the object’s oscillatory period, the mass applied, and the amplitude of the wave (distance of the force applied)? a. The period is proportional to the square root of the mass and is unrelated to the amplitude. b. The period is proportional to the mass and inversely proportional to the amplitude.

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c. The period is inversely proportional to the mass and proportional to the amplitude. d. The period is proportional to the square of the mass and inversely proportional to the amplitude.

Answer: a. The period is proportional to the square root of the mass and is unrelated to the amplitude. Remember that the period is the time for one oscillation.

4. What is the relationship between the frequency of the object in simple harmonic motion, the amplitude of the force, and the force constant? a. The frequency of the simple harmonic motion is proportional to its force constant and the amplitude of its force. b. The frequency of the simple harmonic motion is proportional to the square root of the force constant and unrelated to the amplitude. c. The frequency of the simple harmonic motion is inversely proportional to the square root of the force constant and its amplitude. d. The frequency of the simple harmonic motion is proportional to the square of the force constant and unrelated to the amplitude.

Answer: b. The frequency of the simple harmonic motion is proportional to the square root of the force constant but is not at all related to the amplitude of its force. What this means is that you can pluck a guitar string and will have the same sound when plucking it hard as when plucking it softly.

5. What happens when the damping of an oscillating system reaches critical damping? a. The velocity maximum of the system will increase. b. The equilibrium is achieved the fastest. c. The system will overshoot the equilibrium to the maximal degree. d. The system will gradually undershoot the equilibrium point.

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Answer: b. Critical damping will return the system to equilibrium as quickly as possible without overshooting. An underdamped system will oscillate through the equilibrium position and an overdamped system will move more slowly toward equilibrium than one that is critically damped.

6. What does the application of a force at a frequency that reaches an object’s resonance frequency do? a. Increase the period b. Increase the frequency c. Decrease the frequency d. Increase the amplitude Answer: d. Upon reaching the resonant frequency, the amplitude will be increased. This can be reached in different objects that have different resonant frequencies.

7. What is the velocity of the wave in a wave system? a. The peak to peak time period b. The peak to peak distance horizontally c. The distance from peak to peak vertically d. The distance traveled over a given period Answer: d. The velocity will be the distance travelled over a given period defined by T or the lambda divided by the period.

8. When it comes to the waves of the ocean, what will the time be between crests of the wave be in initials? a. T b. x c. lambda d. v Answer: a. The time between crests of a wave will be the period, which is also determined as the T of the wave. The initial x will be half the

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height from peak to trough, lambda will be the wavelength, and v will be the wave velocity.

9. When two identical waves have the peak of one happening at the trough of another, what happens to the resultant wave? a. The amplitude and wavelength will double. b. The amplitude will be the same but the wavelength will double. c. The amplitude and the wavelength will be zero. d. The amplitude will double and the wavelength will be the same.

Answer: c. The waves completely cancel one another out, which will be referred to as destructive interference. There will be no wave whatsoever.

10. When two waves of opposite direction come together, they will create periods where the wave amplitude is zero. What is this point on a wave? a. Node b. Antinode c. Loop d. Standing wave Answer: a. The node in a standing wave situation is where two waves have destructive interference such that there is complete cancellation of the two waves. The antinode is the point of maximal wave addition and maximal constructive interference.

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CHAPTER 13: ELECTRICAL CHARGES AND ELECTRICAL FIELDS This chapter introduces the physics of electricity by covering electric charges and electric fields. Static electricity is just one aspect of electricity that is well understood by anyone who touches an object and gets an electric shock. Also covered is the topic of electromagnetic force, which is a type of energy that applies to electrical fields. A natural part of the discussion is that of conductors of electricity and insulators of electricity, which are also a part of this chapter.

STATIC ELECTRICITY Static electricity is a common phenomenon. It is seen in dry weather when one touches another object or person and is the phenomenon of plastic wrap clinging. The term “electric” comes from the Greek word for amber, which is known to cause static electricity. Lightning is another phenomenon that is related to static electricity.

There are two types of charge in electricity, which are positive and negative electric charges. Like charges will repel each other, while unlike charges attract each other. The force between charges will decrease with the distance between them. What we know already is that negative charges have a predominance of electrons and positive charges have a dearth of electrons. This is borne out by studies of atoms and their charges.

The simple model of the atom will show a central positively-charged nucleus of protons as well as surrounding negatively-charged electrons. The smallest unit of negative charge is the electron, while the smallest unit of positive charge is the proton. A hydrogen atom will have one proton and one electron. Neutrons are also seen in the nucleus of the atom but will have a neutral charge. There are other charge-carrying particles that can be seen in nuclear decay phenomena and in cosmic rays but these survive only for a brief period of time.

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The charges of electrons and protons are the same in magnitude but will have an opposite sign. All charged particles are just multiples of these simple positive and negative charges. The symbol q is used to describe an electric charge in physics. The SI unit for charge is the coulomb or the letter C. The number of protons it takes to make a coulomb of charge is about 6.25 x 1018 protons or electrons.

From a molecular standpoint, there is no infrastructure to the electron. There is a substructure inside the proton, however. It appears that there are sub-particles called quarks inside protons that are believed to carry fractional charges. Quarks will have charges that are either -1/3rd or +2/3rds. At no time is it considered that charge can be created or destroyed. This is called the law of conservation of charge, which is similar to the law of conservation of energy and the laws of conservation of angular and linear momentum.

According to the law of conservation of charge, there is charge conservation at all times.

The opposite of an electron is called a positron or anti-electron. The total charge will always be zero. In the same vein, there is matter and antimatter, which means that, when matter is created in a particle accelerator, antimatter must also exist so that the charge is zero. When matter and antimatter are brought together, they completely annihilate one another, being converted into energy, which goes by the letter E.

CONDUCTORS AND INSULATORS There are some things that easily allow for the transfer of charges. Salty water will carry charge through it as will metallic substances. Electricity moves not through the transfer of positive charge but always through the transfer of negative charge or “free electrons”.

Any substance that has the ability for free electrons to flow through them will be called conductors. Superconductors are things that will allow the movement of charge without a loss of energy, which is not the case with conductors, which will lose energy because the electrons will collide with fixed atoms and molecules.

Salty water and other salt-containing liquids will conduct electricity because they contain ions. Ions are positively-charged or negatively-charged atoms or molecules in salt solutions. The anion will have a negative charge, while the cat-ion will have a

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positive charge. In all cases, even when electricity moves through a salt solution, there will be an equal number of anions and cations in the solution.

In addition, there are other substances that are considered insulators because they do not conduct electricity. This is because insulators will have fixed electrons and fixed protons; the electrons will not be able to move as easily through an insulator as is true of a conductor. Pure water is considered an insulator as is dry salt mixtures; however, the mixture of the two in solution is considered a conductor.

Charge can be passed directly touching positively-charged substances with negativelycharged molecules in electroscopes. It is not necessary, however, to transfer the electrical charge of something to something else through direct touching. The charge can be passed through close proximity as long as there is no insulator between the two.

This is called the induced polarization of neutral objects. Polarization is the separation of charges in any object that is overall a neutral object. There is no difference in charge across the total area just a difference in charge across the object. This is shown in figure 106:

Figure 106.

Neutral objects can be attracted to any charged object. Think of a plastic comb rubbed through the air that can pick up neutral objects that aren’t very heavy. This happens because a charged object can polarize a neutral substance, making it charged on one side

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of it so that it will be attracted to the charged object. Some molecules, like water, are polar in that they have a partial positive charge on the hydrogen molecule and a partial negative charge on the oxygen molecule. These molecules are said to have a separation of charge.

COULOMB’S LAW While we have been studying electrostatic forces, the existence of two charges, the attraction of charges and repulsion of charges, and the decrease in force with distance, we haven’t covered whether there is a mathematical formula for these phenomena or not. This, as it turns out, can be identified through Coulomb’s law. Coulomb’s law indicates the relationship between the magnitude of the force between two point charges separated by distance r. Figure 107 describes this relationship:

Figure 107.

What this means is that the electrostatic force will be a vector quantity that is expressed in Newtons. The direction of the force will be along the line that joins the two charges.

It will apply to attractive forces and repulsive forces. According to this equation, the electrostatic force between two subatomic particles is far greater than the gravitational force between two particles.

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ELECTRICAL FIELDS Any type of contact force, such as between a tennis racquet and ball, can be explained through interactions of forces of atoms and molecules that are in close proximity to one another. The force that explains this is Coulomb force. This must be contact force because the action of charges between molecules occurs over extremely close distances.

All objects are surrounded by a force field, however, small, that has the ability to act on another object a short distance away. This is how charged objects can act on other objects over a distance.

In this way, the Coulomb force field is the force field around any charged object in space.

Coulomb’s law can be rewritten to be the force being proportional to a point charge, indicated by the capital letter Q, acting on a test charge, indicated by the small letter q, over the distance between them squared. Figure 108 describes this:

Figure 108.

Coulomb’s force fields will still involve repulsive forces with like charges and attractive forces on unlike charges. The magnitude of the force will be greater if the charge on either one of the objects is greater. What this means is that the magnitude will be proportional to the degree of charge on either side. The force field is not unique to any point in space but instead depends on the charge magnitude of either object.

If the test charge is changed and the point charge is held the same, the electric field is the total force (the Coulomb force) divided by the test charge q. The electric field E will

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be a vector in the same direction as the force, which is held to be positive if there is attraction and negative if there is repulsion. The assumption is that the test charge is small compared to the total force. The SI units for electric field are Newtons per Coulomb. Because the test charge is small, the electric field will be a constant multiplied by the point charge divided by the radius squared.

As mentioned, the electric field is a vector, represented by a magnitude and a direction.

In identifying a force field, multiple forces are described over a space rather than single force vector. Again, it is assumed that the test charge is so small so that the force field is mainly determined by the point charge or capital Q. Figure 109 describes the force field of a point charge with a test charge responding to the point charge anywhere it is placed in the electric field:

Figure 109.

According to convention, the path of the field lines points away from a positive charge and toward a negative charge. Its magnitude is directly proportional to the number of field lines per unit area, because the area is proportional to the radius squared. This means the magnitude is inversely proportional to the square area involved. The closer the field lines are in an electric field, the stronger is the field. This measure of strength applies to all types of fields, including electrostatic, gravitational, and magnetic fields.

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The total electric field from multiple charges is the vector sum of the individual fields created by each charge. Figure 110 shows what an electric field would look like when multiple charges are involved in determining the electric field:

Figure 110.

The above situation is what an electric field looks like when a positive charge and a negative charge add their vectors to make a total electric charge. The field would be weaker between like charges as shown by lines being further apart, while when the field is greater, the lines would be closer together. The greater the distance from two like charges, the field becomes identical to what it would look like in a single, larger charge alone. At very large distances, there is no addition of the vectors because the distance is too far away.

To summarize field lines, these things about field lines are true: •

Field lines begin on positive charges and end on negative charges or at infinity for isolated charges (hypothetically).

•

The density of field lines (or how close they are together) depends on the magnitude of the charge.

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•

The strength of the field is proportional to the number of lines per unit square area perpendicular to the lines.

•

The direction of the electric field is tangent to the field line at any point in space.

•

Field lines can never cross one another.

A conductor is something that contains free charges that will easily move. When there is an excess charge on a conductor in a static electric field, the conductor will quickly respond to reach a steady state known as electrostatic equilibrium. The free charges will move until the field is perpendicular to the surface of the conductor. Any force parallel to the surface will not indicate the presence of an electrostatic equilibrium. Again, the direction of the vector will be from positive to negative and will be perpendicular to the charged conductor.

Conductors in an electric field will be polarized so that the charges will shift. The negative charge will go toward the arrow of the vector, while the positive charge in the conductor will go away from the arrow of the vector. The field will become stronger near the conductor but is not present within the conductor itself. This is seen clearly in figure 111:

Figure 111.

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According to the figure, the electric field is zero inside the conductor and the electrical field lines are perpendicular to its surface, ending or beginning on the charges on the surface of the conductor. Excess charge will reside entirely on the surface of the conductor. A uniform electric field can be created by having a field between two plates that are uniform in strength and direction. The excess charge will distribute themselves uniformly, producing field lines that are spaced uniformly and that will be perpendicular to the surfaces of the conductors.

There is a uniform electric field of about 150 Newtons per Coulomb directed toward the earth and surrounding the earth. This electric field is caused by the ionosphere, which is a layer of charged particles about 100 kilometers above the Earth’s surface. The ionosphere will be positive when the weather is good. When storm clouds gather, it creates an increase in the local electric field because the air is ionized, getting discharged in the form of lightning during a storm.

On uneven surfaces, there will be a clustering of charges on the sharpest points of the surface. Excess charges will tend to move off or on the conductor at the sharpest points.

Figure 112 shows the concentration of charge on an uneven surface with an increase in the charge at the sharpest point:

Figure 112.

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This brings us to the issue of a very pointy conductor, such as a lightning rod. They work best when they are pointed because they can produce an opposite charge to the large charge in a storm cloud, attracting the charge and bleeding the charge away from a building rather than allowing that charge to be picked up by. On the other hand, the conductor that is smooth with a large radius will prevent the leakage of charge into the air.

A Faraday cage is a metal shield that will enclose a volume. All electrical charge will reside outside the cage and not in the cage. It is used to prevent stray electric fields from the outside of the cage causing interference with sensitive instruments within the cage. During an electrical storm, it is better to stay inside a car than outside because the car body acts like a Faraday cage so that there is zero electrical field inside the cage. The same is true if an electrical wire falls on a car during driving.

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KEY TAKEAWAYS •

The charge on an electron is negative, while the charge on a proton is positive.

•

The electron and proton are the smallest units of charge in nature, although theoretically, quarks have a partial charge within a proton.

•

The SI unit of charge is the coulomb.

•

Charge is always conserved, according to the law of conservation of charge.

•

An electric field is generated from positive to negative from two charged objects.

•

A conductor will be polarized within an electric field but will have no electric field within it.

•

Two charged objects can produce an electric field, which is the sum of the force vectors between the two objects.

•

A nonuniform conductor will gather and discharge a charge when the conductor is at its pointiest.

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QUIZ 1. What is true of electrical charges and electrical forces? a. Positive charges attract each other with a force that increases with distance. b. Positive and negative charges attract each other with a force that decreases with distance. c. Positive and negative charges repel each other with a force that increases with distance. d. Negative charges attract each other with a force that decreases with distance.

Answer: b. Positive and negative charges attract each other with a force that will decrease with distance. There are just two types of charges, positive and negative with opposites attracting each other. The force of electricity between two things will decrease with the distance between them.

2. What is considered the smallest unit of charge in nature? a. One proton or one electron b. One proton c. One electron d. One coulomb Answer: a. The smallest unit of charge in nature is the charge on one proton or one electron, which have equal charges (although they are opposite in sign). It takes 6.25 x 1018 charges to make a coulomb of charge.

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3. What is the charge on a quark? a. Plus one b. Minus 1/3 c. Minus one d. Plus 2/3 or minus 1/3 Answer: d. The charge on the quark can be either plus 2/3 or minus 1/3. These are partial charges that together make up the plus one charge of a proton. Quarks are the sub-particles that make up the totality of a proton, although they have not truly been “seen”—only suspected to be there.

4. What is not considered conserved in the laws of physics? a. Forces b. Energy c. Charge d. Momentum Answer: a. Each of these are considered conserved, including that of angular and linear momentum with the exception of force and forces, which are not conserved in physics. From an electrical standpoint, electric charges are conserved.

5. What is the charge on an ion in physical chemistry? a. It is neutral b. It can be positive or negative c. It is negative d. It is positive Answer: b. An ion can be positively-charged or negatively-charged.

Anions, by definition, are negatively charged, while cations are, by definition, considered to be positively charged atoms or molecules.

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6. What is the overall charge on a polarized object? a. It depends on the other object that is polarizing it and the charge of the other object. b. It will be negatively charged. c. It will be positively charged. d. It will be neutral.

Answer: d. While a polarized object will have a partial positive charge on one side and a partial negative charge on the other side, it will be neutral overall even though it is polarized.

7. What are the SI units for the electric field? a. Joules b. Coulombs c. Joules per coulomb d. Joule coulombs Answer: c. The SI units for electric field are joules per coulomb as it is defined as the force divided by the test charge.

8. What is the direction and magnitude of an electric field charge? a. The direction is toward the positive charge and directly proportional to the area involved. b. The direction is toward the positive charge and inversely proportional to the area involved. c. The direction is toward the negative charge and directly proportional to the area involved. d. The direction is toward the negative charge and inversely proportional to the area involved.

Answer: d. By convention, the direction of an electric, magnetic, or electrostatic field charge is toward the negative charge and is inversely proportional to the area involved. When the lines of the electrostatic

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field are closer together, the field is considered stronger because the area is bigger.

9. What is not true of an electric field that has a conductor placed within it? a. The field within the conductor itself will be zero. b. The field extends outward from the positive charge of the conductor. c. The field will polarize the conductor. d. The field lines will be parallel to the conductor.

Answer: d. The field lines will be zero in the conductor and will extend outward from the positive charge of the conductor or toward the negative charge of the conductor. The conductor will be polarized by the electric field; however, the field lines will emanate perpendicular to and not parallel to the conductor itself.

10. What creates the negative electric charge around the earth? a. Storm clouds b. The moon c. The ionosphere d. The sun Answer: c. The ionosphere is a layer of positive charge about 100 kilometers around the earth, causing a field with the earth as the negatively-charged object.

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CHAPTER 14: ELECTRIC POTENTIAL AND ELECTRIC ENERGY The focus of this chapter is to extend the understanding of electricity to include electric potential and electrical energy. It introduces aspects of electricity such as voltage and the storage of electrical energy by capacitors. In this chapter, you will find that electrical energy and voltage are not the same thing because small batteries can have the same voltage as large batteries but will not create the same amount of electrical energy. In this chapter, the actual use of electricity in everyday electrical situations is covered as well.

ELECTRIC POTENTIAL ENERGY Whenever a free positive charge, as indicated by the letter q, is accelerated by an electrical field, it is given a particular amount of kinetic energy similar to an object accelerating in a gravitational field. The electric potential energy will be converted into kinetic energy. In such cases, the change in potential energy will be the same as the kinetic energy. The work done will be the negative of the potential energy. The work done will be independent of the path taken if things like friction aren’t taken into account.

Potential energy has units of joules. Whenever work is done, the PE or potential energy will be negative and the kinetic energy will be positive. The work, too, will be positive. If you’ll remember, force will equal the charge times the electrical energy so that the work and change in potential energy will be proportional to the charge. This introduces the concept of electrical potential, identified by the letter V. This electrical potential is the potential energy per unit charge. What’s important isn’t the actual electrical potential but it is the change in potential or potential difference between two points.

Since the difference in potential energy is more important than the actual potential energy, its units are joules per coulomb or volt, in which one volt is the same thing as one joule per coulomb. The term used in common terms is voltage. Voltage isn’t the 242

actual potential energy but the difference in potential energy between two points. Think of a battery having two terminals with the potential difference being the voltage. One point is arbitrarily given a zero-voltage point. This is similar to identifying zero at sea level. The change in potential energy is the charge times the change in voltage of a system.

So, you can see that voltage is not the same thing as energy. It is instead the energy per unit charge. This means that two different batteries can have the same change in potential energy between the terminals but will differ greatly in the amount of energy that can be given because of the storage capacity of the different batteries.

When a battery moves an electric charge, it will be the negative electric charge that is moved. The flow of electricity will be from the negative terminal to the positive terminal. A car battery and a motorcycle battery both are 12-volt batteries but carry different charges, making their energy potential much different.

When work is done by the battery, the change in potential energy is negative and the potential energy of the battery will be decreased by the movement of charge. When it comes to potential, the potential energy of the positive terminal is said to have a higher voltage. Inside the battery, both positive and negative charges will move.

The actual energy per electron is very small when it comes to macroscopic electricity.

Even this small amount, however, can be very effective in biological systems. X-rays for example, will do damage to human tissue and electrons can change voltages across biological membranes.

The electron will be accelerated between two charged plates, given a kinetic energy that essentially moves the electron “downhill”. In such cases, joules will consist of coulombvolts. In small systems, there is a term called the electron volt or eV. This is the energy given to an electron that is accelerated through a potential difference of one volt. One electron volt is considered 1.60 x 10-19 Joules. This is used instead of voltage in submicroscopic processes.

Because there is a conservation of energy, this is true of electrical energy. Mechanical energy as you know is the sum of the kinetic and potential energy of a system, which will be a constant. Any loss of potential energy is an increase in kinetic energy. The 243

potential energy will be the same thing as the electric potential energy. Electrons have a small mass and therefore, it doesn’t take much energy to produce great electron speeds.

Next, we will discuss the relationship between energy and voltage. A uniform electric field E is produced by placing a potential difference or voltage across two parallel metal plates. The change in voltage is not a vector but is instead a scalar quantity, while an electric field E is a vector quantity. The change in voltage is the difference in voltage between that of one plate and another. Figure 113 shows a uniform electric field and the charge that passes between them:

Figure 113.

In figure 113, the work done is the charge multiplied by the change in voltage between two plates of a uniform electric field. The voltage in a uniform electric field only will be the same as energy multiplied by the distance, so to rearrange the equation, the electrical energy in a uniform electric field Is the voltage divided by the distance. In such a case, one newton per coulomb equals one volt per meter.

The faster voltage decreases over a distance, the greater is the electric field. In such a case, the energy is equal to the change in voltage divided by the change in time 244

(although the sign will change because the potential energy is downhill). The electric field is therefore the gradient or slope of the electric potential, which will be downhill from an electrical standpoint.

One can define the electric potential of a point charge. In such cases, the electric potential of the point charge will be a constant multiplied by the charge and divided by the radius from the point charge. The constant is 9 x 109 Newton meters squared per coulomb squared. This means that the voltage decreases with distance, while the electrical field energy will decrease with the square of the distance. The voltage is considered to be taken as zero as the distance approaches infinity. The electric potential of a uniform sphere will be the same as what it would look like from a single point in the center of the sphere.

EQUIPOTENTIAL LINES Voltages can be described pictorially in the same way that electric fields were discussed in the previous chapter. As you can remember, the electric field lines radiate outward from a positive charge and end on negative charges. Equipotent lines can be described in two dimensions and equipotent surfaces can be described in three dimensions. These are places where the electric potential is constant. For a point charge, the equipotent lines are lines that connect potentials that are the same distance from the charge as is seen in figure 114:

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Figure 114.

Equipotent lines are lines in which there is no work necessary to move the charge anywhere along the line. Work is necessary only to move a charge from one equipotent line to another. These lines will always be perpendicular to an electric field or to E. The potential will decrease as one spreads out from the central point. Any motion along an equipotent line must be perpendicular to the electric field.

The electric field of a conductor must be perpendicular to the conductor. This means that the conductor represents an equipotential surface as long as the situation is static.

There is no voltage difference across the surface of a conductor. To fix the conductor voltage at zero volts, the conductor can be grounded to the earth in a process called “grounding”.

Figure 115 shows equipotent lines between positive and negative poles. These will look different, depending on whether the charges are negative and negative (and thus are repelling each other) or when the charges are positive and negative (and thus are attracting each other):

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Figure 115.

As you can imagine, the equipotential lines are evenly spaced and parallel to one another in the case of a uniform electric field and parallel conducting plates. In such a case, the voltage potential at the positive plate will be the highest and the voltage potential at the negative plate will be at the lowest.

CAPACITORS A capacitor is a device that is used to store electric charge. Capacitors do a great deal in electric systems by filtering static in radio reception and will store energy in heart defibrillators. These commercial capacitors will have two conducting parts that are close together but do not touch. Often, an insulator is used between the two plates in order to create the separation.

While the capacitor’s overall charge is, of course, zero, there will be split of charge between two conducting plates. A battery can be used to charge the capacitor into positive and negative plates. The amount of charge that can be stored depends on the capacitor’s size and on the voltage applied. The electric field strength is directly proportional to the overall charge of Q. The greater the charge applied by a battery, the greater the amount of charge it can hold.

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So, what is capacitance? This is proportional to the charge that is stored in the capacitor. The charge stored in the capacitor is the capacitance multiplied by the applied voltage. It is defined as the amount of charge stored per volt applied. The unit of capacitance is the Farad, which goes by the initial F. One Farad is one coulomb per volt. A one-farad capacitor is one that will store one coulomb when one volt is applied.

A one-Farad capacitor would be huge and most are in the range of 1 x 10-12 Farads to 1 x 10-3 Farads.

The parallel plate capacitor will have two identical conducting plates separated by a distance d with no material between the plates. When voltage V is applied to the capacitor, an amount of charge Q is held by it. Capacitance depends on the surface area of the two plates and by distance between them. The bigger the plates, the more charge can be stored. The force between the charges will decrease with distance. What this means is that, the larger the surface area of the capacitor and the closer they are to one another, the more charge can be stored.

The capacitance can be defined as a constant multiplied by the area divided by the distance. The constant is called the “permittivity of free space”, which is 8.85 x 10-12 Farads per meter. This is such a small number because a farad is a very great number.

The problem with capacitors is that they have to be big in order to work unless some modifications are done. Some modifications include using plates that can be rolled up so that they can be bigger and can use up less physical space. Another modification is to introduce a dielectric, which is an insulating material between the plates of the capacitor in order to allow the distance to be as small as possible. These can hold greater electric fields that can be gotten through air without breaking down.

The other benefit of using a dielectric is that the equation doesn’t use the permittivity of free space constant but uses the dielectric constant also referred to by the letter k. This gives the capacitor a greater degree of capacitance. The dielectric constant for air is nearly one, while for a vacuum, it is set at one exactly. Other dielectrics will have different dielectric strengths, above which the material will break down and conduct.

The dielectric strength is the maximal electric field above which the dielectric begins to break down and conduct electricity. The more easily a dielectric is polarized, the greater

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is its dielectric constant. Water is highly polar as a molecule and therefore it has a dielectric constant of 80, which is relatively high. It will increase the Coulomb force between the two plates. The dielectric constant is the ratio of the electric field in a vacuum to that in the dielectric material. It is highly related to the polarizability of the material.

Some molecules are polar and will have a separation of charges within the molecule.

Hydrogen ions are partly positive, while oxygen molecules are slightly negative. This molecule and similar molecules have higher dielectric constants because they are more polarizable. In a sense, water has an electric field and a charge separation. It provides a screening or shield of the electric fields in biological systems. Figure 116 shows the polar water molecule:

Figure 116.

What’s true of capacitors is that they can be linked together in a variety of situations in physics. Many capacitors together can act like a single capacitor with the total capacitance depending on the capacitance of the different capacitors. These can be done in series and parallel or in combination of both.

Remember that the capacitance is the charge divided by the voltage. The combination of capacitors in series resembles a single capacitor with an effective plate separation greater than that of the individual capacitors alone. Large plate separation means a

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smaller capacitance. This will mean that capacitors in series doesn’t add to the capacitance of a system but subtracts from it. One divided by each capacitance and added together leads to one divided by the total series capacitance when in series.

Capacitors in parallel have the same voltage applied to each capacitor with conductors being equipotential from each other. These capacitors have the same charges on them as they would if they would have been connected individually. The total charge is the sum of the individual charges. In such cases, the capacitance is the sum of the capacitance of each capacitor. This leads to the appearance of a much larger capacitor. If there is a combination of parallel and series capacitors, the capacitors in series are added as in the equation, while capacitors in parallel are added together so the sum is given.

Figure 117 shows how to add capacitors in series and in parallel. When adding capacitors in series and in parallel, the C total will be the C in series plus the C in parallel added together:

Figure 117.

In reality, capacitors are used to store energy in things like defibrillators, calculators, and camera flash lamps. This is electrical potential energy of the system. Capacitors start with zero voltage and go up to a full voltage when charged. This makes the change in voltage the same thing as the total voltage on the capacitor. The average voltage on 250

the capacitor during the charging process is half of the total voltage so the energy stored will be the total charge multiplied by the voltage divided by two. Figure 118 shows several ways to describe the energy stored in a capacitor:

Figure 118.

In this case, the energy stored by a capacitor is in Joules and is proportional to the square of the charge while being inversely proportional to the capacitance of the capacitor. The equations in figure 118 are simply rearranging what we know of the values of total voltage, total charge, and capacitance.

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KEY TAKEAWAYS •

One can look at electricity from the standpoint of electrical potential energy and the electrical kinetic energy.

•

Electricity in a battery goes from a negative terminal, through a light bulb, for example, to a positive terminal.

•

The electrical potential of a point charge will decrease with distance from the charge.

•

An equipotent line is a line that has equal potential across the entire line so it takes no work to transfer the charge from one point on the line to any other point on the line.

•

A capacitor is a device that will store charge by splitting it between two plates that are close together and often separated by a dielectric or insulator.

•

Capacitors can be placed in parallel or in series, with different results when this occurs.

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QUIZ 1. In identifying a volt as a change in electrical potential energy, what are its units? a. Joules b. Joule-coulombs c. Coulombs d. Joules per coulomb Answer: d. The units of the volt are in joules per coulomb; these represent the change in potential energy as it relates to the charge of an electrical object.

2. In electricity, what is the best description of the voltage of something? a. A measure of a system’s potential energy b. A measure of its difference in potential energy between two points c. A measure of the potential energy and kinetic energy of the system d. A measure of the potential energy minus the kinetic energy of the system Answer: b. This is a measure of the difference in potential energy between two points with one point given an arbitrary zero voltage point.

3. What is the relationship between the electric potential of a point charge and the distance from the charge? a. The electric potential is proportional to the distance from the point charge. b. The electric potential is proportional to the square of the distance from the point charge. c. The electric potential is inversely proportional to the distance from the point charge. d. The electric potential is inversely proportional to the square of the distance from the point charge.

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Answer: c. The electric potential is inversely proportional to the distance from the point charge. This will lead to zero electric potential as the distance approaches infinity.

4. What is not true of equipotent lines around a point charge? a. The potential will be greater near the point charge. b. No work is required to send a charge anywhere along an equipotent line. c. Work is required to send a charge between equipotent lines. d. An equipotent line will be parallel to the electric field lines.

Answer: d. The equipotent line is a line that has the same potential from a point charge. No work is necessary to send a charge anywhere along an equipotent line. Work is, however, required to send a charge between equipotent lines. These equipotent lines will be perpendicular to the electric field lines.

5. What is a capacitor in electricity? a. It is a device that is grounded to zero. b. It is an insulated conductor. c. It is the same thing as a battery. d. It is a device that will store charge.

Answer: d. A capacitor is a device that will store charge by accepting charge that is separated between two plates.

6. What determines the amount of charge that a capacitor can hold? a. The voltage applied and the size of the capacitor b. The distance between the two plates c. The insulation between the two plates d. The presence of repelling plates Answer: a. The voltage applied to a capacitor and the size of the capacitor are directly related to the amount of charge that a capacitor can hold.

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7. What is not true of a dielectric in a capacitor? a. It will allow for an increased diameter between the two charge-separating plates. b. It is an insulator that can be used between the two charge-separating plates. c. It leads to a dielectric constant instead of a permittivity of free space in determining the capacitance. d. It increases the capacitance of a capacitor.

Answer: a. A dielectric is an insulator that can be used between the two charge-separating plates that can increase the capacitor’s capacitance.

It will allow for a decreased diameter between the plates.

8. What is the relationship between the dielectric constant and the dielectric strength? a. These are essentially the same thing in different units. b. The dielectric constant is based on that of a vacuum, while the dielectric strength depends on experimental evidence. c. These two values are multiples of each other. d. These two values are inversely proportional to one another.

Answer: b. The dielectric constant is arbitrary and is set with a vacuum being 1. The dielectric constant is the amount of voltage that can be applied before the substance begins to break down.

9. What happens to the capacitance of four capacitors connected in series? a. The capacitance will be the average of the four capacitors. b. The capacitance will be less than that of a single capacitor alone. c. The capacitance will be the capacitance of each of the capacitors multiplied by one another. d. The capacitance will be the capacitance of each capacitor added together.

Answer: b. Because the inverse of the capacitances added together will be the inverse of the capacitance total, the total capacitance will be less

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than the capacitance of one of the capacitors alone. It effectively creates a capacitor that has a very wide distance between the plates, which makes a less effective capacitor.

10. What happens to the capacitance of four capacitors connected in parallel? a. The capacitance will be the average of the four capacitors. b. The capacitance will be less than that of a single capacitor alone. c. The capacitance will be the capacitance of each of the capacitors multiplied by one another. d. The capacitance will be the capacitance of each capacitor added together.

Answer: d. When capacitors are aligned in parallel, the capacitances of each capacitor are added together to make the total capacitance of the system.

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CHAPTER 15: ELECTRIC CURRENT AND CIRCUITS This chapter covers the topics of electric currents and circuits. Electric current is defined as the movement of a charge from one place in another over a period of a certain time. Such a thing has the capacity to do work. This then gets into Ohm’s law as it applies to electrical resistance and to the subject of circuits. There are AC current situations and DC current situations, which are things that many have heard about but will now understand from the perspective of physics.

ELECTRIC CURRENT Electric current is the rate at which charge flows through a system. Large currents are necessary to start large things and small currents are required to start small things. The basic definition of electric current, which goes by the letter I, is the change in charge divided by the change in time. The SI units for electric current are amperes, which are coulombs per second. Fuses and circuit breakers are in amperes as are electrical appliances.

In a simple circuit, there will be a battery, a conducting path, and a load, which is referred to as a resistor. The direction of flow is from positive to negative in a conventional current, which is what the flow would be if a positive charge were flowing, even though negative charges are the ones that flow through metal wires. In ionic solutions (such as with salt water), both positive and negative charges will flow. With Van de Graaff generators as seen in figure 119, it is positive charges that flow:

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Figure 119.

The current flows in the direction of the electric field vector, which is the opposite of the flow of electrons. Unlike static electricity, where the conductor in equilibrium cannot have an electrical field, conductors that carry current have an electric field and are not considered to be in static equilibrium. It takes an electric field to supply the energy that moves charges. Electrons do not bump into one another during flow but will flow due to mutually repulsive electrostatic forces.

Electrical signals are known to move very quickly. Think of phone conversations throughout the world that do not have a noticeable delay. The speed of electrical currents is on the order of 108 meters per second. This is much smaller than the speed of light. Individual charges move very slowly (at 10-4 meters per second). How can this be?

This is because the forces between charges act rapidly at a distance. Negative charges push against one another in a much faster way than the actual movement of charges.

This passes the signal on rapidly. In a sense, electricity involves an electrical shock wave from high density to low density and not the flow of a single electron alone.

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A good conductor will have large numbers of free electrons in them. The actual distance an individual electron can flow before bumping into something is small, resulting in the random flow of electrons. There is, however, an electric field in the conductor that causes the electrons to drift in a direction opposite to the field. The drift velocity Vd is the average velocity of the free electrons (or other charges). This velocity is quite small.

The density of the charges will be inversely proportional to the drift velocity. Low density will increase the drift velocity and the flow of electrons.

Things that are good electrical conductors are also good conductors of heat. This is because these electrons can also transport thermal energy. The collisions of electrons will transfer energy to the atoms of the conductor. This will increase the temperature.

An exception to this is the superconductor, which can have a steady flow of electrons without a continued energy supply. This heat energy will increase the temperature of a lightbulb filament, allowing the light to turn on.

The number of free charges per unit volume is given the symbol n. It depends on the material. The current can then be described as the charge on an electron multiplied by the density of electrons, multiplied by the volume and divided by the time. The x or distance divided by time is the drift velocity. Figure 120 shows the current as described by the density of charge:

Figure 120.

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Remember that the drift velocity is not the same thing as electron speed as they flow much faster than the actual drift velocity. In addition, not all electrons are free to drift as some are overly attracted to atomic nuclei. In metals, many electrons are “shared” by nuclei so they have a chance to flow. These are the outer electrons that aren’t so highly attached to nuclei. These free electrons will accelerate when an electric field is applied.

The will collide with atoms in a lattice, generating thermal energy. This type of situation will allow for electron flow, which isn’t the case with insulators.

OHM’S LAW So, what actually drives current? It takes a battery or a wall outlet to do this by creating a potential difference between one aspect of the system to another aspect. The potential difference created forms an electric field, which exerts a force on the charges, causing a current to happen. According to Ohm’s law, the voltage applied will be directly proportional to the current. This is an empiric law and not what happens in real life.

If voltage drives current, there are things that cause an impediment to current. This is the resistance to flow, which is similar to air resistance and friction. This resistance goes by the initial capital R. It involves the collisions of atoms and molecules that will limit the current. A correlate of Ohm’s law is that the current is inversely proportional to the resistance. In putting this together, you get the entirety of Ohm’s law, which is that the current in amperes is equal to the voltage divided by the resistance. Ohmic substances are those that have Ohm’s law applied—like good conductors, for example.

An object with simple resistance is called a resistor, even if the amount of resistance associated with it is small. The Greek capital letter omega is the unit for resistance, called the ohm. The units for ohm are volts per ampere. Resistors themselves can have resistance within them but this will be included in the total resistance of the resistor.

Figure 121 shows a circuit with a resistor but you should recognize that resistance is seen in all parts of the circuit:

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Figure 121.

Resistances can range from extremely small numbers of ohms to very large numbers of ohms. Ceramic insulators have 1012 ohms, while copper wires have resistances of 10-5 ohms. Superconductors are nonohmic because they do not have any resistance at all.

Resistance is related to the shape of an object and what it is made of.

The phrase IR drop refers to the voltage drop across a resistor. If voltage drops across a resistor, this is referred to as the IR drop, which is similar to the drop in fluid pressure that occurs when a pipe has resistance to flow. There is going to be conservation of energy so that, when a resistor has an IR drop, the energy will be dissipated as heat energy. The voltage applies energy that is dissipated as heat. The energy supplied by the voltage source and the energy converted by the resistor will be equal. The IR drop will equal the voltage output in a simple circuit.

ELECTRIC RESISTANCE AND RESISTIVITY The resistance of an object, as mentioned, depends on the material it is made of and its shape. In a simple cylinder, the resistance is proportional to its length, similar to that of flowing water through a pipe. The resistance, too, will be inversely proportional to the cross-sectional area A of the cylinder. The equation is that the resistance is equal to the resistivity multiplied by the length and divided by the area. Figure 122 is the relationships seen in a cylinder that has resistance:

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Figure 122.

The resistivity is an intrinsic property of the material, independent of the shape or size of the object. Objects can be conductors, semiconductors, and insulators, based on their degree of resistivity. Most resistors will have fixed atoms that don’t allow the flow of electrons. Semiconductors have an in-between status, which makes them important in modern electronics.

Resistance varies by temperature. Some superconductors will have zero resistivity at low temperatures. Resistivity increases with increasing temperature in the case of conductors. This is because atoms will have more collisions at higher temperature so that resistivity will be higher. Over a small increment of temperature, there is a temperature coefficient of resistivity, which helps define the difference in resistance with temperature.

Certain alloys will have limited temperature dependence and will have a low temperature coefficient of resistivity. This coefficient will be high and positive when the resistance is temperature dependent and increases with temperature. There may be a negative coefficient of friction, in which the resistance decreases with temperature. This is true of semiconductors. Semiconductors will be more conductive at higher temperatures (and will have low resistivity) because there will be freer electrons at a higher temperature. The coefficient of resistance will not be linear and consistent over greater ranges of temperature.

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ELECTRIC POWER AND ENERGY Power in electricity is the rate of energy conversion or electric power. Lightbulbs have power ratings in Watts, like 25-watt bulbs and 75-watt bulbs. Because the voltage is the same, the resistance of a 75-watt bulb is lower than that of a 25-watt bulb.

Power is the rate at which energy is moved or the potential energy divided by time or the charge multiplied by the voltage divided by time. Because the current is the charge over time, the power is equal to the current times the voltage. The units of electric power are watts or joules per second. In some applications, power can be described as voltamperes or kilovolt-amperes. Figure 123 shows the relationship between power, voltage, resistance, and current:

Figure 123.

What can be determined is that, the lower the resistance connected to a voltage source, the greater the power that is delivered. It means that, when the voltage is doubled to a 25-watt bulb, its power will quadruple, which will probably burn it out.

When it comes to the cost of electricity, the more one uses appliances for the longer period of time, the more electricity will cost. Power is energy divided by time so energy used by a device is its power over a period of time. The energy units on an electric bill is kilowatt-hours, which is consistent with what energy is. You are paying for energy and not necessarily for wattage or current. One kilowatt-hour is equal to 3.6 x 106 Joules of energy. You can reduce an electric bill by changing from incandescent bulbs to

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fluorescent lights (like compact fluorescent lights or CFLs) or white LED lights, which use up less energy and last longer (but often cost more upfront).

ALTERNATING CURRENT AND DIRECT CURRENT Direct current involves the flow of electric charge in just one direction. This is not the kind of electricity seen in common electrical situations, which involve alternating current. Alternating current or AC is the periodic reversal of direction. This will be a sinusoidal flow of electricity that is typical of the power sources one sees in homes and businesses. In such cases, the electrical voltage will flow in a wave-like pattern that has a certain frequency in Hertz. The voltage will be in phase with each other.

In an AC current, the light actually flickers at around 120 times per second; however, you cannot detect that. Sometimes, it can be detected with a fluorescent bulb. The power will always be fluctuating with the power equal to the amps times the voltage.

There will always be power but the current changes direction with an average power being one-half multiplied by the voltage multiplied by the amps. The RMS current and the RMS voltage are the root-mean-square current and the root-mean-square voltage, respectively. These are averages that are quoted when talking about electricity and electrical items you normally think of.

Why is AC used rather than DC? Power is sent from long distances—from large electric plants to one’s homes and businesses. It’s important that the energy losses get minimized during transfer. High voltages can be transmitted with much smaller power losses than low voltages. Because one doesn’t need such high voltages in the home, it is decreased, partly for safety reasons. It is much easier to increase and decrease AC voltages than DC voltages. This is why AC voltages are used. Transformers will step up the voltages at the power plant and decrease them at the point of use.

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CIRCUITS Most circuits will have more than one component, called a resistor, that limits the flow of charge in a circuit. The limit on the charge’s flow is called the resistance. Resistance, like capacitance, can be done in series and in parallel. This is shown in figure 124:

Figure 124.

Resistors in series have the flow of current passing through each resistor in sequence.

The voltage drop in such a case will be additive so the total resistance will be the sum of all of the resistances. Each resistor will dissipate its own amount of energy; however, the total energy will be conserved in the system because of the conservation of energy laws.

The same current will flow through each resistor in series but the individual resistors don’t get the total source voltage. Instead, they divide the voltage.

Resistors in parallel have a different ability to resist the current. Each resistor will have the full voltage of the source applied to it. This is the case of the resistors for different

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parts of an automobile or different parts of the electricity in one’s home. It is the case with circuit breakers, for example. Each resistor will have the full voltage.

Because of the conservation of charge, the current will be the sum of the different currents through each resistor. According to this, the inverse of the total resistance is equal to the sum of the inverses of the resistances, which is similar to capacitors. This means that the resistance in total will be less than the smallest of the individual resistances, increasing the total current to the resistors. Parallel resistors do not each get the total current; they divide it.

In some circuits, there are resistors in parallel and in series. Wire resistance itself is in series, while other resistances are considered to be in parallel. There is a certain resistance that is gotten from the wires themselves that must be one resistance calculated. This must be added to the resistances of those resistors in parallel. In some cases, the wire resistance is negligible but, in other cases, the wire resistance cannot be negated. With worn or long extension cords, the resistance in the wire might be great, reducing the voltages available to the different plug-ins.

ELECTROMOTIVE FORCE There are many different types of voltage sources, including many different types of voltages. There is wind energy, nuclear energy, and solar energy. Each of these will create a potential difference and can supply a current. The potential difference will create an electric field that causes this current. This is referred to as electromotive force, even though it isn’t a force at all. Instead, it is a certain type of potential difference in electricity. The units of EMF are volts, which is the potential difference of a source when no current is flowing. The EMF will decline as the battery is depleted. The output voltage is referred to as the EMF. A larger truck battery that is twelve volts can deliver more current than a 12-volt motorcycle battery. The EMF is the same with both of them but, because of size, there is a smaller internal resistance called r in the larger battery. The internal resistance will block the flow of current from within the source. As a battery is depleted, the internal resistance will increase but it is more complex than that. For example, the internal

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resistance of rechargeable nickel-cadmium cells, depends on how many times and how significantly the batteries have been depleted.

The voltage output of a device is measured by measuring the potential difference across its terminals. This is called the terminal voltage. It consists of the EMF minus the current and internal resistance multiplied by each other. This means that, if the current is high or the internal resistance is high, the terminal voltage will be decreased. If the internal resistance becomes significant, the terminal voltage will be diminished, which is the case when the battery dies down.

Battery testers will actually test internal resistance. Because they use small load resistors to intentionally draw a current from the battery, they check to see if the terminal voltage has dropped below an acceptable level. If the battery is weakening, the internal resistance will be high and the terminal voltage will be low. Battery chargers will pass a current opposite to the current they supply. The voltage output of the battery charge must be greater than the EMF of the battery in order to reverse the current going through it. It replenishes its chemical potential.

Batteries for toys and flashlights are set up in series in order to produce a larger total EMF. They must be put in the right order or the total EMF, which is the sum of the EMFs of each battery, will not add up and the batteries will not work. The disadvantage is, however, that their internal resistances will add up. This makes a series of batteries less effective than one larger battery even though their EMFs will be the same as a larger battery.

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KEY TAKEAWAYS •

The ampere is the unit for electric current.

•

In drawing a circuit, the direction of the current is from the positive terminal to the negative terminal, even though this isn’t the flow of electrons in a typical electrical circuit.

•

The flow of electricity depends on the potential difference between two terminals.

•

AC current is alternating current, while DC current is direct current. AC current will increase and decrease with more ease than is the case with DC currents so this is why it’s used in homes and businesses.

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Resistors can be anything that takes away the current and results in an IR drop or voltage drop across it.

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Resistors in parallel operate differently than resistors in series.

•

The EMF is the potential voltage across two terminals if no current is applied to it.

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QUIZ 1. What are the units for the ampere in physics? a. Coulombs per second b. Joules per second c. Charge per meter squared d. Volts per meter Answer: a. The units for ampere involve the change in charge over the change in time, which are coulombs per second.

2. In any electrical situation, which charges flow and in which direction? a. Only negative charges flow from negative to positive. b. Only positive charges flow from positive to negative. c. Both negative and positive charges flow from their highest energy level to the lowest energy level. d. Either positive or negative charges, or both can flow, depending on the electrical situation.

Answer: d. In metal electrical circuits, only negative charge flows. In ionic and biological systems both negative and positive charges flow, while the flow in van de Graaf generators, is of positive charges only.

3. When we think of flow of electrons, what is true of the density of electrons and their drift velocity? a. The density will be directly proportional to the drift velocity. b. The density will be inversely proportional to the drift velocity. c. The density will be inversely proportional to the square root of the drift velocity. d. The density will be unrelated to the drift velocity.

Answer: b. The density of electrons will be inversely proportional to the drift velocity in that, the greater the density of electrons, the more the

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electrons will bump into one another and will have a decreased drift velocity, which is the velocity of the electrons themselves.

4. According to Ohm’s law, what is true of what drives the current of an electric flow? a. The potential difference or voltage applied b. The kinetic energy that is innate to electrons c. The electromagnetic repulsion of electrons d. The drift velocity of an electron in a conductor Answer: a. The potential difference between two places or the voltage applied is what drives the current of electricity.

5. What system will have the greatest numbers of ohms associated with it? a. Ceramic tube b. Copper wire c. Water d. Superconductor Answer: a. Ceramic does not conduct electricity much at all so this would have the greatest resistance or the greatest number of ohms associated with it. Resistance is based on the shape of an object as well as on its substance.

6. What is referred to as the IR drop when a current travels through a resistor? a. This is the loss of energy from the system as the electrons flow. b. This is the loss of voltage across a resistor. c. This is the kinetic energy increase across a resistor. d. This is the loss of thermal energy across a resistor.

Answer: b. The IR drop is the loss of voltage across a resistor. It is the change of kinetic energy of the circuit to thermal energy in the resistor.

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7. What is the unit for power in electricity? a. Joules b. Amperes per second c. Watts d. Watts per second Answer: c. Watts are joules per second, which is the unit for power in electricity.

8. What are you paying for when it comes to your electric bill? a. Current used b. Power used c. Voltage used d. Energy used Answer: d. You are paying for the energy used, which is in kilowatthours. This equals to the number of joules used with the relationship being 1 kilowatt-hour equaling 3.6 x 106 Joules of energy.

9. What will the total resistance be when two resistors exist in series? a. It will be equivalent to the resistance of the highest resistor and not related to the lowest resistor. b. It will be equivalent to the average resistance of the resistors in series. c. It will be resistances added together. d. It will be the resistances multiplied together.

Answer: c. The total resistance experienced when resistors are in series is equal to the sum of the resistances of the two resistors. The resistors will receive the same current but will split the voltages between them.

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10. When resistors in an electrical system are in parallel, what is the total resistance of the system? a. It will be the sum of the resistances together. b. It will be the resistances multiplied together. c. It will be the average of the resistances together. d. It will be less than the resistance of the smallest resistor.

Answer: d. The inverse of the total will be the additions of the inverses of the individual resistances. What this means is that the total resistance will be less than the resistance of the smallest resistor.

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CHAPTER 16: MAGNETISM This chapter focuses first on magnetism and then on the relationship between magnetism and electricity. Magnetism is common in nature and explains many things related to what we see in nature, such as the magnetic poles on Earth. Magnetism can cause electrical currents to be generated, which is also discussed as part of this chapter.

The chapter also brings into focus the topic of electromagnetism and electromagnetic waves. You will see that there is a vast range of these types of waves that go from those that heat food (which are microwaves) to waves that are much higher in frequency than those a person can see (which is the narrow spectrum of visually-seen electromagnetic waves).

MAGNETS Most people know that magnets attract iron, such as those seen in many metallic objects, and magnets will either attract or repel other magnets. Magnets on earth have two poles. One pole points toward the magnetic north pole of the earth, while the other pole points toward the magnetic south pole of the earth. Like poles always repel like, while unlike polls attract each other. This is identical to what is seen in electrostatics.

The difference is that it is impossible to separate the north and south poles in the way that positive and negative charges can be separated in electromagnetism. Cutting up a magnet will create small magnets that still have north and south-seeking poles—just smaller ones.

FERROMAGNETS Only certain elements will have strong magnetic effects. These include iron, cobalt, gadolinium, and nickel. These are ferromagnetic elements. There are other elements that have weak magnetic effects, which can only be detected using sensitive instruments.

These substances will not only be attracted to magnets, they can be magnetized themselves to have north and south poles. This permanent magnetization can occur

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when heating and then cooling an unmagnetized piece of iron in a magnetic field, for example, or by tapping the iron with a strong magnet.

In each magnet situation, the magnet has individual atoms aligned. The individual atoms act like a tiny bar magnet with the positive and negative aspects of each atom attracting each other. Heating and cooling or “tapping” will align the atoms completely in a permanent way. Each atom acts like its own magnet within a larger magnet.

Increased heat allows movement of the atoms so that they can more easily align.

There is a specific temperature for each type of substance that will not magnetize the metal, called the Curie temperature that, for iron, is 1043 degrees Kelvin. This is when thermal agitation is so great that it doesn’t carry a magnetic charge consistently.

ELECTROMAGNETS Electrical currents can cause magnetic effects so that a compass needle can be deflected by a current-carrying wire. Electromagnetism is the use of an electric current to make magnets. This is a temporary phenomenon that makes electromagnets but they are extremely useful. These types of magnets are used in magnetic resonance imaging in medical circles and are used to lift cars in wrecking yards. When you take iron filings, for example, they create a pattern around a magnet and around an electrical current that looks the same in both situations. Electric current is, in fact, the source of all magnetism.

Strong magnetic effects can be created with a ferromagnet and an electromagnet. These two enhance one another. Magnetic information can be stored on videotapes and computer hard drives are applications of magnetism in common use. A CD player will have a rotating magnetic disk as well as a read/write disc. These will incorporate digital theories by having magnetic regions and nonmagnetic regions.

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MAGNETIC FIELDS The magnetic forces of a magnet will act over a distance. The direction of the magnetic field lines is defined to be the direction in which the north end of the compass needle points. This magnetic field is generally known as the B field. If the field lines could be seen in a magnet as is seen in figure 125, they would be seen close together and in parallel to one another. Note that the arrow goes from the north pole to the south pole.

Figure 125.

A field can be described as a way of mapping forces surrounding an object such that the object can influence something without a physical connection. Magnetic fields will map magnetic forces, just as gravitational fields map gravitational forces, and electric fields map electrical forces. The direction of the magnetic field will be tangent to the field line at any point along its line. The strength of the field is proportional to the areal density of the lines. Like electricity lines, magnetic lines never cross. These lines are continuous, never beginning and never ending.

MAGNETIC FORCES All magnetism is caused by a current or the flow of charges. Magnetic fields exert forces on charges that are moving. This is how they exert forces on other magnets. Remember the right-hand rule in determining the direction of angular momentum? This is the same thing that applies to magnetic forces. The magnitude and direction of magnetic 275

force F on charge q that is moving at a speed v in a magnetic field that has a strength called B can be described as shown in figure 126:

Figure 126.

Figure 126 shows the Lorentz force, which is the force on a charged particle in a magnetic field. The SI unit for magnetic field strength is the Tesla, as defined by the letter T. One tesla is one Newton-amps per meter. The gauss, as identified by G is 10-4 Tesla. The earth’s magnetic field is 0.5 gauss although much higher degrees of magnetism are possible with very strong magnets.

According to one right-hand rule as it applies to current, the direction of the magnetic force can be described by wrapping the fingers in the direction of the field B, in which the force is the perpendicular thumb pointing upward. Figure 127 is the direction of the magnetic field around a wire with current I:

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Figure 127.

The direction of the force will be perpendicular to the plane formed by the velocity and the direction of the magnetic field. The fingers represent the direction of the field lines, while the thumb points in the direction of the current. In the absence of a current, the right palm faced upward will have the fingers point along the field lines and the palm of the hand facing the force on the negative charge. This is described in figure 128 and is called the right-hand rule one:

Figure 128.

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There is no magnetic force on static charges—only on moving charges. Electric fields do not affect magnets unless the charges are moving because only moving charges exert forces on magnets. According to the right-hand rule, if the velocity is parallel to the direction of the magnetic field B, the force will be zero, meaning that charged particles follow magnetic field lines rather than cross them.

Magnetic forces can cause a charged particle to move in a circular or, more commonly, a spiral path. This is what happens to cosmic rays that approach the earth and what happens in proton accelerators. As we have seen, magnetic force is going to be perpendicular to velocity, doing no work on the charged particle so that the particles speed and energy will remain constant. This is uniform circular motion in a magnetic field. The magnetic force F supplies the centripetal force Fc on the charged particle. Any motion parallel to the field will have a zero magnetic force. This produces a spiral, rather than completely circular, motion.

The properties of charged particles in magnetic fields are related to what’s seen with the Aurora Borealis and Aurora Australis. The particles will get trapped into spiral orbits around the lines of a magnetic field rather than crossing them. This causes them to be energized and to glow. Remember that the north magnetic pole in earth is actually what’s referred to as the south magnetic pole in a regular magnet.

When it comes to the earth’s magnetic field, there are two field lines from the south pole to the north pole above the earth that have trapped magnetic charges in them. These make up the inner van Allen belt (300 kilometers above the earth) and the outer van Allen belt (16,000 kilometers above the earth). These particles come in from outer space and get trapped in the belts—avoiding the poles because of the strong magnetic field strength at the poles.

As we’ve talked about, there is a relationship between a magnetic field and free moving charges. A similar effect is seen when a magnet is applied to a conductor. Moving electrons will feel a magnetic force toward one side of the conductor with the field being perpendicular to the electron drift velocity. There will be a net positive charge on one side of the conductor and a net negative charge on the opposite side of the conductor, leaving a voltage, which goes by epsilon, called the Hall emf. This is referred to as the

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Hall effect. The Hall effect is not along the length of the conductor but across its width instead. Figure 129 shows the Hall effect on a conductor:

Figure 129.

The Hall effect is used to determine if a conductor is carrying a positive charge or a negative charge. In the case shown, it is a positive value if the electrons are carrying the charge (which is the normal way in electricity) or if a positive charge is moving (as is seen in certain semiconductors). The electric field E across the conductor will be equal the epsilon (the Hall emf) multiplied by the width of the conductor. The Hall EMF will produce an electric force that balances the magnetic force on the moving charges.

Another way to say this is that the magnetic force causes a charge separation that must be balanced by the electric force on the conductor.

The Hall effect can be used to make Hall probes, which can measure the magnetic field strength B. These can be placed within the field without disrupting it and can be calibrated to measure the strength of the field. It can also be used to measure the flow of fluid that has no charges in it. A magnetic field can be applied perpendicular to the flow direction in order to produce a Hall emf across it. The hall EMF will equal the field magnitude multiplied by the diameter of the tube multiplied by the average velocity of the fluid. This can be used in the measurement of blood flow.

Because charges cannot escape a conductor, the magnetic force on the charges within the conductor will be transmitted to the conductor itself. The force on a length of wire in a uniform magnetic field B will equal pi multiplied by the magnetic field B multiplied by the sine of theta, where theta is the angle between the force and the direction of the 279

current. If this is a perpendicular field, the angle will be 90 degrees and the sine of 90 degrees is 1 so it doesn’t factor in.

What this means is that the magnetic force on current-carrying wires can be used to convert electric energy to do work. This is the basis of the work done by motors, which will be discussed in a few minutes. Magnetism can be used to drive pumps that move fluid without any moving mechanical parts. This is called MHD or magnetohydrodynamics. This has practical applications when it is important to drive fluids without the impact of mechanical parts, such as in nuclear reactors.

MOTORS Motors are a common application of magnetic force. Motors have loops of wire within a magnetic field. When current is passed through the loops, the magnetic field causes torque to be applied by the loops, which rotates a shaft and transfers electrical energy into mechanical work. Figure 130 shows what this looks like in an actual motor:

Figure 130.

Torque will be defined as the distance from the pivot point times the force applied times the sine of theta, where theta is the angle between the force applied and the line to the pivot point. Figure 131 will show the torque applied when a coil is placed around a shaft:

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Figure 131.

The magnitude force will change as the angle theta changes; the direction of the torque will reverse direction as the angle comes around past 90 degrees. Motors have multiple loops of wire so as to magnify the torque applied. Ultimately, it is the electrical energy that uses magnetism to drive the motor. This causes an oscillation of the coil with automatic switches referred to as brushes. The brushes will reverse the direction of the current in order to keep the torque going in the same direction.

METERS These are represented by the analog fuel gauges in a vehicle and utilize magnetic torque on a current-carrying loop. The meter will have magnets that make the magnetic field proportional to the loop as much as possible. The meter is calibrated with a needle showing certain torques applied to the loop because the needle balances the torque on the loop. Only a small amount of revolution takes place on the pivot point, which leads to a change in the needle on the scale as is seen in figure 132:

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Figure 132.

MAGNETIC FIELD AND CURRENTS What you have learned so far is that currents and magnetism are related. Currents themselves can create a significant magnetic field as can be demonstrated by trying to use a compass near overhead electric power lines. Current loops create magnetic fields, as has been demonstrated in the way motors work. What about a straight wire and the current applied to it? Will this cause a significant electric field. Look back at figure 127.

It shows the direction of the magnetic field as it applies to a wire and a current. This is referred to as the right-hand rule two. The current will wrap around counterclockwise if the current is going upward as is identified by the curvature of the fingers and the direction of the thumb.

For a long, straight wire with a current applied, the magnitude of the magnetic field B will be proportional to the current inversely proportional to r, which is the shortest distance to the wire. The actual equation is shown in figure 133. It involves a constant called the permeability of free space, which is related to the speed of light. The assumption is that the wire is very long with the constant being four pi x 10-7 Teslameters per ampere.

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Figure 133.

So, what about a current-carrying circular loop? There will be a field produced in the center of the loop that has a certain strength. At the center the magnetic field will be the permeability of free space multiplied by the current divided by two times the radius of the circle enclosed by the loop. It is only valid at the center of the loop but it is similar to the findings seen in a straight wire. The larger the loop, the smaller will the field be at the center of the loop.

A solenoid is a long coil of wire that can create a strong, uniform magnetic field with very little field created outside of the coils. A battery is applied to the solenoid, creating a field inside the tube of coils. Only at the ends of the tube will the field strength be diminished. This will be a uniform field as is seen in figure 134:

Figure 134.

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In the case of a solenoid, B equals the permeability of free space constant multiplied by the charge and multiplied by the number of loops per unit length. Note again that this is not related to the diameter of the coils or any distance inside the coils as this is uniform.

This can be a large magnetic field strength that is utilized by the medical field in magnetic resonance imaging or MRI scanners. Superconducting wires are used so that not a lot of heat is generated by the current necessary to create large magnetic fields.

ELECTROMAGNETISM What is true of magnetic currents and electricity is that it is the change in magnetic field that generates a current and not the absolute value of the magnetic field. If you put a magnetic bar through a circular coil, you will get a galvanometric reading that will change, depending on the rate of change of the bar’s magnetic field within the coil. If the magnetic bar is held stable, there will be no current and no reading on the galvanometer. This ability to cause an emf and a current in a wire is called induction.

This leads to the concept of magnetic flux. The magnetic flux is the magnetic field multiplied by the area it is applied to multiplied by the cosine of theta, which is the angle of the field as it is applied to the area. It is greatest when the field is perpendicular to the area. The change in magnetic flux is what produces the EMF. The EMF is directly proportional to the change in flux and inversely proportional to time. The faster the change in flux, the greater will the EMF generated be. If the coil has N turns, the EMF will be N times greater than for a single coil. This leads to an EMF that is N multiplied by the change in flux divided by the change in time. This is called Faraday’s law of induction. The units for EMF are in volts, which is typical for EMF. There is a minus sign in this law because the emf will create a current and a magnetic field that oppose the change in flux. This opposition is laid out in the form of Lenz’s law.

Lenz’s law is an example of the conservation of energy. The induced emf produces a current in direct opposition to the change in flux because a change in flux means a change in energy. The energy in the system can enter or leave but there is resistance to that. If Lenz’s law wouldn’t be true, and if the induced EMF were in the same direction

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of flux, energy would increase without an apparent source and the conservation of energy would be violated.

Motional EMF is induced whenever a conductor moves in a magnetic field or whenever a magnetic field moves relative to a conductor. This can create a sort of current loop in the conductor called the “eddy current”. These currents can produce a significant drag (called magnetic damping) on the motion involved. If a metal pendulum is passed through a magnetic field, there is a significant drag on the bob of the pendulum that isn’t present if the bob is made from an insulating material. There will be drag in both directions of the pendulum. This only happens as the pendulum is entering and leaving the magnet’s field in keeping with induction and the change in magnetic flux causing a current in the metal bob. The bob needs to be a solid plate in order to generate a circular current within the bob; it isn’t seen as much when the bob is slotted and can’t be induced as much.

These eddy currents and magnetic damping are used in trash separation at recycling centers. The metals and nonmetals are passed down a ramp, the metals will be dragged down by metal damping. Nonmetals will not be affected. Metal detectors work best when they are continually moved because they detect the eddy current from a metal near the detector.

Magnetic braking works for large trains but cannot stop the train altogether because they become less effective at slow speeds. The magnetic eddy effect is seen only when there is movement. Braking of roller coasters happens with magnetic braking systems.

Induction cooktops use a varied magnetic field to produce an eddy current that heats up the cooking pot but only if the cooking pot is metallic. It won’t work if the pot is ceramic.

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INDUCTANCE Inductance is the ability to induce an electromagnetic current. Mutual inductance is the induction of one device upon another. The current passing through one set of coils can transmit energy to a second coil next to it. This is what happens in a transformer, although transformers are more efficient. The cause of the induction is the rate of change of the current through one device acting on another. The equation for this is seen in figure 135:

Figure 135

The larger the mutual inductance, the more effective is the induction between the two.

The units for mutual inductance are the Henry, which is ohm-seconds. Transformers are efficient inductors so there is a large mutual inductance. Electric dryers have too high a mutual inductance so there is counter-winding of the coils that cancel out the magnetic field produced.

Self-inductance can also occur. When the current through a coil changes, it creates a magnetic field and flux, inducing a counter emf as is required by Lenz’s law. The induced emf is related to the geometry of the device as well as the rate of change of the current. Any device that can self-induct is called an inductor. Induction always opposes the changes in current. This means that there is opposition to rapid change within an

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inductor. This means that energy can temporarily be stored in an inductor, similar to that of a capacitor.

ELECTROMAGNETIC WAVES Maxwell’s equations put together the facts related to magnetism and electricity into a series of statements and equations that are outlined here: •

Electric field lines start on positive charges and end on negative charges with an electric field defined as a force per unit charge, with the strength of the force related to the permittivity of free space.

•

Magnetic field lines are continuous and have no beginning or end. The strength of the magnetic field is related to the magnetic constant called the permeability of free space.

•

Changing magnetic fields will induce an electromotive force, which creates an electric field. This relates to Lenz’s law, which means that the emf opposes the change in magnetic field.

•

Magnetic fields are generated by moving charges or by changing electric fields so that electric current changes will create magnetism.

What these theories put together is the idea that electric fields and magnetic forces are not separate but are different manifestations of the same thing—which is electromagnetic force. These changing fields will propagate from their source like waves. These are now defined since the time of Maxwell to be EM waves or electromagnetic waves. These waves are capable of exerting a force over great distances from their source. How fast do these waves move? When calculated, it turns out that they move at 3 x 108 meters per second, which is the speed of light.

Whenever a current varies, there will be variable electric and magnetic fields that move out from the source like waves. The electric field surrounding the wire is produced by the charge distribution on the wire. The magnetic field B propagates outward as well.

Both of these propagate as an electromagnetic wave, such as when a broadcast antenna sends out a signal in a radio or TV station. These waves are similar to other waves we’ve

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talked about, with wavelengths identified by lambda and frequencies identified by f, which are inversely proportional to one another.

These electromagnetic waves are transverse waves that have a peak and a trough. The electric wave put out by an antenna will be in phase with the magnetic wave but these will be perpendicular to one another. Charges radiate whenever they are accelerated.

Antennae have a varying charge distribution that form standing waves with a certain resonant frequency determined by the dimensions of the antennae. This is why one tunes a radio to get a certain frequency, which is the resonant conditions for a specific antenna.

An antenna is necessary for receiving signals as well. Receiving antennas are designed to resonate at specific frequencies. When the radio is turned on, the electrical components will pick up the signal, converting the signal to audio formats. The same thing is seen with TV signals and the electromagnetic waves that are generated from the station.

The stronger the E-field or electrical field, the greater the current and the greater the Bfield created. The current is directly proportional to voltage, so the voltage is proportional to the field strength E. The ratio of the E field strength and the B field strength is a constant, equal to the speed of light. The E field is much greater than the B field generated because of this ratio being so large and constant.

There are different categories of electromagnetic waves, such as radio waves, infrared waves, and ultraviolet waves. Each wave has a frequency and a wavelength associated with it, and each wave travels at the speed of light. Because they all propagate at the speed of light, the speed of light is equal to the frequency multiplied by the wavelength.

The greater the frequency, the smaller the wavelength.

The longest waves are radio waves, with AM wavelengths being longer than FM waves.

TV waves are similar to FM waves in wavelength. Microwaves have a shorter wavelength than many radio waves and infrared waves are shorter than that in wavelength. Light waves have a very small bandwidth between infrared waves and ultraviolet waves. Higher than these are x-rays and finally gamma waves, which have

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the highest frequency and lowest wavelength. Figure 136 shows the electromagnetic spectrum:

Figure 136.

In general, these things can be made true of electromagnetic waves. High-frequency waves are more energetic and can penetrate more than low-frequency waves. Highfrequency waves carry more information per unit time than low-frequency waves. The shorter the wavelength, the smaller the detail it can resolve.

Radio waves can be produced by the current in wires and circuits. These are used to create microwaves, AM radio waves, FM radio waves, cell phones, and TV pictures.

Power lines will generate extra-long wavelengths of electromagnetism as long as many kilometers in wavelength. While power lines do give off waves, there is no evidence that these waves cause disease.

ELF waves or extremely low frequency waves are used as means of communication in submerged submarines. These can penetrate salt water farther than waves of shorter wavelengths. It takes these long wavelengths so the waves aren’t disturbed as much by salt water.

AM radio waves are called this because they are “amplitude modulation” waves that have a particular frequency emanating from the radio station. The wave received varies in amplitude but not in frequency. These are tuned in by antennas at the detection site.

FM radio waves are called “frequency modulation” waves, which is another way of carrying information. There is a carrier wave sent out by the radio station that is modulated in frequency by the audio signal producing a wave of constant amplitude but

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varying frequency. FM waves are less subject to noise from stray sources because the amplitude stays the same.

Television waves require a higher bandwidth to carry visual and audio information. The entire FM radio band is within the range of the TV bandwidth. TV channels are divided into VHF (very high frequency) and UHF (ultra-high frequency) waves. Cable TV and satellite TV transmission is ultra-high frequency, which can carry more information in the high-definition or HD format. The higher the frequency of these types of waves, the more direct the wave transmission and reception must be because they can’t travel around structures.

Microwaves are the highest-frequency EM waves that can be produced by an electric circuit. They are called microwaves because of their comparison to the frequency of radio and TV waves. Microwaves can be generated by the thermal agitation of atoms and molecules at any temperature above absolute zero. These will transmit information from communication sources but require a clear line of sight between the transmitter and receiver. Radar is an example of the use of microwaves.

Microwaves are used to generate an alternating electric field with waves that get absorbed by food in a microwave oven. Polar molecules like water will absorb microwaves, resulting in dielectric heating or an increase in water’s temperature.

Because it is water and similar molecules that absorb the radiation, the plate does not heat up. Rotating turntables will spread out the hot spots. Microwave diathermy uses microwaves to heat body areas affected by strains and sprains.

Infrared regions will overlap a little with microwave regions. Infrared radiation is produced by the thermal motion and the vibration and rotation of atoms and molecules.

The frequency is just below the red visual light region, which is why it’s called infrared radiation. Human bodies will give off infrared radiation so that night-vision scopes can detect it and convert it to visible light. Infrared emissions are proportional to the fourth power of the absolute temperature of an object. About half the radiation arriving on earth from the sun is infrared radiation, with a lesser amount being visible light. A small amount is ultraviolet waves.

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Visible light is a very narrow band of waves on the spectrum. It is produced by vibrations and rotations of atoms and molecules as well as by electronic transitions within atoms and molecules. Colored light has different wavelengths between 400 and 750 nanometers. Red light has the lowest frequency and the longest wavelength, while violet has the highest frequency and shortest wavelength.

UV light or ultraviolet light, which has a higher frequency than violet light is produced by molecular motions and electron transitions within atoms and molecules. These wavelengths are invisible to the human eye. The sun has UVA radiation, UVB radiation, and UVC radiation (which have increasing frequencies). Most UVB and UVC waves are absorbed by the ozone in the atmosphere, making 99 percent of this radiation of the UVA type.

UVB radiation is what causes skin cancer. All UV radiation will damage collagen fibers, resulting in wrinkle formation. Sunburn is caused by large exposures, while skin cancer is caused by repeated exposures. Tanning is a defense mechanism in which the body makes pigments to prevent future exposure to skin from the sun. UVB exposure is linked to cataracts, especially in the equatorial regions of the body.

X-rays are created by high-voltage discharges. An electron can be accelerated in a vacuum tube by a high voltage that strikes a metal plate, producing x-rays, thereby ionizing an atom. X-rays can penetrate the skin (unlike ultraviolet rays), causing damage to atoms in cells and making x-rays both causative of and treatment for cancer.

X-rays penetrate differently depending on tissue density, which makes them helpful in medical applications. These types of rays can also precisely identify the shapes of molecules, using a technique called x-ray diffraction.

Gamma rays are extremely high-frequency electromagnetic waves, which is the electromagnetic radiation emitted by a nucleus, which is why it’s called nuclear radiation, used in weaponry and nuclear reactors. These are nearly identical at some frequencies to x-rays, differing only in the source. At higher frequencies, these can be very damaging to human tissue. It is the type of radiation used in nuclear medicine.

Gamma radiation is used to protect food from spoilage by killing the microorganisms.

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No one knows if this damages the food. X-rays and gamma rays are used to scan luggage.

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KEY TAKEAWAYS •

Magnetic field lines are continuous, going through the magnet itself.

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The units for magnetic field are the tesla.

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Motors are devices that make use of electrical energy and loops of wire that generate a magnetic force, putting torque on a shaft.

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There is a strong relationship between the movement of charge and the creation of a magnetic field.

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The magnetic field can cause a deflection on a galvanometer but only when the magnet is moving.

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Inductance involves the generation of an emf by changing the current of a nearby device affecting another device.

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Electromagnetic waves involve a wide range of frequencies, indicating a substantive similarity between electricity and magnetism.

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All electromagnetic waves travel at the speed of light.

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QUIZ 1. What determines the direction that two freely hanging magnets of different sizes (two inches and one inch each in size) next to each other will hang? a. The larger one will point to the north-seeking end to the north pole and the south-seeking end to the south pole while the smaller one will point opposite to the larger one. b. The two magnets will point east-west with opposite poles of the magnets attracting each other. c. Both will hang with their north-seeking end to the north pole but will hang as far apart from one another as they can. d. The magnets will swing around a central axis and neither will reach an equilibrium and hang without motion.

Answer: c. The earth’s magnetic field will be strong enough that they will hang in a north-south position but they will hang as far apart from one another as they can because their like poles will repel each other as long as the repulsive forces between them are strong enough.

2. How do you separate the north and south poles of a cylindrical magnet that is two inches in diameter? a. Cut it in half b. Cut it lengthwise down the center of the cylinder into small slivers c. Cut it down to a single atom in length d. It is not possible to separate the poles Answer: d. It is not possible to separate the poles of a magnet, even if you cut it down to a single atom in length because even the iron atom will have a magnetic north and magnetic south pole.

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3. What happens to the forces on a charged particle in a magnetic field? a. The forces will be greatest parallel to the direction of the field. b. The forces will be greatest toward the south pole of a magnet. c. The forces will be zero parallel to the direction of a magnetic field. d. The forces make it easier to cross magnetic lines than to travel along them.

Answer: c. The forces will be zero parallel to the direction of a magnetic field, which make charged particles more likely to travel along the field lines rather than cross them.

4. What is the Hall effect in magnetism? a. It is the separation of charge across a conductor because of a magnetic field across it. b. It is the movement of charges along a conductor because of a magnetic field applied. c. It is the alignment of charges along magnetic field lines from positive to negative. d. It is the force that drives electrons and protons apart in atoms placed in an electric field.

Answer: a. The Hall effect is the separation of charge across a conductor because of a magnetic field applied across the conductor.

5. It has been established that wires can generate a magnetic field. What is the proportionality of the magnitude of this electric field? a. It is proportional to the current and proportional to the distance from the wire. b. It is proportional to the current and inversely proportional to the distance from the wire. c. It is proportional to the square of the current and inversely proportional to the distance from the wire. d. It is proportional to the square root of the current and the distance from the wire.

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Answer: b. There is a proportionality constant called the permeability of free space but the overall proportionality is that it is proportional to the current applied and inversely proportional to the distance from the wire.

6. What is the magnetic field like inside and outside a solenoid? a. It is uniform and large inside the solenoid, while it is nearly zero outside of the solenoid. b. It is strongest in the middle of the solenoid and just outside of the solenoid, decreasing with distance. c. It is strongest at each end of the solenoid. d. It is independent of the number of turns of coil in the solenoid and is the same inside and outside the solenoid.

Answer: a. The magnetic field inside the solenoid will be uniform and large inside the solenoid, while it is nearly zero outside of the solenoid.

It is dependent on the number of turns of coil in the solenoid.

7. What is measured in Henry units? a. Electrical resistance b. Mutual inductance c. Change in current over time d. Magnetic field strength Answer: b. The unit of the Henry involves mutual inductance, in which a high mutual inductance means that the change in current of one device can significantly change the emf of another device.

8. What is not true according to Maxwell’s equations? a. A static charge will produce a magnetic field. b. Magnetic fields have no beginning and no end. c. Electric fields start at the positive end and end at the negative end. d. EMFs are created by changing magnetic fields.

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Answer: a. These are true statements except that it is not true that a static charge will produce a magnetic field. It is the current and not the charge itself that produces the magnetic field.

9. Which type of electromagnetic wave has the highest frequency and the ability to carry more information? a. AM radio waves b. Cable TV waves c. FM radio waves d. TV antenna waves Answer: c. Cable and satellite TV waves will have higher frequencies and can deliver HD or high-definition TV by carrying more information in the wave.

10. What type of wave is used in the practical application of radar? a. AM radio waves b. Microwaves c. Gamma waves d. Infrared waves Answer: b. Radar is done by using microwaves that are transmitted and bounce back from weather systems; these types of waves can be seen in many practical applications around the earth.

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CHAPTER 17: LIGHT AND OPTICS This chapter gets into electromagnetic waves in the spectrum of visible light and its properties. Light comes in rays from various sources and is subject to reflection, refraction, and diffraction—each being aspects of waves not unique to light waves; however, they are unique phenomena seen in everyday physics and in life. The properties of light as it relates to passing it through a lens and the properties of light as it strikes a mirror are discussed in this chapter as well as the physics of light optics.

GEOMETRIC OPTICS According to what we know of electromagnetic waves, light is a wave that moves in a straight line as a ray or “light ray”. Unless it is acted on by things like refraction and reflection, it will travel at the speed of light, which is 3 x 108 meters per second, but it will not go around corners. This leads to the idea that light can be explained through geometry and simple trigonometry. This aspect of optics is referred to as “geometric optics”. The two main laws that are covered in this chapter are the law of refraction and the law of reflection, which involve mathematical principles you already understand.

REFLECTION Looking into a mirror or seeing an image on a lake both involve the process of reflection.

The same is true of large telescopes, which use reflection to form images of astronomical objects. As light reaches a reflective surface at angle theta between the ray and the perpendicular to the surface, the reflected ray will bounce off the reflective surface at the same angle theta just opposite the perpendicular. This is seen in figure 137:

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Figure 137.

Light can reflect also from irregular surfaces; however, the light will strike and bounce off at different angles, making the light appear more diffused. Diffused light is what allows a person to see what’s written on a piece of paper from many different angles.

Diffusion happens with many things, with the exception of mirrors. According to the law of reflection, the angle of reflection equals the angle of incidence, as is seen in figure 137.

REFRACTION Refraction can be seen when looking into a fish tank. There is distortion of the image because light will change directions or bend when it passes through the water. This bending of water is referred to as refraction. Refraction accounts for many phenomena seen when looking at light images and is the process light goes through in a lens, in water, and in optical fibers or fiber optics.

The speed of light, defined by the letter c, is a key concept in Einstein’s theory of relativity. It turns out that light will change speed when traveling from one medium to another. The traditional speed of light is that which it travels in a vacuum. It does not travel at the same speed through water, glass, and other mediums. This leads to the index of refraction, called by the small letter n, which is specific to the material and is defined as the speed of light divided by the velocity of light in the material. Because of this ratio and the fact that light is slower in all media, the value of n will always be greater than one.

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Some light will be reflected as it reaches a different medium but, unless the medium is opaque, light will get through and will be refracted according to the law of refraction.

The larger the index of refraction, the slower light will travel through it. The angle of the bending of light will depend on its index of refraction.

The law of refraction called “Snell’s law”, stated in equation form, is that the index of refraction for one medium multiplied by the sine of theta one is equal to the index of refraction of the second medium multiplied by the sine of theta two. This is seen in figure 138:

Figure 138.

The incoming ray is referred to as the incident ray and the outgoing ray is called the refracted ray, with the two angles called the incident angle and the refracted angle.

There is a list of the index of refraction for various substances that you can look up for these types of equations.

A good quality mirror may reflect more than 90 percent of the light that originally falls on it but it would absorb the rest. Total reflection can be achieved using certain aspects of refraction. If the index of refraction for medium two is less than the index of refraction for medium one, the ray will bend away from the perpendicular. The largest angle of refraction for angle two is 90 degrees. The critical angle theta c for a combination of material is the angle at which the angle of refraction will be 90 degrees.

If angle one is greater than this critical angle, all of the light is reflected back into the first medium. This is called total internal reflection. This critical angle is seen in figure 139: 300

Figure 139.

There can only be total internal reflection when the index of refraction in the second medium is less than the index of refraction than the first.

Fiber optics makes use of total internal reflection and is widely used in the transmission of cable TV signals, the internet, and telephone signals. This involves the transmission of light down fibers made from plastic or glass. The fibers are thin so that light strikes the inside surface at an angle greater than the critical angle, allowing the light rays to be totally reflected. Fibers are coated with a substance having an appropriate refractive index. This allows the transmission and “bending” of light across large distances and over angulated fibers. Cladding or “coating” of the fibers prevents light from being transmitted between fibers in a bundle.

A light ray that strikes an object that has two mutually perpendicular reflecting surfaces, will come back exactly parallel in the direction from which it came. This is independent of the angle of incidence. This type of object is referred to as a corner reflector because light bounces from its inside corner. Simple safety reflectors operate as corner reflectors. These are seen in periscopes and in binoculars as well. Two perpendicular mirrors will be typical corner reflectors.

Diamonds sparkle because of total internal reflection. The critical angle for diamonds in air is only 24.4 degrees, making it difficult for light to exit the diamond. The facets on diamonds are particularly good at enhancing this property. Light will exit only at specific places, leading to the sparkle seen in diamonds. Zircon is a gemstone that has a

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similar but less index of refraction, making it less “sparkly” when compared to diamonds.

DISPERSION Dispersion is what is seen when one looks at a rainbow or at the colors seen in a prism.

Six main colors are seen in a rainbow: red, orange, yellow, green, blue, and violet, with indigo sometimes seen. These colors have different wavelengths of light. White light is a mixture of all visible wavelengths. Dispersion can be defined as the spreading of white light into its full spectrum of wavelengths by the process of changing white light’s direction in a manner that depends on wavelength into its full spectrum of wavelengths.

Red has the longest wavelength, followed by orange, yellow, green, blue, and violet. The range is from about 750 nanometers to just under 400 nanometers of wavelength.

Figure 140 shows the spectrum of visible light:

Figure 140.

What this means is that the angle of refraction of light depends somewhat on its wavelength so that the n or index of refraction is increased with decreasing wavelength so it is greatest for violet light. This means that violet light will bend more than red light. Figure 141 shows the light bending in a prism:

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Figure 141.

Rainbows are made by a combination of reflection and refraction of light, which results in dispersion of the light. In order to make a rainbow, there needs to be water in the atmosphere that will bend light according to the differences in the way light bends by wavelength. It is seen to a greater effect when the background is dark as with dark clouds in stormy weather.

LENS AND LIGHT SYSTEMS Lenses are found in many different types of optical instruments. They make use of the law of refraction in order to do many things that affect light. Lenses are traditionally thought of as being convex, meaning they have a curvature that is outward in the middle. Convex lenses are referred to as converging lenses. The index of refraction of a lens will be greater than that of air.

As you can see from figure 142, there is a point at which the rays of light cross. This is called the focal point or big letter F of the lens. The distance from the center of the lens to its focal point is called its focal length, shown by the small letter f.

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Figure 142.

A more powerful lens has a greater effect on light rays. A powerful converging lens will focus parallel light rays closer to itself and will have a smaller focal length. The light will focus onto a smaller spot in a powerful lens. The power P of a lens is the inverse of its focal length or one divided by its focal length. The unit of power in a lens is called the diopter, given by the big letter D. This is different from the power seen in the rest of physics and is described as such: One diopter equals 1 meter to the minus one power.

Lenses can also be diverging lenses, in which they are concave. Light will bend away from the axis so that light appears to originate from the same point, which is the focal point of the lens. There is also a focal distance, which are defined to be negative because these are found before the lens and before light enters it, rather than after light exits the lens. The power of the lens will be negative in diopters as well. Figure 143 shows a diverging lens:

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Figure 143.

One can trace the paths that light rays take through a lens or through any matter. A thin lens is one in which light will refract but will not have a large amount of dispersion or aberrations. An ideal thin lens has two refracting surfaces but the lens is so thin that it can be assumed that light rays will bend only once. A thin lens will have the same focal length on either side of the lens.

There are two distances to note when looking at a lens. There is the object distance and the image distance. The image distance is the distance of the image from the center of a lens. This can determine the height of the object before and after the image is seen through the lens. The height of the image divided by the height of the object in real life is referred to as the magnification, the small letter m, of the lens. This is seen in figure 144:

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Figure 144.

The image will be magnified when the size of the image past the lens is greater than the actual image as light from it passes through the lens. Real images, such as those produced by a movie projector, are formed by converging lenses and when an object is further away from the lens than its focal length. The image will be inverted by a converging lens when the object being looked at is further away than the focal length of the lens as is seen in figure 144. A virtual image is an image on the same side of the lens as the object and cannot be projected onto a screen. This will be an upright image and will be seen in convex lenses used to treat nearsightedness.

MIRROR IMAGES Images seen in flat mirrors are the same size as the object and will be seen as being “behind the mirror”. A variety of images can be gotten from a mirror, including magnification of an image in makeup mirrors and rear-view mirrors of automobiles.

Security mirrors do the opposite, forming small images compared to the object. Each of these phenomena can be explained by the law of reflection. The image seen in a mirror is considered a virtual image because it cannot be projected. This is obvious because, if you walk behind a mirror, you cannot see the image.

Concave mirrors are mirrors that are small compared to their radius of curvature, acting similar to a parabolic mirror. They act very much like the “thin lens” that best 306

approximates a perfect lens but is not perfect. These small, concave mirrors will have a fairly well-defined focal point at a distance from the center of the mirror. The focal length will be positive as these are considered converging mirrors. Figure 145 shows a converging mirror:

Figure 145.

The shorter the focal length, the more powerful the mirror is with power being one divided by the focal length. The more curved the mirror, the greater is its power because it will have a short focal length. The focal length, using simple geometry and the laws of reflection, is half the radius of curvature of the mirror.

Convex mirrors are diverging mirrors, in which the light from the mirror appears to originate from a focal point behind the mirror. Both the focal length and power of the mirror will be negative as is the case with diverging lenses. The rules of ray tracing and the law of reflection apply to determining what these images look like when an object is placed in front of the mirror. Figure 146 shows a diverging or convex mirror:

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Figure 146.

With a diverging lens, objects will appear larger when looked at in the mirror, while the reverse is true of a converging lens.

WAVE OPTICS Light is basically a type of electromagnetic wave that obeys the equation that the speed of light is equal to the frequency times the wavelength. As long as light interacts with larger objects, it behaves like a ray and only runs into interference when interacting with small objects. Because of the wave quality of light, it is subject to addition and subtraction (or interference) like any other wave. This means that, if passing light through a small slit, the light will behave like a wave and not like a slit.

The speed and wavelength of light will change as the light goes through a medium, but the frequency will be the same. The velocity of light in a medium is the speed of light divided by the index of diffraction.

If light is passed through a slit that is smaller than the actual wavelength of the light, you can see that light bends around the slit, similar to sound, which can be heard around corners when sound passes through a door. This bending of a wave around the edges of an opening or an obstacle is referred to as “diffraction”. This is a characteristic of all waves that have all wavelengths. Figure 147 shows diffraction of light as it passes through five slits:

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Figure 147.

Slits like those shown in the figure demonstrate the light is a wave that diffracts differently depending on its wavelength, experiencing augmentation and interference of the waves, according to the physics of light behaving as a wave.

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KEY TAKEAWAYS •

Light travels at the speed of all electromagnetic waves, which is about 3 x 108 meters per second.

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Light is a wave that behaves like a ray when considering the difference between the wavelength of light and the objects involved, which tend to be much bigger than the wavelength of light.

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The laws of reflection and refraction refer to the behavior of light as it passes through different media.

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Light undergoes dispersion into different colors of the rainbow because different colors of light will have different indices of refraction in different media because of their wavelengths.

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There are convex lenses and concave lenses that will be convergent or divergent.

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The magnification of a lens is the image height divided by the object height.

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Mirrors will create virtual images that will be the same as the image, larger than the image, or smaller than the image, depending on the concavity and convexity of the mirror.

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Light behaves as a wave that will bend when passed through a slit that is small enough to impact the path of the wave.

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QUIZ 1. What is the speed of light? a. 2 x 104 meters per second b. 7 x 106 meters per second c. 5 x 107 meters per second d. 3 x 108 meters per second Answer: d. The approximate speed of light and of any electromagnetic waves is 3 x 108 meters per second.

2. What most increases the diffusion of reflected light? a. A rough surface for reflection b. High intensity light (high amplitude) c. A smooth surface for reflection d. Light at the highest frequency Answer: a. The diffusion of reflected light is greatly increased by a rough surface, in which light strikes the surface at different angles, getting bounced off at different angles. This leads to diffused light compared to light striking a smooth surface.

3. The bending of light in different mediums is called what? a. Reflection b. Diffraction c. Refraction d. Dispersion Answer: c. The refraction of light is the bending of light in different mediums. This is particularly common when viewing light and objects in the water or through a watery medium. Reflection is light bouncing off of a surface.

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4. What property of light is referred to in Snell’s law? a. Reflection b. Diffusion c. Deflection d. Refraction Answer: d. Snell’s law is the law of refraction, which states that light bends at the sine of angle theta depending on its index of refraction.

5. What is the critical angle of refraction? a. When the angle of refraction is 90 degrees. b. When there is total reflection of the light at the second medium. c. When there is no refraction between the two mediums. d. When there is 50 percent reflection and 50 percent refraction.

Answer: c. The critical angle is when the angle of refraction is 90 degrees. Any angle greater than the critical angle will involve total reflection of the light at the interface between the two mediums.

6. What phenomenon best explains what is seen in a rainbow’s colors? a. Dispersion b. Deflection c. Refraction d. Reflection Answer: a. A rainbow is seen because of the dispersion of light; this dispersion is seen because of different indices of refraction for the different wavelengths of light.

7. Which color of light has the longest wavelength? a. Green b. Violet c. Yellow d. Red

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Answer: d. Red has the longest wavelength of light, while violet has the shortest wavelength. What this means is that red light bends the least and violet light bends the most when traveling through a prism.

8. What is a diopter in optics? a. The measure of the index of refraction of a lens b. The measure of the power of a lens c. The measure of the thickness of a lens d. The measure of the focal length of a lens Answer: b. The measure of the power of a lens is the diopter, which is the inverse of the focal length of a lens in meters. This is often used in optics in order to determine how powerful a given lens is.

9. Where is the focal point of a diverging lens? a. It will be before the light enters the lens b. It will be within the lens itself c. It will be after the light enters the lens d. It does not exist with this type of lens Answer: a. The focal point will be before the light enters the lens, with the focal distance being negative and the diopters or “power” also being negative.

10. When determining the magnification of a lens, how is this determined? a. It is the difference between the focal point before and after the lens. b. It is the ratio of the focal distance of the object and the focal point of the image. c. It is the ratio of the height of the image and the height of the object. d. It is the percentage increase in size of an object after it passes through the lens.

Answer: c. The magnification of the lens is the height of the image divided by the height of the object.

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CHAPTER 18: QUANTUM PHYSICS This chapter briefly introduces quantum physics, which is the physics that involves the behavior of very small things. While most properties of physics can be explained on a macroscopic scale, quantum physics describes physical principles in ways that could not have been understood in the early days of physics. As it turns out, when breaking matter down into its smallest states, the study of physics becomes very different from that which is understood on a macroscopic scale. Issues that come up when looking at matter at an atomic level are the basis of this chapter.

QUANTIZATION As it turns out, energy can be considered quantized in certain situations. This means that the system is allowed to have only certain energies and not a continuum of energies as you have already studied. This is the equivalent of having an automobile that could only go at certain speeds and not at others. What it means in quantum physics is that energy is transferred in discrete lumps. Just like matter is broken down into atoms and subatomic particles, energy can be broken down into discrete units.

One place where energy is quantized is in the absorption of electromagnetic radiation.

The electromagnetic spectrum radiating from a hot solid is directly linked to the temperature of the solid. We talked about this in an earlier chapter when we discussed radiation and absorption of energy. A jet-black radiator has an emissivity of one at all wavelengths, emitting radiation called blackbody radiation. The intensity of the radiation varies according to the fourth power of its absolute temperature. The peak of its spectrum will shift to shorter wavelengths at higher temperatures. While it seems continuous, it is the curve of the spectrum of intensity versus its wavelength that shows the energy is, in fact, quantized.

What has been understood over time is that atoms and molecules absorb and emit radiation while behaving like tiny oscillators. The energy of these oscillators needs to be quantized in order to describe the way that blackbody radiation behaves. Max Planck

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uncovered that the energy of an oscillator with a frequency f must be the sum of n plus one-half, multiplied by the frequency and Planck’s constant. In this case, n is any integer that isn’t negative and Planck’s constant is 6.626 x 10-34 Joule seconds. This means that the energy can only change in discrete steps related to the object’s oscillating frequency.

Does this have any corollary in macroscopic situations? Think about the harmonics in sound waves so that sounds only sound good together at certain frequencies or the fact that movement of a person down a hill or up a hill involves stepping up or down a staircase rather than gliding up continuously. Each step changes the potential energy of the person in discrete numbers.

Planck’s constant is an extremely small number so, for the infrared frequency being emitted by a blackbody, which is 1014 Hertz, the difference between energy levels is Planck’s constant times the frequency, which is 0.4 electron volts. This is close to the energy seen in atoms themselves. Because this is so small, the results seen in quantum physics are too small to make a big difference in what is seen macroscopically.

The sun is a body that contains gases, which emits electromagnetic radiation—some of which can be seen as visible light. Studies of the emissions of hot gases have shown that the information from these emissions could determine the type of gas and the temperature of the gas. It turns out that electromagnetic emissions come from electrons in atoms that are transitioning between energy levels within the individual atoms of the gas. They can be seen as the atomic spectrum of a specific gas. For a specific gas, only certain wavelengths or frequencies of radiation can be emitted. This is seen as the line spectrum of a gas. The line spectrum is unique to the gas and indicates that the electron energies being emitted are quantized. Figure 148 shows the emission spectra of different elements:

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Figure 148.

Electricity is passed through a substance with the atoms and molecules absorbing energy that is re-emitted as electromagnetic radiation. It is different for the different atoms, revealing differences in their atomic spectra.

PHOTOELECTRIC EFFECT When light strikes a substance, it can eject electrons from the substance. This is referred to as the photoelectric effect. This refers to the idea that light produces electricity. This has been proven on a macroscopic scale by the idea that solar cells can use the energy from the sun to power things. Einstein took this information and showed that the photoelectric effect can only be explained if electromagnetic radiation itself is

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quantized. The unit of light energy is called the photon. The energy of a photon is proportional to the frequency of electromagnetic radiation. Photon energy is equal to Planck’s constant multiplied by the frequency.

While this equation is similar to that of blackbody radiation, the difference is that it identifies the fact that electromagnetic radiation is itself a quantized thing. These waves are not continuous but have their energy absorbed and emitted in clusters. Because there are so many photons in common light sources, the energy of individual photons is completely unnoticed. A higher intensity of light means there are more photons involved in the light itself. White light has light energy of multiple frequencies.

Based on the photoelectric effect, the following things necessarily are true, particularly if we assume there is just one color of light involved because then all the photons have the same energy: •

For a given material, there is a threshold frequency of f-zero, below which no electrons are ejected, regardless of intensity. If the photon energy is too small (not the light’s intensity) to break the electron away, no electrons will be able to be ejected. This wouldn’t be the case if the radiation was a wave function.

•

Once electromagnetic radiation falls on a material, electrons are ejected immediately. Once a high-enough frequency is reached, the electron is ejected.

This wouldn’t be the case if the radiation was a wave function.

•

The number of electrons ejected over a specific time is proportional to the intensity of the radiation. High intensity equals high numbers of photons per unit area.

•

The maximum kinetic energy of the ejected electrons is independent of the intensity of the radiation. There are simply more electrons ejected rather than high-intensity electrons.

•

The kinetic energy of an ejected electron is equal to the energy of the photon minus the binding energy of the electron to its atom. This means that some energy is necessary to break the electron away with the rest going to kinetic

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energy of the electron being ejected. The binding energy is Planck’s constant multiplied by the threshold energy of the material.

What you will come to see is that many aspects of electromagnetic radiation can only be explained by the fact that there are photons, such as why UV light can be dangerous to human skin. A DNA molecule can be damaged by about 1 electron volt of energy, so that radiation of high enough frequencies will potentially have biological effects. It explains why higher frequencies of EM radiation have a greater potential to damage biological tissues.

IONIZING RADIATION A photon is considered one quantum of electromagnetic radiation. What we have seen is the energy is equal to Planck’s constant multiplied by the frequency, which is also equal to Planck’s constant multiplied by the speed of light and divided by the wavelength of light. The greater the wavelength, the lower the photon energy.

The binding energy of a tightly bound molecule is about 10 electron volts, while that of a loosely bound molecule is about 1 electron volt. The energy it takes to ionize an atom, called the ionization energy is greater than these—at 10 to 1000 electron volts. Gamma rays from nuclear and cosmic EM radiation have the highest frequencies and the highest photon energies. This is enough to ionize all types of materials, capable of damaging biological tissues. This type of radiation affects rapidly-dividing cells the most and they can penetrate solid objects.

ULTRAVIOLET RADIATION Ultraviolet radiation has electron voltages of between 4 and 300 electron volts. This overlaps into the lower end of the energy of some x-rays. UV radiation comes from the de-excitation of atoms of hot solids or gases. A UV photon has enough energy to ionize atoms and molecules, which makes them it different from the photons of visible light.

UV light can cause skin cancer and is used as a sterilizer because of these properties. It takes several UV photons in order to disrupt cell production or to kill a single bacterium.

UV light can trigger the production of vitamin D in the skin, which cannot be done with 318

visible light because it has insufficient energy per photon to alter the molecules causing its production.

VISIBLE LIGHT Visible light has photon energies of about 1.63 to 3.26 electron volts per photon. This can affect the outer electron shells in atoms and molecules. A single photon of light can trigger the retina to respond but, according to quantum theory, it takes a certain frequency of light for a photon to affect a certain type of molecule. The energy step of the molecule must equal the energy of the photon for it to be absorbed. Violet flowers do not absorb violet light because there is no energy step available for them to absorb this frequency of light.

Red light does not have the energy levels to expose black-and-white film, which is why red light is used in dark rooms. Non-violet dyes fade faster in light than violet colors because they absorb light of higher energies to a greater degree (violet-colored light) and break down more easily. Green, yellow, and red dyes absorb higher energy light, causing them to break down. The higher the energy frequency, the greater is the ability of the EM radiation to be absorbed, which is why UVC light and most UVB light gets absorbed by the molecules in the atmosphere.

LOW-ENERGY PHOTONS Infrared radiation will have very low photon energies when compared to visible light.

This is why it can’t alter atoms and molecules to a sufficient degree. Water, however, will have many states that are separated by energies within the IR and microwave energy ranges. This is why skin appears black in the IR range, with an emissivity near one. It is also why microwaves are absorbed by water molecules and not by other types of molecules. When these types of photon energies are hazardous, it is because of a lot of photons acting together and not to the accumulation of photons. One difference is that at extremely high intensities of these types of radiation frequencies, strong electric and magnetic fields can be created, which can be ionizing in themselves.

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PARTICLE-WAVE DUALITY Photons are massless particles, which means that light has no weight. We know that light behaves as a wave but how does it act both like a particle and a wave? It is called particle-wave duality, which is a feature of everything to some extent. Stated again, all things in the universe have particle-wave duality.

According to theory, the wavelength of something is Planck’s constant divided by its momentum. This is called de Broglie wavelength. Since Planck’s constant is extremely small, large things have very short wavelengths. This means that things like macroscopic objects do not experience constructive and destructive interference as is seen in small things that behave more like waves.

HEISENBERG UNCERTAINTY PRINCIPLE Because of the duality of particles and waves, it is difficult to nail down the exact position of things as small as electrons. Instead of finding the exact position of an electron, what happens is that you get a probability distribution rather than an exact location. If you try to follow the path of a particle, the exact location of it is no longer the same. If an electron’s position is assessed repeatedly, there will be a probability of finding it in a certain location but it cannot be located directly.

According to Heisenberg Uncertainty Principle, it is impossible to measure the location x of an electron and its momentum p at the same time. This is because the momentum and the wavelength are inversely proportional to one another. One of these can’t be determined with certainty without hopelessly increasing the uncertainty of the other.

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KEY TAKEAWAYS •

Quantum physics comes into play when looking at very small things.

•

All electromagnetic energy is in packets called quanta. Photons represent the particle aspect of electromagnetic radiation.

•

The higher the frequency of an electromagnetic wave, the greater is its photon energy and the greater its ability to ionize molecules.

•

All things in the universe have particle-wave duality.

•

According to the Heisenberg Uncertainty Principle, it is impossible to tell the location and momentum of an electron or other small particle at the same time.

This leads to a probability distribution for electrons rather than an exact location.

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QUIZ 1. Why are the findings seen in quantum physics not seen in classical physics? a. The physics of large objects is different from that of small objects. b. Large objects do not oscillate, which is necessary for quantum physics to be valid. c. The masses of large objects makes the physical principles seen in quantum physics impossible. d. The mathematics applies to both types of physics; however, quantum energy changes are very small.

Answer: d. The mathematics does apply to both types of physics. It’s just that Planck’s constant is very small and the energy quanta seen appear to be continuous in large-scale classical physics.

2. What is not a basic theorem based on the photoelectric effect? a. Light energy can cause electricity. b. Electromagnetic radiation is quantized. c. Photons are positively charged. d. Photons can eject electrons from atoms.

Answer: c. Each of these is a true feature of the photoelectric effect except that photons themselves do not have a charge on them.

3. What determines the energy of a photon? a. The frequency of the light energy. b. The mass of the photon. c. The brightness of the light. d. The speed of light.

Answer: a. The energy of a photon is based on the frequency of the light energy. Because the energy equals Planck’s constant multiplied by the frequency, the energy will vary with frequency.

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4. Based on photon energies and the photoelectric effect, which types of electromagnetic radiation is more hazardous to biological systems? a. Microwaves b. Gamma waves c. Infrared waves d. Light waves Answer: b. The energy in high frequencies of EM radiation is greater than that of radiation at lower frequencies. This means that gamma rays have a greater ability to eject electrons from biological material than waves of lower frequencies.

5. Which energy is considered the highest? a. The rotational energy of a molecule b. The binding energy of a molecule c. The vibrational energy of a molecule d. The ionization energy of a molecule Answer: d. The ionization energy of a molecule is on the order of 10 to 1000 electron volts, which is the highest amount of energy. It means it takes high frequencies of EM radiation in order to cause a molecule to ionize.

6. What can be said of high intensity light of anywhere on the light spectrum? a. Higher intensities of light lead to higher photon energies. b. Higher intensities mean higher numbers of photons per square area. c. Higher intensities mean a greater density per unit volume of photons. d. Higher intensities of light lead to increased ability to damage biological tissues.

Answer: b. Higher intensities of light mean higher numbers of photons per unit area. It isn’t the volume but the area that matter but the intensity is unrelated to the photon energy. High intensity light still does not have the photon energy to damage biological tissues.

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7. Based on quantum theory, what determines the color of something? a. The color of light on the object must not be able to be absorbed by the object’s molecules. b. The color of the object is the color of light that is best absorbed by the object. c. The color of the object is the frequency of the light is rejected by the object. d. The energy photon level that determines the color is weakest for the color of the object.

Answer: a. The color of light on the object must not be able to be absorbed by the object based on its preferred energy level changes.

This light is instead reflected, giving the color of the object.

8. What is the mass of a photon? a. It is the same mass as an electron. b. It is equivalent to Planck’s constant multiplied by the frequency of the photon. c. It is the same as the mass of a proton. d. It has no mass.

Answer: d. Photons are considered massless entities.

9. Which things in nature can be said to have particle-wave duality? a. Radio waves b. Electrons c. Photons d. All substances have duality Answer: d. All substance, however large or small, have particle-wave duality even though most things are said to be one or the other. This, according to quantum physics, is not the case.

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10. How is the wavelength of matter related to its size? a. The wavelength is proportional to the object’s mass. b. The wavelength is inversely proportional to the object’s momentum. c. The wavelength is inversely proportional to the object’s mass. d. The wavelength is proportional to the object’s momentum.

Answer: b. According to de Broglie’s principle, the wavelength of something is Planck’s constant divided by its momentum. This means that large objects have very short wavelengths.

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SUMMARY The intention of this course was to explain all of the major topics one would learn in a typical college physics course. Physics is the study of the physical attributes of matter and energy. Matter can consist of small particles and large objects—as small as subatomic particles or as large as planetary bodies and other celestial bodies. Regardless of size, everything must follow specific physical laws and principles, which have become better understood by you as you studied the material in this course. There are also mathematical aspects of the nature, applicable forces, and seemingly invisible energies involved in physical structures that were covered in this course. While there have been many mathematical relationships discussed in this the study of physics, the topics we talked about were hopefully understandable in both mathematical and nonmathematical ways.

Chapter one in the course introduced the subject of kinematics, which is the study of the motion of objects without regard to the objects’ masses and without the consideration of the particular forces that may have caused the movement of the objects.

Objects are always in motion—even if they do not appear to move as there is the continuous vibrations of molecules and atoms that make up the object. In this chapter, we looked at the basics of movement, including velocity, acceleration, and the acceleration of a body that is free-falling on earth.

The focus of chapter two was the motion of an object in two dimensions. Many things do not simply go in a straight line or in an up-and-down fashion. These include celestial objects in orbit, automobiles that travel around a curve, and the arcing of a ball. There are different equations and different vectors that apply to these types of movements, which need to be studied and memorized. Three-dimensional kinematics is very similar to two-dimensional kinematics, except that the x, y, and z axis are part of this discussion. Many of the physics principles you have learned in the course apply to motion in more than one direction, allowing for more types of situations involving the motion of objects to become solvable.

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Chapter three looked at Newton’s Laws of Motion as well as applications of these laws. It went further in the discussion of motion to outline things like friction, drag, and elasticity when it comes to motion. Chapters one and two looked at motion in its purest form without an understanding of the forces behind the motion. What necessarily followed in chapter three was a discussion of “dynamics”, which are the forces that affect the movement of the different objects and systems. It turns out that Newton’s Laws are the foundation of the study of dynamics. These laws were uncovered in 17th Century but still apply today on earth as well as in space. They apply to the study of classical mechanics, meaning that they apply to speeds less than light and sizes of objects greater than molecules.

Chapter four in the course dealt with uniform circular motion, which is defined as motion in a circular path at a constant speed. It involves things like pure rotational motion, which occurs when an object is traveling a path that is centered around a single point. This is different from pure translational motion, which is motion that has no rotation associated with it. There is mixed motion as well, with circular and rotational components. In addition, related components discussed in the chapter were the Coriolis effect and Kepler’s laws of planetary motion, which also apply to circular motion.

The topics of chapter five were the concepts of work, energy, and power. Work involves the process of getting something done using forces or the transfer of energy from one state to another. Energy, as you may know, cannot be created or destroyed; it can only be transferred from one form to another. The chapter also introduced the topic of power, which is a closely related term that is the rate (energy amount per time period) at which work is done or energy converted. The relationship between energy, work, and power were covered as part of this chapter.

Chapter six in the course dealt with the subjects of linear momentum and collisions. It involved the topic of linear momentum, which is the velocity of something and the product of its mass. Impulse was also covered, which is the change in momentum of an object in a system. Momentum leads an object toward collisions with other objects.

There are elastic collisions and inelastic collisions that differ in their apparent conservation of momentum. Each of these topics was able to build upon things that

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were learned in the previous chapters on force, velocity, and mass, as well as Newton’s laws.

Chapter seven opened up the topics of statics, torque, and rotational motion. While motion was primarily covered in the beginning of the course, there is an entire branch of physics associated with nonmoving forces, which are collectively referred to as statics.

Torque involves forces that act in a twisting fashion in order to cause motion or the potential for motion. This led to the study of rotational motion, including rotational acceleration and motion that is not necessarily uniform.

Chapter eight in the course looked into the subjects of fluid statics and fluid dynamics.

The liquid state represents a state of matter in which there is some cohesion of the molecules that is different from that of solids and gases. There are characteristics of fluids, such as density, pressure, and other factors that were explained as part of the chapter. In addition, there are aspects of this state of matter that specifically touch on the dynamics or flow properties of liquids in physics and biology. The flow of fluid can be relatively laminar or turbulent, depending on a variety of factors, including the viscosity of a particular liquid, which were discussed as part of this chapter.

The focus of chapter nine was temperature and the properties of substances related to temperature, such as evaporation, humidity, and phase changes of a given substance.

This led to a discussion of kinetics and kinetic theory as it applies to gases. The ideal gas law was covered, with some attempt to link ideal gases with real gases. As it turns out, all substances are affected by their own temperature and the temperature of their surroundings, with expansion occurring in solids, liquids, and gases to varying degrees.

This chapter combined theories and influences of both physics and physical chemistry as they apply to molecular systems and macroscopic substances.

Chapter ten in the course looked at issues related to heat, which is itself a form of energy. Heat can be stored in a substance or transferred from one substance to another.

Quite often, heat is not recognized until it is in transit from one thing to another. There are different types of heat transfer methods, including convection, conduction, and radiation. There are fundamental issues in physics related to heat transfer and specific laws that apply to the transfer of heat energy, which were covered in this chapter.

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Chapter eleven in the course was also concerned with temperature and heat transfer; however, it expanded on these topics further and went into the laws of thermodynamics that apply to heat as it relates to energy and work. In earlier chapters, the topics were on heat as a pure form of energy transfer, while this chapter was about the ability of heat transfer to perform work. Heat is like any other form of energy. In many systems you studied in this chapter, heat transfer has the ability to do things like run engines and allow machines to function. The laws of thermodynamics are not just laws of physics; they have practical applications that are seen in everyday life.

Chapter twelve introduced the topic of oscillatory motion, which is movement back and forth between two points. There are many systems that oscillate, some of which create waves. Waves can be visual, such as the waves in a swimming pool or ocean.

Other waves that aren’t commonly seen as waves include sound waves and light waves.

Waves create disturbances that carry energy, from small waves that carry light energy to large waves that create tsunamis and earthquakes. Waves, as it turns out, have the energy to augment each other or to interfere with one another, which was covered in this chapter.

Chapter thirteen in the course introduced the physics of electricity by covering electric charges and electric fields. Static electricity is just one aspect of electricity that is well understood by anyone who touches an object and gets an electric shock. Also covered was the topic of electromagnetic force, which is a type of energy that applies to electrical fields. A natural part of the discussion is that of conductors of electricity and insulators of electricity, which were also a part of this chapter.

The focus of chapter fourteen was to extend the understanding of electricity to include electric potential and electrical energy. It introduced aspects of electricity such as voltage and the storage of electrical energy by capacitors. In this chapter, you discovered that electrical energy and voltage are not the same thing because small batteries can have the same voltage as large batteries but will not create the same amount of electrical energy. In this chapter, the actual use of electricity in everyday electrical situations was discussed as well.

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Chapter fifteen in the course covered the topics of electric currents and circuits.

Electric current is defined as the movement of a charge from one place to another over a period of a certain time. Such a thing has the capacity to do work. This then got into Ohm’s law as it applies to electrical resistance and to the subject of circuits. There are AC current situations and DC current situations, which are things that many have heard about but now understand from the perspective of physics.

Chapter sixteen focused first on magnetism and then on the relationship between magnetism and electricity. Magnetism is common in nature and explains many things related to what we see in nature, such as the magnetic poles on Earth. Magnetism can cause electrical currents to be generated, which was also discussed as part of this chapter. The chapter also brought into focus the topic of electromagnetism and electromagnetic waves. You have seen that there is a vast range of these types of waves that go from those that heat food (which are microwaves) to waves that are much higher in frequency than those a person can see (which is the narrow spectrum of visually-seen electromagnetic waves).

Chapter seventeen in the course looked into electromagnetic waves in the spectrum of visible light and its properties. Light comes in rays from various sources and is subject to reflection, refraction, and diffraction—each being aspects of waves not unique to light waves; however, they are unique phenomena seen in everyday physics and in life. The properties of light as it relates to passing it through a lens and the properties of light as it strikes a mirror were discussed in this chapter as well as the physics of light optics.

Chapter eighteen in the course briefly introduced quantum physics, which is the physics that involves the behavior of very small things. While most properties of physics can be explained on a macroscopic scale, quantum physics describes physical principles in ways that could not have been understood in the early days of physics. As it turns out, when breaking matter down into its smallest states, the study of physics becomes very different from that which is understood on a macroscopic scale. Issues that come up when looking at matter at an atomic level were the basis of this final chapter.

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COURSE QUESTIONS AND ANSWERS 1. In remembering the SI units for measuring units, what is not the correct SI unit? a. Meter b. Second c. Ampere d. Gram Answer: d. The SI units are all correct, except that kilograms are used in place of grams for the measurement of weight in the SI system.

2. In knowing the SI units and prefixes, which prefix represents the smallest unit of measurement? a. Mega b. Kilo c. Femto d. Milli Answer: c. The term “femto” stands for 10-15 of something, making it the smallest unit of those listed.

3. A person starts at 1 meter from a reference point and goes to 5 meters from a reference point. They then travel 2 meters toward the reference point in one direction. What is the distance? a. +5 meters b. -3 meters c. 2 meters d. 6 meters Answer: c. The distance is the difference between the starting and ending points and is different from the distance traveled. The distance traveled would be 4 meters plus 2 meters; however, the distance from 1 331

meter and three meters (which is the endpoint) is just 2 meters so this is the correct answer. It is always a positive number so no sign is necessary.

4. A professor walks to the right 5 meters, to the left 7 meters and to the right 1 meter. What is the magnitude of the displacement? a. Negative one meter b. 1 meter c. 13 meters d. 6 meters Answer: b. In such situations, the magnitude of the displacement is the distance between the beginning and end point without a sign or direction. It is the absolute value of the change in placement of an object, which will always be a positive number.

5. What is the SI units defining velocity? a. Miles per second b. Meters per second c. Miles per hour d. Kilometers per hour Answer: b. The SI units representing velocity is meters per second.

Because it represents the displacement divided by the time, it can be represented as a negative number and is itself a vector quantity.

6. If a person starts at home and goes 15 miles, then returns home in two hours, what is the average velocity? a. Zero miles per hour b. Fifteen miles per hour c. Thirty miles per hour d. 7.5 miles per hour Answer: a. The velocity is a vector so if a person starts and ends at the same point, the displacement will be zero so the velocity will be zero

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miles per hour, regardless of how far the person traveled and the time it takes to travel.

7. An object is going along the x axis to the right at a certain velocity and the acceleration is a negative number, what is this called? a. Positive acceleration b. Negative acceleration c. Negative displacement d. b. Deceleration Answer: d. Acceleration that is in the opposite sign of the velocity is referred to as deceleration. In this case, the velocity is positive because the object is traveling to the right on the x-axis. The acceleration is a negative number; this means that deceleration is happening. Positive acceleration can occur in the opposite direction of the velocity and negative acceleration can occur in the same direction as the velocity.

8. Two people are running. Person one is running at 4 miles per hour and person two is running at six miles per hour. How much further will person 2 be compared to person 1 after thirty minutes? a. 4 miles b. 1 mile c. 2 miles d. 6 miles Answer: b. The person going at 6 miles per hour will go 3 miles in thirty minutes, while the person going at 4 miles an hour will go 2 miles in thirty minutes. The difference between the two after thirty minutes is 1 mile.

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9. What factor determines the acceleration of an object falling on the moon? a. Air resistance b. Mass of the object c. Mass of the moon d. Distance of the fall Answer: c. There will be differences in the acceleration of an object on the moon versus the earth due to the mass of the moon and not to anything else.

10. What is the value of “g” or the force of gravity? a. 1.66 meters per second squared b. 3.9 meters per second squared c. 6.7 meters per second squared d. 9.8 meters per second squared Answer: d. The force of gravity is 9.8 meters per second squared, although it does vary slightly depending on where on the earth a person is.

11. The “constant” of the force of gravity is actually different slightly, depending on all but what factor? a. Mass of the object b. Latitude on the earth c. Altitude of the falling object d. Geological formations in the earth Answer: a. The constant of the force of gravity does not depend on the mass of the object but is dependent on each of the other factors.

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12. In the situation of throwing an object upward, what will generally be the sign of the velocity and what will generally be the sign of the force of gravity? a. Both of these will be positive values b. Both of these will be negative values c. The velocity will be negative and the force of gravity will be positive d. The velocity will be positive and the force of gravity will be negative Answer: d. In general, an object thrown upward will have the force of gravity be negative and the velocity will be positive. That being said, you can choose for the reverse to be true if you wish to change the coordinates. The values of g and the velocity will be oppositely signed because they naturally oppose one another in this situation.

13. In traveling in a horizontal and vertical fashion, what affects the horizontal component of the motion? a. The sum of the vertical and horizontal vectors. b. The hypotenuse of the horizontal and vertical vectors. c. The sum of the horizontal vector and the square root of the vertical vectors. d. The magnitude of the horizontal vector only.

Answer: d. The magnitude of the horizontal vector only affects the horizontal component of the vector. There is complete independence of the horizontal and vertical vectors when something travels in two directions. The same is true of the vertical vector, which will be independent of the horizontal vector.

14. In determining the final vector in a 2-dimentional situation, where does the first vector begin? a. At the 0,0 point on the graph b. At the point of the horizontal vector c. At the point of the vertical vector d. At the final displacement point

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Answer: a. In graphing the vectors in 2 dimensions, you need to start at the 0,0 point on the graph and begin with any vector you wish, although the convention is to start with the horizontal vector first. If a vector is not horizontal or vertical, you will need to use a protractor in order to get the angle correct.

15. You have a vector of a magnitude A and a vector of magnitude -1/2A and you are adding these vectors on a graph. The angle of the vector A is 45 degrees.

What would the angle and the direction be of the resultant vector? a. Angle of -45 degrees and magnitude of 1.5A b. Angle of 45 degrees and a magnitude of 1/2A c. Angle of 135 degrees and a magnitude of 1/2A d. Angle of 45 degrees a magnitude 1.5A Answer: b. The angle of the vector will be the same at 45 degrees. This is basically subtracting a vector using the magnitude A and the magnitude -1/2A. The final magnitude will be 1/2A; however, as they have the same angle, this will not change.

16. You are traveling from point A to point B along vector AB and then go twice as far in the same direction. In calculating this displacement, what are you doing mathematically? a. Multiplying a vector by a scalar b. Multiplying two scalars c. Adding two vectors d. Subtracting two vectors Answer: a. In such cases, the simplest explanation is that you will be multiplying a scalar and a vector. This will lead to an answer that has a direction and a magnitude that will be the multiplication of the values of the vector and scalar.

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17. In studying the motion of an object, what is the actual path of the object referred to as? a. Projectile b. Displacement vector c. Gravity d. Trajectory Answer: d. The actual path of the object is referred to as the trajectory.

This can be seen by looking at the path of the thrown object.

18. In the study of the motion of an object, what is the object referred to as? a. Projectile b. Displacement vector c. Gravity d. Trajectory Answer: a. The object being thrown is referred to as the projectile.

Remember that it can be thrown in any direction and that it follows the law of gravity after that.

19. You are shooting a projectile out of a cannon and you want the maximum range. The effect of air resistance being negligible, what is this angle? a. 25 degrees b. 38 degrees c. 45 degrees d. 63 degrees Answer: c. Without the effect of air resistance, you need to be shooting the projectile at 45 degrees. This will provide the greatest range. The number is not going to be 0 degrees, for example, because gravity will kick in and will drop the projectile to the ground before it reaches a maximal range.

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20. You are shooting a projectile out of a cannon and you want the maximum range; however, there is air resistance to take into account. What will the optimal angle be to get the maximum range? a. 25 degrees b. 38 degrees c. 45 degrees d. 63 degrees Answer: b. Because air resistance on earth plays a role, the optimal angle of shooting something out of a cannon will be 38 degrees.

21. What is not a relationship between the time of flight and the following variables when it comes to a projectile? a. It is inversely proportional to the force of gravity b. It is directly proportional to the initial velocity c. It is proportional to the square of the initial velocity d. It is related to the sine of the angle the projectile is shot at Answer: c. Each of these is true except that the time of flight is not proportional to the square of the initial velocity but is instead directly proportional to the velocity in the equation.

22. In discussing relative velocity on a plane, what is this definition? a. The velocity relative to the plane b. The velocity relative to the earth c. The velocity relative to person on the plane d. The velocity relative to whatever you decide it is Answer: d. The relative velocity in any two-dimensional situation is whatever you decide it is. There will be relative velocity associated with any of these things.

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23. A boat is traveling a 100-meter river at 1 meter per second that has a current of 2 meters per second. How far down the river will the boat travel when it reaches the opposite shore? a. 100 meters b. 133 meters c. 200 meters d. 340 meters Answer: c. The boat will cross the river in 100 seconds. In the same point in time (100 seconds), the current will take the boat 200 meters downstream.

24. What factor most determines the inertia of an object? a. The force of gravity b. The object’s roughness c. The object’s mass d. The friction of the surface Answer: c. Strictly speaking, mass is the determining factor in the inertia of something. Things with a heavier mass will have greater inertia. Because inertia can involve a non-moving object, things like roughness or friction of the surface to not apply to the object. Mass is a scalar quantity while weight involves the force of gravity, making weight a vector and mass a scalar quantity.

25. You are sitting on a boat and push on the bow of the boat at maximal force.

How far will you move the boat? a. You will not go in any direction. b. You will move the boat in the direction of the bow at a maximal force. c. You will move the boat at less than the maximal force. d. The amount you will move depends on your total mass.

Answer: a. Because you are in the boat, you will not provide an external force on the boat so the boat will not move in any direction.

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26. What is the accepted SI unit for force? a. Kilograms b. Grams per second squared c. Pounds d. Newtons Answer: d. The SI unit for force is the Newton. One Newton is described as 1-kilogram meter per second squared. It is the force necessary to take a kilogram to an acceleration of a meter per second squared. The pound is also a unit of force used in the US. 27. According to Newton’s law, what is the relationship between mass, acceleration, and force? a. Force is directly proportional to mass and acceleration b. Force is inversely proportional to mass and acceleration c. Force is proportional to mass and inversely proportional to acceleration d. Force is inversely proportional to mass and proportional to acceleration Answer: a. Because force is equal to mass times acceleration, force is considered directly proportional to mass and acceleration.

28. Rocket thrust is a type of force that propels a rocket. What would happen if the rocket thrusted in a vacuum? a. It would propel in the same way. b. It would not propel at all. c. It would propel to a lesser degree. d. It would propel to a greater degree.

Answer: d. A rocket in a vacuum would propel to a greater degree. This is because the rocket propulsion does not cause thrust by pushing against air or against the ground. It will propel better in a vacuum because it does not have to deal with the friction due to air in the system.

29. Why is it important to define the system when dealing with forces?

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a. Because different systems will have different masses that will affect the system as a whole. b. Because a system should be the simplest object in any given grouping of objects. c. Because there needs to be an external force in order to have movement of an object. d. Because you need to encompass the greatest number of components in order to define the system.

Answer: c. There needs to be an external force in order to have movement of an object in the system. If you include things that just involve internal forces, the forces will cancel each other out and there will be no net movement within the system.

30. What factor is not true of Normal Force? a. It is a force that is perpendicular to the surface. b. It opposes the weight of an object on a surface. c. It is a force that applies to a surface. d. It is a force that is perpendicular to the ground.

Answer: d. The normal force applies to a surface and is perpendicular to the surface, opposing the weight of the object on the surface. It is not necessarily perpendicular to the ground because the surface itself can be an angular surface that is of itself not parallel to the ground.

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31. You are looking to solve an equation regarding placing a weight on an incline.

How do you set the x and y axes on a coordinate system? a. The x-axis is the earth, while the y-axis is perpendicular to that. b. The x-axis is the incline, while the y-axis is perpendicular to that. c. The x-axis is the downward force, while the y-axis is perpendicular to that. d. The y-axis is the downward force, while the x-axis is perpendicular to that.

Answer: b. The x-axis has been set as the incline, while the y-axis is considered to be perpendicular to that. This is different from the coordinates in other force equations. In addition, the angle theta is different and is the angle of the incline, which is ninety degrees minus the angle between the downward weight vector and the x axis.

32. What is not true of the force of tension? a. It is described as the force pushing on an object. b. It is described as the force exerted by a flexible connector. c. It exerts a force that is parallel to an object’s length. d. It can include a cable, cord, or similar connector.

Answer: a. While this is generally a flexible connector like a cable or cord, it is not necessarily elastic. It can only exert a force that is parallel to its length. It only has the capacity to pull an object; it cannot push on an object.

33. In the case of a person standing on a tightrope slightly to the left of the end of the tightrope, which vector or force will be the smallest in magnitude? a. Tension on the left side b. Tension on the right side c. Weight of the person d. Net force on the system Answer: d. The net force must be zero because the person is not moving. In fact, the net force is the sum of each of the other forces.

34. Which type of force in physics is only attractive?

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a. Weak nuclear forces b. Strong nuclear forces c. Gravitational force d. Electromagnetic forces Answer: c. Each of these forces has attractive and repulsive characteristics except for gravitational force, which is only attractive.

35. What is the tensile strength of an object? a. The elasticity of the object b. The force that will cause deformation of an object c. The force that will break an object to cause permanent deformation d. The resistive force of an object against an external force Answer: c. The tensile strength is the force that will break an object, ultimately causing permanent deformation of the object.

36. What is the relationship between stress and strain of an object being compressed or stretched? a. Stress and strain are inversely proportional to one another b. Stress and strain are the same thing c. Stress is proportional to the square of the strain d. Stress is directly proportional to strain Answer: d. Stress and strain are proportional to each other with stress equaling the Young’s modulus multiplied by the strain with Young’s modulus being a constant depending on the substance being acted upon.

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37. Which force is exerted perpendicular to the length of a nail? a. Shear force b. Compression c. Tension d. Elastic force Answer: a. Shear force is a force that is exerted perpendicular to the length of a cylindrical object, such as a nail. Compression and tension are applied parallel to the length of the nail or cylindrical object.

38. How many radians exist in 180 degrees of a rotation angle? a. Pi b. 2-Pi c. One-half pi d. One-third pi Answer: a. The totality of the number of radians in 180 degrees is defined by the circumference of the circle will be pi because 180 degrees is half of 2-pi.

39. How many degrees is one radian? a. 24.5 b. 57.3 c. 72.1 d. 90 Answer: b. One radian is the equivalent of about 57.3 degrees. This is based on the calculation that 2-pi is 360 degrees and that pi is about 3.14.

40. What is the SI unit for angular velocity? a. Meters per second b. Degrees per second c. Meters per minute d. Radians per second 344

Answer: d. The SI unit for angular velocity is radians per second. This is different from the linear velocity, which is the arc length traveled per second, which is in meters per second.

41. What is the direction of the angular speed of an object going in a uniform circle? a. Tangent to its path in the circle b. In a clockwise direction c. In a counterclockwise direction d. In either a clockwise or counterclockwise direction Answer: d. The angular speed can only be in two directions, clockwise or counterclockwise. This is measured in radians per second.

42. What is the direction of the centripetal acceleration of something traveling in a circle? a. In the direction of the circle’s velocity b. In the tangent of the circle c. Outward from the center of the circle d. Inward toward the center of the circle Answer: d. The direction of the centripetal acceleration vector is always in the direction toward the center of the circle.

43. What is the relationship between the linear velocity of a point in a circle and the centripetal acceleration? a. Centripetal acceleration is proportional to the square of the linear velocity. b. Centripetal acceleration is inversely proportional to the square of the linear velocity. c. Centripetal acceleration is proportional to the linear velocity. d. Centripetal acceleration is inversely proportional to the linear velocity.

Answer: a. The relationship between linear velocity and the centripetal acceleration in the movement around a circle is that the centripetal acceleration is proportional to the square of the linear velocity.

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44. What is the direction of the centripetal force vector? a. Inward toward the center of the circle b. Outward away from the center of the circle c. In the direction of the linear velocity of a vehicle d. In the direction of the angular velocity of a vehicle Answer: a. The centripetal force vector is located inward toward the center of the circle. It is in the same direction as the centripetal acceleration but is related to the mass of a vehicle or object.

45. There is an ideal banked angle for a frictionless surface of a car going around a curve that will keep the vehicle from spinning out. How does this angle relate to the mass of the vehicle? a. The angle is proportional to the mass of the vehicle b. The angle is proportional to the sine of the mass of the vehicle c. The angle is proportional to the cosine of the mass of the vehicle d. The angle is unassociated with the mass of the vehicle Answer: d. The ideal angle is related to the negative one tangent of the velocity squared divided by the radius and divided by the force of gravity. It is unrelated to the mass of the vehicle.

46. According to the Coriolis effect, what direction do hurricanes and cyclones rotate in the Northern and Southern Hemisphere? a. Clockwise in both hemispheres b. Counterclockwise in both hemispheres c. Counterclockwise in the northern hemisphere and clockwise in the southern hemisphere d. Clockwise in the northern hemisphere and counterclockwise in the southern hemisphere Answer: c. The direction of curvature of the hurricanes and cyclones is counterclockwise in the Northern Hemisphere and clockwise in the

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Southern Hemisphere. This is because of the Coriolis effect on wind patterns on earth.

47. Of the four basic forces on earth, which is the weakest force? a. Strong nuclear forces b. Weak nuclear forces c. Electromagnetic forces d. Gravitational forces Answer: d. Gravitational force on earth is the weakest force of the four basic forces. Remember that it is based on the mass of the earth and the distance from it.

48. What is not a reason why a person would weigh differently on different parts of the earth? a. The earth is not perfectly round. b. The gravitational pull is based on the actual center of the earth. c. Mass of the person will differ at the equator than on the north pole. d. Mineral deposits beneath the earth will affect the weight of the person.

Answer: c. Each of these will affect the weight of a person on earth, including things like the presence of mountains that change the distance of a person from the center of the earth, although the mass of a person does not change, regardless of where the person is on earth— or anywhere in the universe. The gravitational pull is based on the center of the earth and the earth isn’t perfectly round.

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49. What affects the tides of the oceans the most? a. The angles of the sun, moon, and earth at any given point in time. b. The distance between the sun and the earth. c. The distance between the moon and the earth. d. The rotation of the earth over a 24-hour period of time.

Answer: a. The angle of the sun, moon and earth at any given point in time will affect the tides the most. The moon exerts a greater effect on the earth’s tides than the sun but both will have an effect on tides. The tides are greatest when the sun, moon, and earth are in alignment with each other.

50. What do Kepler’s laws mainly refer to? a. The orbits of planets around the sun b. The orbit of the moon around the earth c. The orbits of satellites around the earth d. The orbits of any of these things in space Answer: d. Kepler’s laws refer to the elliptical orbit of any celestial body around any other celestial body, which can be the earth around the sun, the satellites around the earth, and the moon around the earth. In fact, it applies to any celestial body orbiting around a larger celestial body.

51. What must be true in order to have force engage in work? a. The force must be at least to some degree in the desired direction. b. There must be sufficient force in order to overcome gravity. c. The force must be opposing another force. d. The force must be in a horizontal direction.

Answer: a. Force in the desired direction exerts work. There can be many forces associated with a system; however, if there is no component of force that is in the desired direction, there can be no work exerted.

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52. What is the SI unit for energy? a. Newtons b. Kilocalories c. Calories d. Joules Answer: d. The SI unit for energy is Joules, which is in the form of Newton-meters. This is the same unit used for work because, in many ways, energy and work are related, although they are not strictly the same thing.

53. Which energy unit is considered the smallest? a. Calorie b. Food calorie c. Joules d. Kilojoules Answer: a. One calorie is the smallest unit, equal to slightly more than 4 Joules. A food calorie is the same thing as a kilocalorie from a scientific perspective.

54. You recognize that it takes 100 Newtons of force to push an object one meter.

What is the kinetic energy involved in this? a. One joule b. Ten joules c. 100 Joules d. This cannot be determined without knowing the mass of the object Answer: c. In its simplest form, kinetic energy equals force times the distance so that things like mass are already in the equation. In this case, 100 Newtons times 1 meter equals 100 Joules.

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55. When looking at the different forces acting on a system that together determine the kinetic energy of a system, what is not a force involved? a. Weight of the item b. Applied force c. Force of friction d. All of these apply Answer: d. All of these are forces that may apply to a system, depending on the system involved. Remember that weight is a force related to gravity and might apply to a system if work is used in the direction of or opposite to the direction of gravity.

56. When is an item or object said to be at zero gravitational potential energy? a. At the earth’s surface b. At the top of a table or other surface c. At its maximum height in the system d. At its minimum height in the system Answer: a. It’s important to recognize that zero gravitational potential energy is at the earth’s surface but that, when doing work, this setpoint is arbitrary and the important thing to remember is that the difference in two potential energies is more important than the absolute value of its gravitational potential energy.

57. What is not a conservative force? a. The force of a spring that is compressed. b. The force of a wind-up toy that is wound up. c. The force of a forklift raising a box. d. The weight of an object on the edge of a cliff.

Answer: c. The conservative force must be a potential energy of some sort. Each of these represent a form of potential energy except for the raising of a box on a forklift, which is a form of kinetic energy and not potential energy.

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58. In a system where there are only conservative forces, what value remains constant? a. Mechanical energy only b. Kinetic energy only c. Potential energy and kinetic energy each are a constant d. Potential energy only Answer: a. Mechanical energy is potential energy plus kinetic energy.

These will remain a constant and will just change in form from kinetic to potential energy and vice versa.

59. What is true of nonconservative forces? a. They oppose the conservative forces. b. They do not change the mechanical energy of a system. c. They depend on the path of the object. d. They decrease the total energy of a system.

Answer: c. Nonconservative forces depend on the path of the system.

They may or may not oppose the conservative forces and, by definition, they change the mechanical energy of a system but do not change the total energy because energy cannot be created or destroyed.

60. Which type of energy is not considered a form of “other energy” that must be quantified if an energy equation is to be complete so that energy is not created or destroyed? a. Radiant energy b. Gravitational energy c. Thermal energy d. Chemical energy Answer: b. Gravitational energy is not considered an “other energy” because this is basically potential energy. The others are energy forms that are added or take away from potential and kinetic energy so that,

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in accurate energy statements, these must be considered to make sure that energy is not created or destroyed.

61. What is the SI unit for power? a. Horsepower b. Joules c. Watts d. Calories Answer: c. The SI unit for power is watts, which is the same thing as saying joules per second. Horsepower is a non-SI unit for power, which is equal to 746 watts.

62. About how many watts go into one horsepower? a. 4 watts b. 130 watts c. 322 watts d. 746 watts Answer: d. There are 746 watts in one horsepower. Horsepower is sometimes used to define units of power as it is more practical with larger systems but this does not represent an SI unit for power.

63. What units are electricity bills listed in when it comes to the energy you pay for? a. Kilowatts b. Horsepower c. Kilowatt-hours d. Watts per minute Answer: c. When paying an electric bill, you are paying for energy used and not for power. This means that you are paying for power multiplied by time or power consumption, which has the units of kilowatt-hours.

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64. What is the proportionality between momentum and velocity? a. Momentum is inversely proportional to velocity b. Momentum is proportional to velocity squared c. Momentum is proportional to velocity d. Momentum is proportional to acceleration but not to velocity Answer: c. In simplest forms, momentum equals mass times velocity, making it directly proportional to the velocity of something.

65. What is the best definition of impulse in the study of physics? a. The length over which momentum is provided b. The length over which force is provided c. The time in which momentum occurs d. The time over which a force is applied Answer: d. Impulse is a change in momentum or the time over which a force is applied. It means that small forces need a longer time in which to cause a change in an object. Airbags will lengthen the time over which a force is applied, decreasing the impulse.

66. Does using an airbag change the impulse that occurs during a crash? a. Yes, because it changes the forces b. Yes, because it changes the time period c. Yes, because it changes the time period and the force d. No, it does not change the actual impulse Answer: c. The airbag will change the force as it spreads it out over a larger square area and the time period for the force to be applied so it will change the impulse on both aspects of what impulse means.

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67. You calculate the change in momentum of two objects and do not find experimentally that momentum has been conserved. Which is not a reason why this might be? a. This is because there has been an external force that is not a part of the system in question. b. This is because the system chosen was not large enough. c. This is because forces like friction were not taken into account in the calculation. d. This is because conservation of momentum only applies to masses of the same size.

Answer: d. There is always conservation of momentum if the system size is large enough, if no external forces might be in play, and because forces like friction need to be taken into account.

68. Under what condition is a collision more likely to be elastic? a. Two vehicles on a gravel road b. Two hockey pucks on an air table c. A football player hitting a goal post d. An electron and a proton Answer: d. True elastic conditions exist mostly in situations of the collisions of subatomic particles; however, when it comes to macroscopic conditions, the condition of two hockey pucks on an air table is the next-best form of an elastic condition as friction is less likely to be a problem.

69. What is not an example of an inelastic collision? a. Objects that stick together during the collision process b. Objects that deform during the collision process. c. Objects that collide on a frictionless surface. d. Objects that explode during a collision.

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Answer: c. The inelastic collision will have the objects sticking together, deforming, or exploding during the collision, each of which take the internal kinetic energy out of the system or add kinetic energy. In a frictionless system, there is generally an elastic situation rather than an inelastic situation.

70. In the situation where an object strikes another object in the x-direction and both objects travel off in opposite directions, what is the velocity component in the direction of angle theta1 of the object that goes at that angle (theta1) from the x-axis? a. Velocity after the collision multiplied by the sine of theta1. b. Velocity after the collision multiplied by the cosine of theta1. c. Velocity before the collision multiplied by the sine of theta1. d. Velocity before the collision multiplied by the cosine of theta1.

Answer: b. The velocity will not be the same as the velocity before the collision because of the collision. As the object heads of to an angle of theta1, its velocity in the x-axis will be the velocity after the collision multiplied by the cosine of theta1.

71. In calculating equations in which objects collide in two directions, what is something that does not have to be assumed? a. The objects cannot rotate b. The objects are considered to be point masses c. Momentum must be conserved d. Kinetic energy is distributed equally between the two objects Answer: d. The equations depend on the fact that the objects themselves cannot rotate and that they are considered point masses.

As always, momentum must be conserved; however, the kinetic energy will not be distributed equally between the two objects as it depends on the objects’ weights and their starting velocities.

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72. In the collision of two objects in two dimensions, what aspects of momentum will be conserved assuming there is no rotation of the objects? a. Total momentum, momentum in the x-direction, and momentum in the ydirection b. Total momentum only c. Momentum only in the direction of the collision, which is the x-direction in most cases d. Momentum is not conserved in all cases Answer: a. The momentum is going to be preserved and, because momentum is a vector, it will also be preserved in the x- and ydirections.

73. What is the main difference between elastic and inelastic collisions? a. Elastic collisions have momentum and kinetic energy conserved and inelastic collisions have only kinetic energy conserved. b. Elastic collisions involve rotational momentum and inelastic collisions do not. c. Elastic collisions have momentum and kinetic energy conserved and inelastic conditions have only momentum conserved. d. Elastic collisions involve similarly-sized objects, while inelastic collisions involve large differences in size of the objects.

Answer: c. The elastic collision will have momentum and kinetic energy conserved while inelastic collisions have only momentum conserved (kinetic energy can be added or taken away).

74. What is the best definition of torque? a. Torque is the angle that force is applied to an object. b. Torque is the force that causes angular acceleration of a circular object. c. Torque is force that opposes circular motion. d. Torque is a force that results in rotation around an axis.

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Answer: d. The correct definition of torque is a force that results in rotation of something around an axis. It is a force applied linearly as a vector that causes spinning or rotation.

75. What is the proportionality of torque to the radius from the pivot point and the force applied? a. Torque is directly proportional to the radius and the force applied. b. Torque is directly proportional to the radius squared and the force applied. c. Torque is inversely proportional to the radius and proportional to the force. d. Torque is inversely proportional to the radius and proportional to the force squared.

Answer: a. Torque is directly proportional to the radius and the force applied. It can be defined as force times the perpendicular radius or force times the radius times the cosine of theta if the force isn’t applied perpendicular to the end of the object.

76. What are the SI units for torque? a. Newtons per meter b. Newton-meters c. Newtons per second d. Kilogram meters Answer: b. The SI units for torque are Newton-meters, in which Newtons is the force applied and meters is the perpendicular radius.

77. A pencil lying flat on the surface of a flat table is in what type of equilibrium? a. Neutral equilibrium b. Limited neutral equilibrium c. Stable equilibrium d. Unstable equilibrium Answer: b. This is an object in limited neutral position because it can be moved in one direction that would cause it to roll but it cannot be

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moved via torque with the pivot point along its length because it is already lying flat.

78. A marble in the bottom of a bowl is said to be in what type of equilibrium? a. Neutral equilibrium b. Limited neutral equilibrium c. Stable equilibrium d. Unstable equilibrium Answer: c. This is in stable equilibrium because any action against it will bring it back into equilibrium so it cannot get out of this state unless continuous force is applied to it.

79. A ball is on the very top of a hill. What is its equilibrium status? a. Neutral equilibrium b. Limited neutral equilibrium c. Stable equilibrium d. Unstable equilibrium Answer: d. This is in unstable equilibrium because any force applied to it will topple the ball as it sits on the pivot point, which is the top of the hill.

80.In a lever or fulcrum situation, what is the proportionality to input force and input length of the lever as it relates to the mechanical advantage? a. The mechanical advantage is inversely proportional to the input force and inversely proportional to the input length. b. The mechanical advantage is inversely proportional to the input force and directly proportional to the input length. c. The mechanical advantage is directly proportional to the input force and directly proportional to the input length. d. The mechanical advantage is directly proportional to the input force and inversely proportional to the input length.

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Answer: b. The mechanical advantage can be described in terms of ratio of Force out and Force in as well as the ratio between the length in and the length out. This means that the mechanical advantage is inversely proportional to the force input and directly proportional to the length input so the longer the length of a lever, the greater is the mechanical advantage.

81. An object is lifted using two pulleys that exert two pounds of force each.

What is the mechanical advantage of each pulley? a. One b. Two c. Four d. One-half Answer: a. The mechanical advantage of one pulley will be one so the total mechanical advantage on the system is two. The more pulleys used, the greater is the mechanical advantage. Each pulley just changes the direction of the force but not the magnitude of the force.

82. What are the units for angular acceleration? a. Meters per second b. Kilogram meters per second squared c. Radians per second d. Radians per second squared Answer: d. The angular acceleration is equal to radians per second squared, because it is the change in angular velocity divided by the change in time.

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83. In understanding the dynamics of rotational movement as in trying to open a door, what is the relationship between force, angular acceleration, and the mass of the door? a. Angular acceleration is proportional to the force applied and the mass of the door. b. Angular acceleration is proportional to the force applied and inversely proportional to the mass of the door. c. Angular acceleration is inversely proportional to the force applied and proportional to the mass of the door. d. Angular acceleration is inversely proportional to the force applied and inversely proportional to the mass of the door.

Answer: b. If you think about it, the greater the force applied, the greater is the angular acceleration. In addition, the more massive the door (the greater its mass), the less is the angular acceleration when opening the door.

84. In looking at the moment of inertia of a rotating hoop, what is the equation for this value? a. Inertia equals the mass times angular acceleration b. Inertia equals mass times the force of gravity c. Inertia equals mass times the radius squared d. Inertia equals mass times the linear velocity Answer: c. Inertia equals mass times the radius squared. In a hoop situation, it involves the mass of the entire hoop and the radius of the hoop.

85. In looking at the moment of inertia of a rotating disc around its axis, what is the equation for this value? a. Inertia equals one-half mass times the radius squared b. Inertia equals mass times the radius squared divided by pi c. Inertia equals mass times the radius

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d. Inertia equals mass time the radius squared divided by twelve Answer: a. In looking at a simple rotating disc, you get the inertia equal to one-half of the mass times the radius squared. Note that this is slightly different from the hoop situation. You’ll find inertia equations for cylinders and spheres as well that are similar but differ in proportionality.

86. Why does a skater have greater angular velocity when she pulls her arms in? a. She increases the external torque on the system b. She increases the angular momentum in the system c. She increases her center of mass d. She decreases her moment of inertia Answer: d. The angular momentum will always be conserved. The net external torque will be minimal on ice and the center of mass will be the same regardless of her position. The moment of inertia is inversely proportional to the square of the radius so decreasing the radius will decrease the moment of inertia, increasing the angular velocity because the inertia times the angular velocity will stay the same.

87. What will the net force be on the pivot point of a lever, such as a tennis racket in a hand, at the percussion point? a. It will be zero. b. It will be negative and related to the mass and velocity of the incoming ball. c. It will be positive and related to the angular momentum of the lever after it is hit. d. It will be positive and related to the moment of inertia of the system when the ball hits the lever.

Answer: a. At the percussion point, the net force on the pivot point will be zero because of its distance from the pivot point of the lever.

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88. You spin a disc in the counterclockwise direction. Where will the direction be of the angular momentum? a. Counterclockwise b. Tangential to the direction of rotation c. Upward along the axis of rotation d. Downward along the axis of rotation Answer: c. The direction of angular momentum and angular velocity will be upward along the axis of rotation, using the right-hand rule.

89. What are the SI units for pressure? a. Pascals b. PSI or pounds per square inch c. Mm Hg or millimeters of mercury d. Newtons Answer: a. Pressure is defined as the force applied to a square area, which in SI units is the Pascal or Newtons per meter squared.

90. What is not considered the atmospheric pressure at sea level? a. One atmosphere b. 760 mm Hg c. 1.01 Kilopascals d. 1 torr Answer: d. The atmospheric pressure at sea level is equivalent to about 760 torr, 760 mm Hg, 1.01 kilopascals, or 1 atmosphere.

91. According to Pascal’s principle, what is true? a. Pressure of a fluid in an open system will increase when force is added. b. Total pressures in an enclosed system of different substances are additive. c. The pressure in an enclosed system of fluid diminishes in the interior of the system. d. Total pressures in an enclosed system of different substances will multiply when force is applied.

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Answer: b. The pressure in an enclosed system under pressure will transmit undiminished and will be additive if there is more than one substance in the system, according to Pascal’s principle.

92. What is the relationship between the force on a piston, the pressure, and the surface area of the piston? a. The pressure exerted is force times the surface area. b. The pressure exerted is surface area divided by force. c. The pressure exerted is force divided by surface area d. The pressure is inversely proportional to the force and to the surface area.

Answer: c. The equation is pressure equals force divided by surface area. It is used to drive pistons because a small force over a small surface area can cause a large force over a larger surface area when the pressure is the same.

93. What is the specific gravity of something traditionally? a. The ratio of an object’s density to that of the density of water. b. The ratio of the submerged volume to the total volume of an object. c. The percent of an object that is floating above the surface of water. d. The percentage of an object that is submerged in water.

Answer: a. The specific gravity of an object is the ratio of its density to that of the density of water. In urine evaluations, the specific gravity of urine will be higher than one because the urine is a solution that is denser than plain water.

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94. What device will measure the specific gravity of a fluid? a. Manometer b. Barometer c. Hydrometer d. Sphygmomanometer Answer: c. A hydrometer will measure the specific gravity of a fluid by submerging a weighted tube in the fluid. The tube is calibrated to measure the specific gravity. Manometers measure pressure as do sphygmomanometers, which measure blood pressure. Barometers measure atmospheric pressure.

95. What are the SI units for the surface tension of a liquid? a. Kilograms per meter squared b. Newtons c. Kilograms per meter d. Newtons per meter Answer: d. The surface tension is related to the force per linear meter so its units are Newtons per meter.

96. What property of water gives it its greater surface tension than alcohol? a. Density b. Cohesive forces c. Adhesive forces d. Mass Answer: b. The cohesive forces of a liquid like water are the forces that exist between molecules of the same type. This is greater in water, which is a polar molecule. Alcohol is not polar and has a lesser cohesive force between its molecules.

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97. What is the SI unit for the flow rate of a liquid? a. Meters per second b. Liters per second c. Liters per minute d. Cubic meters per second Answer: d. The flow rate for a liquid is the volume of liquid flowing divided by the time. It is equal to liters per minute or liters per second; however, these are not the SI units for fluid flow, which are cubic liters per minute.

98. How does the flow rate of a fluid in a conduit compare to its cross-sectional area and the velocity of the fluid? a. The flow rate is inversely proportional to the cross-sectional area and proportional to the velocity. b. The flow rate is proportional to the cross-sectional area and proportional to the velocity. c. The flow rate is proportional to the cross-sectional area and inversely proportional to the velocity. d. The flow rate is inversely proportional to the cross-sectional area and inversely proportional to the velocity.

Answer: b. As flow rate is the cross-sectional area times the velocity, these are directly proportional to the flow rate.

99. Because of the continuity equation for fluids, what is the relationship between the cross-sectional area and the speed of flow of an incompressible liquid in a conduit? a. The speed of flow must remain constant, regardless of the cross-sectional area. b. The speed will increase in proportion to the cross-sectional area. c. The speed will decrease in proportion to the cross-sectional area. d. The square of the speed will be proportional to the cross-sectional area.

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Answer: c. Because of the continuity equation for incompressible liquids, the speed will decrease as the cross-sectional area increases, thus they are inversely proportional to one another.

100.

According to the Bernoulli’s equation, the change in pressure of a fluid is

related to what aspect of its density and velocity when the potential energy is not changed? a. The change in pressure is proportional to its density and its change in velocity squared. b. The change in pressure is proportional to its density and its change in velocity. c. The change in pressure is inversely proportional to its density and proportional to its velocity. d. The change in pressure is unrelated to its density and proportional to its velocity squared.

Answer: a. The change in pressure equals one-half times its density multiplied by the change in velocity squared. What this means is that, when the change in velocity is squared, this is proportional to the change in pressure.

101.

What will not affect the resistance to flow of a fluid?

a. Viscosity of the fluid b. Length of the tube it flows in c. Pressure differential across a tube d. Diameter of the tube Answer: c. Each of these will affect the resistance. Viscosity and the length of the tube will increase the resistance and the increased diameter of the tube will decrease resistance and turbulence will increase resistance. The pressure differential across a tube will not affect resistance but will affect the flow rate.

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102.

What is the relationship between the resistance, the viscosity of a fluid and

diameter of the tube with regard to the flow through a tube of a fluid, according to Poiseuille’s law? a. The resistance is proportional to the radius squared and the viscosity. b. The resistance is proportional to the viscosity and inversely proportional to the radius squared. c. The resistance is inversely proportional to the viscosity and to the radius cubed. d. The resistance is proportional to the viscosity and inversely proportional to the radius to the fourth power.

Answer: d. There is Poiseuille’s law, which states that the resistance through a tube is proportional to the viscosity and inversely proportional to the radius to the fourth power. This means that resistance is markedly different when the radius of the tube is increased or decreased but only linearly related to the coefficient of viscosity.

103.

What does the Reynolds number predict in the flow of fluid going through

a tube? a. The pressure of the fluid b. The laminar versus turbulent flow of the fluid c. The viscosity of the fluid d. The rate of flow of the fluid Answer: b. The Reynolds number predicts the laminar versus turbulent flow of the fluid so that the lower the number, the higher is the chance of the flow being laminar.

104.

At high Reynolds numbers for an object traveling through a viscous fluid,

such as water or even air, what is the drag force related to when it comes to speed? a. The drag force is inversely proportional to the square of the speed

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b. The drag force is inversely proportional to the speed c. The drag force is proportional to the square of the speed d. The drag force is proportional to the speed Answer: c. At high Reynolds numbers, the drag force is proportional to the square of the speed but at low Reynolds numbers, it is proportional to the speed only and not the square of the speed. This means that things like headwind will greatly affect the drag force of cyclists riding in a race.

105.

What is the best definition of osmotic pressure in a solution?

a. The pressure between two membranes when the net transfer of water is zero. b. The pressure between two membranes when the net transfer of solute is zero. c. The net pressure between two membranes when there is more water on one side than the other. d. The pressure exerted on a membrane by solutes on one side of the membrane.

Answer: a. The pressure between two membranes when the net transfer of water is zero is the osmotic pressure. It depends on the concentration of solute on both sides of the membrane and, in many cases, one side of the membrane has pure water in it and the other has water and solute in it.

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106.

What is the difference between one degree in Celsius and one degree in

Kelvin? a. The difference is that Celsius is 9/5 of a degree Kelvin b. The difference is that Celsius is 5/9 of a degree Kelvin c. The difference is that Celsius degrees are twice as big as Kelvin degrees d. There is no difference between one degree in Celsius and one degree in Kelvin Answer: d. One degree in Celsius measurements is the same thing as one degree in Kelvin measurements. The main difference is that they differ in their starting points and ending points but not in the absolute value of what a degree difference means.

107.

What is the difference between one degree in Celsius and one degree in

Fahrenheit? a. The difference is that Celsius is 9/5 of a degree Fahrenheit b. The difference is that Celsius is 5/9 of a degree Fahrenheit c. The difference is that Celsius degrees are twice as small as Fahrenheit degrees d. There is no difference between one degree in Celsius and one degree in Fahrenheit Answer: a. There is a substantive difference between a degree in Celsius and a degree in Fahrenheit. A degree in Celsius is 9/5 or 1.8 of a degree in Fahrenheit.

108.

What temperature in Celsius is said to be “standard temperature”?

a. 0 degrees Celsius b. 10 degrees Celsius c. 25 degrees Celsius d. 50 degrees Celsius Answer: c. Standard temperature is the same thing as “room temperature”, which is about 25 degrees Celsius. Many chemical

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processes are said to occur at standard temperature, which should be said to be this temperature unless specified otherwise.

109.

In kinetic theory, what is something that does not have to be an

assumption? a. Gas molecules have elastic collisions with no energy lost in the collision. b. Transfer of kinetic energy is given off as heat. c. Atoms of gas have no mass. d. There is random motion of gas particles with wide spaces between molecules.

Answer: c. Each of these is an assumption that must be made with the exception of the assumption that molecules or atoms of a gas have no mass.

110.

In the Maxwell-Boltzmann distribution related to kinetic theory, what

does this distribution relate to? a. The mass of gas molecules in a given system. b. The probability of velocities of gas particles. c. The temperature equilibrium of gases in a container. d. The pressure differential within a mixture of gases.

Answer: b. The probability of velocities of gas particles in a container is what the Maxwell-Boltzmann distribution relates to.

111.

According to the Maxwell-Boltzmann distribution, what is not described as

part of this? a. The pressure of gases will vary with temperature. b. The heavier molecules will move slower than lighter molecules. c. The speed of molecules increases with temperature. d. There is a lesser range of speed with lighter molecules than with heavier molecules.

Answer: a. The Maxwell-Boltzmann distribution is a distribution probability molecule that indicates that heavier molecules will move

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more slowly than lighter molecules but will have a wider probability of distribution. In addition, the speed of molecules will increase with temperature as will the distribution range.

112.

What is the average translational kinetic energy of a molecule most related

to when it comes to a gaseous situation? a. Pressure b. Temperature c. Molecular mass d. Volume Answer: b. The average translational kinetic energy of a molecule is most related to the temperature and will vary with the temperature of a molecular situation. It is independent of the mass of the molecule and volume and is only related to pressure because pressure is temperature-related.

113.

What is the linear coefficient of expansion of a solid most related to?

a. The composition of the solid b. The temperature c. The linear length of the solid d. The mass of the solid Answer: a. While the coefficient of expansion is slightly related to temperature, it is mostly related to the composition of the solid. The change in length will be equivalent to its total length, the coefficient of expansion, and the change in temperature. For large changes in temperature, the average coefficient of expansion is used in the calculation.

114.

At what temperature is water its densest?

a. 25 degrees Celsius b. 0 degrees Celsius c. 4 degrees Celsius

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d. Minus 40 degrees Celsius Answer: c. Water has unique properties and will increase volume and decrease density above four degrees Celsius. It will also decrease density and increase volume below this temperature, making water its densest at 4 degrees Celsius.

115.

Under constant pressure, what happens to an ideal gas when the

temperature is changed? a. There would be increased interaction between the molecules themselves. b. The volume will contract. c. The volume will expand. d. The volume will remain the same.

Answer: c. The volume would expand with a constant pressure. In an ideal gas, there is presumed to be no interaction between the molecules themselves.

116.

What is Avogadro’s number?

a. The number of molecules of hydrogen divided by the atomic mass of hydrogen. b. The volume of an ideal gas at zero degrees Kelvin. c. The atomic mass of a molecule divided by Boltzmann constant d. The number of molecules in a mole of a substance.

Answer: d. The number of molecules in a mole of a substance is considered to be Avogadro’s number. A mole is defined as the number of atoms or molecules found in 12 grams of carbon-12.

117.

What is the universal gas constant in SI units?

a. 8.31 b. 1.99 c. 0.082 d. 6.022

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Answer: a. The universal gas constant can be in many forms; however, the SI units are in Joules per mole per degree Kelvin, which is 8.31.

118.

The transfer of something in the solid phase to the gaseous phase is called

what? a. Condensation b. Melting c. Vaporization d. Sublimation Answer: d. Sublimation is the direct transfer of something from the solid phase to the gaseous phase without getting into a liquid phase. In fact, this happens to a great deal of the snow that falls.

119.

When it comes to water in air, what happens at the dew point?

a. Vapor pressure equals the partial pressure b. Fog will occur c. Relative humidity is 100 percent d. Each of these is true Answer: d. At the dew point of water in air, the relative humidity will equal 100 percent and there can be the formation of fog, which is air droplets in suspension. The vapor pressure will equal the partial pressure of water in air so that, at any lower temperature, there will be a prominence of condensation.

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120.

What is the best definition of a calorie?

a. It is the energy stored in food that ultimately does work. b. It is the amount of work done by a gram of water over one degree Celsius. c. It is the amount of energy necessary to raise a gram of water by one degree Celsius. d. It is the amount of energy transferred resulting in the loss of one degree Celsius of an object.

Answer: c. Strictly defined, a calorie is the amount of energy necessary to raise a gram of water by one degree Celsius.

121.

What is the main difference between heat transfer and work?

a. Work implies force done over a distance, which is not necessarily the case with heat transfer. b. Work involves a transfer of energy and heat transfer does not actually change energy. c. Work is spontaneous and yet heat transfer is not spontaneous. d. Work units are not in joules, while heat units are in joules.

Answer: a. It is a bit picky but, strictly speaking, work is defined as force acting over a distance, while heat transfer does not always imply a distance nor does it imply a force acting on something. Rather, it is the spontaneous transfer of energy that comes about just because of the differences in two objects’ temperatures.

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122.

What is the relationship between Q, which is heat transfer, and the specific

heat? a. The Q is proportional to the specific heat. b. The Q is inversely proportional to the specific heat. c. The Q is proportional to the square of the specific heat. d. The Q is inversely proportional to the square of the specific heat.

Answer: a. The Q is proportional to the specific heat, in which Q equals the mass times the specific heat times the change in temperature. The specific heat cannot easily be calculated but is looked up in tables. It is the amount of heat necessary to raise a kilogram of a substance by one degree Celsius.

123.

What is the relationship between specific heat and the temperature of a

substance? a. The specific heat is unassociated with the temperature. b. The specific heat is directly proportional to temperature. c. The specific heat is more related to the temperature of gases than the temperature of liquids and solids. d. The specific heat is inversely proportional to temperature.

Answer: c. The specific heat is somewhat temperature-related but this is more the case with gases than with liquids and solids. It is not a direct correlation and is more dependent on phase than on the absolute temperature of something.

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124.

What process takes the most energy to do (assuming a kilogram of water)?

a. Raise the temperature of water from 10 degrees to 11 degrees Celsius b. Raise the temperature of water from 80 degrees to 81 degrees Celsius c. Vaporize water d. Melt ice Answer: c. It takes more than 2000 kilojoules of energy to vaporize water, which involves no actual temperature change than it does to do any of the other things. Vaporizing water and melting ice involve breaking intermolecular bonds between molecules, and thus the energy required to do these things.

125.

What process takes the least amount of energy to do (assuming a gram of

water)? a. Raise gaseous water up a degree Celsius b. Raise liquid water up a degree Celsius c. Raise ice up a degree Celsius d. Change ice into water Answer: a. The energy required to raise gaseous water up a degree Celsius occurs at a rate of 0.482 calories per grams per degree Celsius, which is less than it takes to raise ice and liquid water up a degree and far less than the energy it takes to change a gram of ice into water.

126.

What is the mechanism involved in the conduction of heat between two

objects? a. Electromagnetic radiation is given off the hotter object to the colder object. b. Heat energy flows from one object to another. c. Heat is given off in the vicinity of the hot object, heating the surface area of the cold object. d. The molecular kinetic energy in hot areas is transferred to the lower kinetic energy of the cold areas.

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Answer: d. The two objects in conduction are in direct contact with one another so that the molecular kinetic energy in hot areas to the lower kinetic energy (on a molecular level) in colder areas until the temperature change is zero and the objects have the same temperature.

127.

What is the proportionality of heat transfer rates in the process of

conduction to the temperature difference and the thickness of the objects? a. The rate of transfer does not depend on the temperature difference and is inversely proportional to the thickness. b. The rate of transfer is proportional to the temperature difference and inversely proportional to the distance. c. The rate of transfer is proportional to the temperature difference and proportional to the thickness. d. The rate of transfer is inversely proportional to both the temperature difference and inversely proportional to the thickness.

Answer: b. The rate of transfer will be greater when the temperature difference is bigger between two objects and will be inversely proportional to the thickness of the two objects.

128.

What type of heat transfer is mostly involved in a forest fire?

a. Visible light radiation b. Infrared wave radiation c. Convection d. Conduction Answer: b. While there is heat transfer associated with the visible light seen in a forest fire, this plays little role in the transfer of heat energy.

Convection plays some role when it comes to wind and rising air from heating the air at the surface of the fire and the decreased density of hot air. Infrared energy radiation plays the major role in the transfer of heat energy in this situation.

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129.

What is true of the color of the object and the radiation of electromagnetic

radiation? a. Black is a poor absorber and a poor emitter of electromagnetic radiation b. Black is a good absorber and a poor emitter of electromagnetic radiation c. Black is a poor absorber and a good emitter of electromagnetic radiation d. Black is a good absorber and a good emitter of electromagnetic radiation Answer: d. What’s true is that things that are good absorbers of electromagnetic radiation are also going to be good emitters of electromagnetic radiation. What it means is that black, which absorbs this type of radiation well will also be a good emitter of this type of radiation.

130.

In determining the rate of heat radiating from an object, how does this

relate to the temperature of the object in degrees Kelvin? a. The heat radiating is directly proportional to the temperature. b. The heat radiating is directly proportional to the square of the temperature. c. The heat radiating is directly proportional to the cube of the temperature. d. The heat radiating is directly proportional to the fourth power of the temperature.

Answer: d. Interestingly, the rate of heat transfer in radiant heat is proportional to the fourth power of the temperature of the object in degrees Kelvin, according to the Stefan-Boltzmann law of radiation.

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131.

In looking at heat energy and work energy in a system, what defines these

energies to be positive or negative? a. Heat energy out of the system is positive and work energy out of the system is positive. b. Heat energy into the system is positive and work energy out of the system is positive. c. Heat energy into the system is negative and work energy into of the system is positive. d. Heat energy out of the system is negative and work energy into the system is positive.

Answer: b. In determining the internal energy of a system, the heat energy put into the system is said to be positive, while the work energy out of the system is said to be positive. Work and heat can go either way in a given system.

132.

In the human body, what is achieved by eating more and exercising less?

a. The potential energy will be increased and the kinetic energy will be decreased. b. The potential energy will be decreased and the kinetic energy will be increased. c. The potential energy will be increased and the kinetic energy will be increased. d. The potential energy will be decreased and the kinetic energy will be decreased.

Answer: a. In eating too much the potential energy will be increased, while exercising too little will decrease the kinetic energy of the body.

Exercise involves increasing work on the body, which increases the kinetic energy. Eating adds chemical potential energy to the body.

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133.

In a pressure/volume curve where work is calculated in a thermodynamic

system, what is the process called when the change in volume is zero? a. Adiabatic b. Isochoric c. Isobaric d. Isothermal Answer: b. Anytime the change in volume is zero, there will be no work done in a pressure/volume system and the process is called isochoric.

134.

In a thermodynamic system, what is the system called when the pressure

is kept the same and work is done based on the force and the change in volume? a. Adiabatic b. Isochoric c. Isobaric d. Isothermal Answer: c. An isobaric system is one in which the pressure is held at a constant and work represents the force on the system and a change in volume.

135.

How many cycles are looked at in the Otto cycle involving an engine that

turns heat into work? a. Two b. Four c. Six d. Eight Answer: b. There are four cycles in this type of engine, which is referred to as a four-stroke engine. The four cycles are intake-exhaust, compression of gases, ignition, and power.

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136.

In a four-stroke engine, what is the order of the four-stroke cycle?

a. Intake-exhaust, compression, power, and ignition b. Intake-exhaust, compression, ignition, and power c. Ignition, power, compression, intake-exhaust d. Compression, intake-exhaust, power, ignition Answer: b. In a four-stroke engine, there must be intake first (in which exhaust also occurs), followed by compression. Then there is ignition of gases, followed by power that drives the engine.

137.

How much power can be done by a Carnot engine?

a. The power is the difference between the two temperatures in the system. b. The power will be the maximum possible for any engine. c. The power will be roughly double that of a four-cycle system. d. The power will be zero.

Answer: d. The power in this type of system would be zero because the engines cannot possibly be done at absolute zero.

138.

When is the maximal efficiency of an engine possible?

a. When the temperature of the high-temperature reservoir is as high as possible and the temperature of the low-temperature reservoir is also as high as possible. b. When the temperature of the high-temperature reservoir is as low as possible. c. When the temperature of the high-temperature reservoir is as high as possible and the temperature of the low-temperature reservoir is as low as possible. d. When the temperature of the low-temperature reservoir is as high as possible.

Answer: c. The maximal efficiency of a system like an engine occurs when the high-temperature reservoir is as high as possible and the temperature of the low-temperature reservoir is as low as possible.

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139.

How can entropy be determined for real systems?

a. It can be determined for reversible systems only. b. It depends solely on the absolute temperature, which can be determined. c. It is a state-dependent thing, independent of the route, so it can be determined. d. There are complex equations that lead to the answer for real systems.

Answer: c. Entropy is a state-dependent factor so it can be determined in real, irreversible, or reversible systems.

140.

What are the SI units for entropy?

a. Joules b. Joules per degree Kelvin c. Joules per degree Celsius d. Calories per degree Celsius Answer: b. The units for entropy are in energy units per degree so, for the SI system, the units are in the form of Joules per degree Kelvin.

141.

How does entropy relate to the second law of thermodynamics?

a. It restates it to say that entropy is conserved in any process. b. It restates it to say that entropy increases when energy is put into it. c. It restates it to say that entropy decreases when energy is put into it. d. It restates it to say that entropy increases or is conserved but never decreases.

Answer: d. Entropy and the second law of thermodynamics are related in that entropy increases or is conserved in any system but never decreases.

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142.

How does the entropy of a seed and earth get decreased in order to make a

plant? a. In such cases, the entropy of the system does not get decreased at all. b. The overall entropy gets recovered when the plant gets eaten or decayed. c. The entropy of the earth must increase in response. d. The decrease in entropy requires an input in energy from the sun but the overall entropy increases.

Answer: d. The decrease in entropy of this system only requires an input in energy from the sun. The actual entropy of the universe will be increased.

143.

What happens to the energy of the system when a force is applied to a

guitar string? a. Some of it becomes potential energy and some of it is given off as sound energy b. The force in Joules gets stored in the string as potential energy c. The force is given off as sound energy d. The force is displaced along the length of the string Answer: a. There are different types of energies involved in this, including the potential energy of the string as well as sound energy given off by the displacement of the string. In some cases, there might be complete storage of the force as potential energy but, in the case of a guitar string, sound energy is also given off. The potential energy is brief because of the restoring force of the string.

144.

What is the units for a Hertz?

a. Joules per second b. Cycles per second c. Cycles per minute d. Meters per second

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Answer: b. Hertz is a unit that is only based on things that are oscillating. The units are in cycles per second.

145.

What is the relationship between the frequency of oscillations and the

period? a. The frequency is proportional to the period b. The frequency and the period are unrelated c. The frequency is inversely proportional to the square of the period d. The frequency is inversely proportional to the period Answer: d. The frequency is inversely proportional to the period with the actual relationship being that the frequency is 1 divided by the period.

146.

What will the maximal velocity of a wave be related to when it comes to

the total displacement? a. It is proportional to the displacement b. It is inversely proportional to the displacement c. It is proportional to the square root of the displacement d. It is inversely proportional to the square root of the displacement Answer: c. The maximal velocity of a wave will be proportional to the square root of the total displacement.

147.

Under what situations will a pendulum behave like a simple harmonic

oscillator? a. Anytime the arc length is less than 90 degrees b. At any time c. Anytime the arc length is less than 15 degrees d. At no time Answer: c. The pendulum will behave like a simple harmonic oscillator when the arc length is less than 15 degrees. This is because the sine of theta will be roughly equal to theta so the force will be a constant multiplied by the displacement distance.

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148.

If there is a minor damping force on an oscillating object, what will be the

factor that decreases the most in dampening the oscillation? a. Period b. Frequency c. Potential energy d. Amplitude Answer: d. The amplitude is going to gradually decrease in a waveform of an oscillating object with the frequency and the period being unchanged until the oscillating object eventually reaches a zero amplitude. The period and frequency will be the same until the object stops.

149.

What will the period T be of a given wave?

a. The peak to peak time period b. The peak to peak distance horizontally c. The distance from peak to peak vertically d. The rate of the wave propagation Answer: a. The period will be in time so it will be the peak to peak time period.

150.

In a wave system, what will the wavelength lambda be?

a. The peak to peak time period b. The peak to peak distance horizontally c. The distance from peak to peak vertically d. The rate of the wave propagation Answer: b. The peak to peak distance horizontally is considered the wavelength lambda and is in distance.

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151.

There are transverse or shear waves as well as longitudinal or compression

waves. Which type of wave is exclusively a longitudinal or compression wave? a. Ocean waves b. Visible light waves c. Sound waves d. Electromagnetic waves Answer: c. Each of these is a transverse wave except for sound waves, which are longitudinal or compression waves in air and in water.

152.

When two identical waves of the same peak and the same troughs add

together, what happens to the resultant wave? a. The amplitude and wavelength will double. b. The amplitude will be the same but the wavelength will double. c. The amplitude and the wavelength will be zero. d. The amplitude will double and the wavelength will be the same.

Answer: d. In such a case, the amplitude will double but the wavelength will stay the same.

153.

The energy of a given wave is related to its amplitude in what way?

a. The energy is directly proportional to the amplitude b. The energy is directly proportional to the square of the amplitude c. The energy is inversely proportional to the amplitude d. The energy is inversely proportional to the square of the amplitude Answer: b. The energy of a given wave is proportional to the square of the amplitude of the wave. This is because it is related to force, which is a force constant multiplied by the amplitude squared.

154.

What is the beat frequency of two waves heard together?

a. The difference between the two wave frequencies b. The sum of the two wave frequencies c. The average of the two wave frequencies d. The ratio of the two wave frequencies 386

Answer: c. The beat frequency is the average of the two wave frequencies. This is the beat that piano tuners look to eliminate when tuning a piano or the beat that is eliminated when tuning a 12-string guitar or a mandolin.

155.

What is the unit used to identify sound intensity?

a. Watts b. Watts per meter squared c. Joules per meter squared d. Decibels Answer: d. Intensity of any wave will be the number of watts applied per meter squared, which is the established SI unit for intensity of a wave. For sound energy, however, the intensity is done in decibels.

156.

What aspect of an atom has the least amount of charge?

a. Proton b. Neutron c. Quark d. Electron Answer: b. The neutron inside an atom will have no charge whatsoever, while electrons and protons will have a charge of -1 or +1 respectively.

The quark will have a partial charge on it that is either positive or negative.

157.

What is the SI unit for electric charge?

a. Watts b. Kilowatts c. Joules d. Coulombs Answer: d. The coulomb will be the SI unit for charge in electricity.

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158.

What actually moves when a conductor allows for the movement of

charges? a. Protons and electrons b. Protons only c. Protons or electrons but not both d. Electrons only Answer: d. The movement of charge through a conductor involves the movement of electrons only and never to the movement of protons.

159.

What is the difference between a conductor and a superconductor?

a. A superconductor is made from metal and a conductor may not be made from metal b. A superconductor conducts electricity faster than a conductor would c. A superconductor conducts electricity without loss of energy, which is not true of conductors d. There is no difference between a conductor and a superconductor Answer: c. The major difference between a conductor and a superconductor is that a conductor will have a loss of energy because of electrons bumping into fixed atoms and molecules, while a superconductor will conduct electricity without a loss of energy.

160.

According to Coulomb’s law, what is the relationship between the force

between two charges and the distance between the charges? a. The force is directly proportional to the distance between them. b. The force is inversely proportional to the distance between them. c. The force is inversely proportional to the square of the distance between them. d. The force is directly proportional to the square of the distance between them.

Answer: c. As you have come to know, the force between two charges is inversely proportional to the distance between them but not just to the

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distance but the square of the distance between them. This is according to coulomb’s law.

161.

What is the direction of the Coulomb force and the electric field vectors

between two charged objects? a. The Coulomb force will be positive if two objects are attractive and the electric field will be positive. b. The Coulomb force will be positive if the two objects are repulsive and the electric field will be positive. c. The Coulomb force will be positive if the two objects are attractive and the electric field will be negative. d. The Coulomb force will be positive if the two objects are repulsive and the electric field will be negative.

Answer: a. The Coulomb force will be a vector that is positive if the two objects are attractive and the electric field, being just the ratio of the force and the magnitude of the test charge, will be positive.

162.

How does one determine the totality of an electric field when there is more

than one charge in close proximity to one another? a. The total electric field vectors are the two vectors multiplied by one another. b. The total electric field vectors are the sum of the two vectors added to one another. c. The total electric field vectors are the difference between the two vectors. d. The total electric field vectors are the ratio between the larger charge and the smaller charge.

Answer: b. The total electric field vectors are the sum of the two vectors added to one another. Each charge has its own vector field, which is added to the field of another charge, whether it be positive-to-positive, negative-to-negative, or positive-to-negative charges in proximity to one another.

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163.

What is a false statement about electric field lines?

a. Field lines cross between positive and negative charges. b. The density of field lines or the magnitude is proportional to the square area the inhabit. c. Field line direction goes from positive to negative. d. The direction of the force in a field line is tangent to the direction of the line at any point.

Answer: a. Each of these is a true statement about field lines, except for the statement that field lines cross because field lines never cross.

164.

When looking at an uneven conductor, what is a true statement?

a. The charge will be equal throughout the conductor, regardless of its shape. b. The charge will discharge more likely on a pointed part of a conductor. c. The charge will collect more on the smooth surfaces of the conductor. d. The conductor will not polarize if it is uneven in any way.

Answer: b. The charge will discharge more likely on a pointed part of a conductor and will be more spread out on the flatter surfaces or convex surfaces of the conductor.

165.

What happens to the charge inside a Faraday cage?

a. The charge will be increased inside the cage versus the outside. b. The charge will be uniformly spread out inside the cage. c. The charge will be higher closer to the cage versus further away from the cage. d. The charge will be effectively zero inside the cage.

Answer: d. The charge will be effectively zero inside the cage because the cage collects the charge and prevents it from building up or getting to the inside.

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166.

Why is it safer inside a vehicle during a thunderstorm?

a. The car acts like a Faraday cage, preventing charge inside the vehicle. b. The tires will ground the car so no charge can reach the vehicle. c. The charge is smooth so no charge can be attracted to it. d. The vehicle is an insulator of charge so that it will not attract lightning.

Answer: a. The car will attract a charge if made from metal but will act like a Faraday cage, preventing the buildup of charge inside the vehicle.

167.

In a battery, the charge will be from what area to what area and what

charge is transferred? a. Negative charge is transferred from the positive terminal to the negative terminal. b. Positive charge is transferred from the positive terminal to the negative terminal. c. Negative charge is transferred from the negative terminal to the positive terminal. d. Positive charge is transferred from the positive terminal to the negative terminal.

Answer: c. The negative charge is what’s transferred in the form of electrons from the negative terminal to the positive terminal with a light bulb or car engine somewhere in between in order to have the charge pass through these items, causing them to run.

168.

What is the definition of an electron volt?

a. It is the energy to take an electron through one volt of potential difference. b. It is 1-thousandth of a volt c. It is the potential energy of an electron in an atom. d. It is the charge assigned to a single electron.

Answer: a. An electron volt is the energy it takes to take an electron through one volt of potential difference. In such cases, it is just 1.60 x 10-19 Joules.

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169.

You have two charges at a distance d apart and one is positive, while the

other is negative. Where is the electrical potential considered to be the most negative? a. Near the negative charge b. Halfway between the two charges c. Near the positive charge d. Near the positive and negative charge (i.e. both places) Answer: a. By convention, the negative charge will have the least potential because the potential will be the most negative there. The potential will be the greatest near the positive charge.

170.

What does the process of grounding do to a conductor?

a. It increases the voltage of any conductor that is grounded. b. It partially decreases the voltage of any conductor that is grounded. c. It reduces the voltage of any conductor to zero with respect to the earth. d. It insulates the conductor.

Answer: c. Grounding will reduce the voltage of any conductor to zero with respect to the earth. It takes the electrons stored in the conductor and grounds them to the earth.

171.

What is the definition of a Farad?

a. The amount of work that a capacitor can do. b. The number of coulombs per volt applied to a capacitor. c. The potential energy stored in a capacitor. d. The number of coulombs a capacitor can store.

Answer: b. A farad defines the number of coulombs per volt applied to a capacitor.

172.

In a parallel plate capacitor, what is the relationship between the

capacitance of the capacitor and its area and distance between the plates? a. The capacitance is proportional to the area of the plates and the distance between them.

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b. The capacitance is inversely proportional to the area of the plates and inversely proportional to the distance between them. c. The capacitance is proportional to the area of the plates and inversely proportional to the distance between them. d. The capacitance is inversely proportional to the area of the plates and inversely proportional to the distance between them.

Answer: c. The capacitance is proportional to the area of the plates and will be inversely proportional to the distance between them so the closer the plates are to one another, the greater the capacitance.

173.

In chemistry, water is a polar molecule. What is not true because of this

nature? a. It will have a partial negative charge on the oxygen atom b. It will have a high dielectric constant c. It will have a partial positive charge on the hydrogen atom d. It will be a conductor of electricity Answer: d. Each of these is true of the polar nature of the oxygen molecule, except that it is not a good conductor of electricity. In fact, water is a poor conductor of electricity unless it has salt in it, which makes it conduct electricity to a greater degree.

174.

If you want a large capacitance, what will make a larger capacitance when

you add four capacitors together? a. Adding four capacitors in series b. Adding two capacitors in series and two capacitors in parallel c. Adding four capacitors in parallel d. It doesn’t matter how they are added Answer: c. Four capacitors in parallel will add to the capacitance more than four capacitors in series or two capacitors in series and two capacitors in parallel.

175.

What are the units of potential energy stored in a capacitor in SI units?

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a. Farads b. Volts c. Joules d. Coulombs Answer: c. Joules are the universal units for energy of any kind. Even though electricity is in different units in many ways, its potential energy will be always in joules. When energy is given by a defibrillator, it is a certain number of joules that are given.

176.

In drawing an electrical circuit, what do you draw the electrical circuit to

look like? a. From the battery’s negative terminal to the circuit to the resistor and back to the positive terminal. b. From the battery’s positive terminal to the circuit to the resistor and back to the negative terminal. c. From the resistor to the negative terminal of the battery to the circuit and back to the positive terminal. d. From the battery’s negative terminal to the resistor to the circuit to the battery’s positive terminal.

Answer: b. The direction of the circuit as drawn is from the battery’s positive terminal to the circuit to the resistor and back to the negative terminal—even though this is not actually how the electricity flows. It flows from the negative terminal to the positive terminal as electrons.

How you draw the circuit is by convention and not the reality of how it flows.

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177.

What allows for the fast movement of electricity in a wire?

a. The electron doesn’t weigh much so it can move rapidly. b. There are so many electrons that, together move very quickly as a unit. c. The electronegativity between electrons forces a flow of electrons from an area of high density to an area of low density. d. Electron movement happens at a rate approaching 108 meters per second.

Answer: c. The electronegativity between electrons is a force—one that forces a flow of electrons from an area of high density to an area of low density, which involves a fast flow of electrons that is faster than the movement of a single electron.

178.

What is true of the relationship between electric current and resistance?

a. The current is inversely proportional to the resistance squared b. The current is proportional to the resistance squared c. The current is inversely proportional to the resistance d. The current is proportional to the resistance Answer: c. A correlate to Ohm’s law is that the current will be inversely proportional to the resistance against the flow of current.

179.

What are the units for ohm?

a. Joules per volt b. Joules per second c. Volts per meter squared d. Volts per ampere Answer: d. The units for ohm are volts per ampere because it represents the voltage applied per ampere of electrical flow.

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180.

In a cylinder, what will the resistance be with regard to the area of the

cylinder and the length of the cylinder? a. The resistance will be proportional to the area and the length of the cylinder b. The resistance will be inversely proportional to the area and inversely proportional to the length of the cylinder. c. The resistance will be proportional to the area and inversely proportional to the length of the cylinder. d. The resistance will be inversely proportional to the area and proportional to the length of the cylinder.

Answer: c. The equation is the resistance equals the resistivity multiplied by the area and divided by the length of the cylinder.

181.

What is not considered true of the temperature coefficient of resistivity?

a. It is zero at low temperatures with superconductors. b. It is negative with conductors. c. It is not necessarily linear (the same constant) over large temperatures. d. It is negative with semiconductors.

Answer: b. This will be positive with conductors because the atoms will have increased kinetic activity, which causes increased collisions and greater resistivity at high temperatures. This means that the temperature coefficients of resistivity will be positive with conductors.

182.

What happens with direct current versus alternating current?

a. Alternating current has lower voltages than direct current b. The current will reverse direction with an alternating current versus a direct current c. The current will come from more than one source in alternating current but not direct current d. It takes more power to run alternating current than it does direct current

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Answer: b. The current in alternating current will reverse direction, when it doesn’t change direction in direct current.

183.

What is not a reason why AC power is used for homes versus DC power?

a. AC power involves high voltage wires that can’t be done with DC wires. b. It is harder to decrease the voltage on a DC current wire compared to an AC current wire. c. It is harder to increase the voltage on a DC current wire compared to an AC current wire. d. The purpose is to have the ability to have less power loss as power is transported from place to place so high voltages that can be varied are necessary.

Answer: a. There is a necessity for high voltage wires for the transportation of electricity and for both increasing and decreasing the voltage at the source and at the destination. It is increased and decreased easier with AC current than DC current. AC power and DC power can both be transported under high tensions.

184.

What happens to the current and voltage when resistors are set up in

parallel to one another? a. The current is the same throughout but the voltages will be divided. b. The current and the voltages will be divided. c. The current will be divided but the voltage each one gets is the same. d. The current and voltages will be the same throughout the system, regardless of the resistors and will be decreased as if just one resistor is present.

Answer: c. The current will be divided between the different resistors but the voltage each one gets is the same. This is how the systems are set up in one’s home and in automobiles.

185.

What are the units for the electromotive force in electrical systems?

a. Amperes

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b. Volts c. Ohms d. Farads Answer: b. The EMF or electromotive force is the voltage or potential difference that a battery can deliver when the battery isn’t delivering a current. The units are in volts.

186.

What is not a true statement regarding the internal resistance of a battery

source? a. The internal resistance will increase when the battery wears down in most cases. b. The internal resistance will affect the terminal voltage. c. The internal resistance will be less with larger batteries. d. The internal resistance will be the same for all batteries of the same voltage.

Answer: d. The internal resistance will be less with larger batteries and greater when the battery wears down in most cases. It will affect the terminal voltage by the equation that the terminal voltage will be the EMF minus the product of the current and the internal resistance.

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187.

Putting smaller batteries half the size in series in order to operate a toy will

result in what happening to the total EMF and total internal resistance of the system versus a single double-sized battery that is the same voltage? a. The EMFs will add up to the same as larger battery’s EMF but the internal resistance total will be greater. b. The EMF total will be less than the larger battery and the internal resistances will be greater. c. The EMF total will be greater with the batteries in series than a single battery and the internal resistances will be less. d. b. The total EMF will be greater than the larger battery but the internal resistances will be greater as well.

Answer: a. The EMFs will add up to the EMF of the larger battery but the internal resistances in total will also be greater than with a larger battery.

188.

Which substance would be least able to be magnetized?

a. Carbon b. Nickel c. Cobalt d. Gadolinium Answer: a. Nickel, iron, cobalt, and gadolinium are all magnetizable and can be made into magnets themselves. Carbon is not considered a ferromagnetic substance.

189.

What can be said about magnetic field lines that is true about them?

a. They arbitrarily start at the south pole and end at the north pole. b. They arbitrarily start at the north pole and end at the south pole. c. They start at the most charged pole and end at the least charged pole. d. Magnetic field lines do not end but form a continuous loop.

Answer: d. Magnetic field lines do not end at all but go through poles.

They are within and outside of the magnet, forming a continuous loop.

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190.

What is the SI unit for the strength of a magnetic field?

a. Gauss b. Tesla c. Newtons per meter d. Newtons Answer: b. The unit for the strength of a magnetic field is the tesla, with a gauss being 10-4 tesla. This is the same thing as Newton-amps per meter and defines the strength of a magnetic field put out by a given magnet.

191.

In a motor, what supplies the energy to the motor when magnetism is

involved? a. The magnetic field applied to the loops of wire. b. The electrical energy that creates a magnetic field on loops of wire. c. The fuel that drives the pistons in the motor. d. The magnet that causes electricity to flow.

Answer: b. The motor is ultimately driven by electrical energy that creates a magnetic field, driving a shaft by exerting torque on it.

192.

What causes the shaft in a motor to rotate in one direction, clockwise or

counterclockwise? a. The shaft does not rotate in one direction but it goes back and forth. b. The magnetic charge reverses itself after half of a rotation of the shaft. c. The angle of force changes so that the torque is kept in one direction. d. The direction of the current changes after half of a rotation of the shaft.

Answer: d. There is reversal of the current after half of a rotation of the shaft that keeps the shaft going in the same clockwise or counterclockwise direction.

193.

How does an analog meter work?

a. There is current that rotates a shaft that spins and turns a dial over many revolutions.

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b. There is a magnetic field applied that exerts a force which turns a shaft attached to a needle. c. There is a small current applied that only rotates the shaft in the magnetic field a part of a revolution. d. The magnetic field generates a current that rotates a needle in the system.

Answer: c. An analog meter has a small current applied that only rotates the shaft in the magnetic field over a small revolution distance that causes a needle to turn accordingly.

194.

What most determines the ability of a bar magnet to produce a current

when placed inside a coil of wire? a. The stronger the magnet, the greater the current will be when it is place within the coil. b. It will produce a current only in one direction. c. It is the change in magnetic field that produces the current and not the field itself. d. The current is inversely proportional to the strength of the magnet.

Answer: c. It is the change in magnetic field or the change in magnetic flux that produces the current and not the field itself. If the magnet is stationary, it will not produce a current, even if it is a large magnet.

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195.

What is the science behind a metal detector?

a. It is a strong magnet that can detect iron-containing materials through attracting the metal. b. It detects the change in magnetic flux that is created between the detector and the conductor. c. It detects the magnetic fields created by metallic substances. d. It detects the magnetic drag when the magnet is moved over a metallic substance.

Answer: d. The metal detector is a device that detects the magnetic drag that exists when a magnet is moved over a metallic substance. It is based on the movement of the detector over the metal rather than on the magnetic field of the magnet in the detector.

196.

What can be said about the speed of electromagnetic waves?

a. The waves with the largest period move the slowest. b. The waves with the smallest period move the slowest. c. The waves propagate depending on their frequency. d. The waves all move at the speed of light.

Answer: d. All electromagnetic waves move at the speed of light, regardless of their frequency and wavelength.

197.

What is the type of electromagnetic wave that has the shortest

wavelength? a. Power line waves b. AM radio waves c. Microwaves d. Cell phone waves Answer: a. Power line waves are created by power lines and have very long wave lengths—sometimes wavelengths that are kilometers long.

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198.

Which type of electromagnetic radiation arrives the most to the earth from

the sun? a. Visible light b. Microwaves c. UV waves d. Infrared waves Answer: d. Infrared waves make up the most of the light absorbed by the earth from the sun, responsible in part for the heating of the earth.

199.

Which type of electromagnetic radiation from the sun has the highest

frequency? a. UVA b. UVB c. UVC d. Visible light Answer: c. UVC radiation gets mostly absorbed by the ozone layer and has the highest frequency of all of these electromagnetic waves reaching the earth from the sun.

200. Which type of electromagnetic rays are closest to the frequencies of gamma rays? a. X-rays b. Microwaves c. Ultraviolet rays d. Infrared waves Answer: a. Gamma-rays are considered closest in frequency to x-rays, with some actual overlap between the two. Both types of rays are used to scan luggage in airports as they can penetrate solid objects.

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