PHY 123. PHYSICS FOR ENGINEERS _ FLUIDS AND HARMONICS BSCPE, 1ST YEAR, Section CPE 1A/1B/1C COURSE C

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Each liquid particle does not have a definite path and the paths of individual particles also cross each other

MODULE No. 6 FLUIDS AND HARMONICS 1ST YEAR

PROGRAM:

BSCPE

Year Level:

COURSE CODE:

NSCI02

DESCRIPTION:

Section

CPE 1A/1B/1C

PHYSICS FOR ENGINEERS

LEARNING OUTCOMES At the end of the period, the students should be able to: 1. Apply the principles and law of fluid mechanics to analyze and solve Physics scenarios 2. Analyze the motion of an oscillating system by identifying its period and frequency 3. Solve problems involving fluid mechanics 4. Determine the variables that affect the period of a simple pendulum Prepared: RUJEVI S. BADAGUAS Instructor Recommending Approval:

Reviewed and Checked: ENGR. MARIA THERESA B. PRENDA Program Head Approved:

DR. JOEVELL P. JOVELLANO, CSAS Vice President for Academic Affairs

TANAUAN CITY COLLEGE

NSCI02 /MODULE NUMBER 6

PROF. MICHAEL E. LIRIO, CPA, MMPA President and College Administrator

PHYSICS FOR ENGINEERS

FLUIDS AND HARMONICS Program: Course: Code

BS COMPUTER ENGINEERING PHYSICS FOR ENGINEERS NSCI02 Module #: 6

I. Preliminaries Introduction to the Module Objective

Topic: Instructor: Week #:

FLUIDS AND HARMONICS RUJEVI S. BADAGUAS 12-13 # of Pages:

19

In this module we will be discussing about fluids and harmonics. Included here are the characteristics of fluids, principle involved in analyzing fluids, and some of example problems.

Also included here is the discussion about oscillation motion, the types of harmonic motion, pendulums, as well as its types and formulas.

Assessment/ Evaluation

Section Topics

Learning Outcomes

Modality

Section 1: Density and Fluid Pressure

1. Apply the principles and law of fluid mechanics to analyze and solve Physics scenarios

Discussion/participation

2. Analyze the motion of an oscillating system by identifying its period and frequency

06 Laboratory Experiment 1: Properties of Liquids

3. Solve problems involving fluid mechanics

06 Laboratory Experiment 2: Simple Pendulum

06 Quiz 1

Power Point Presentation via MOODLE

Section 2: Fluid Principles Section 3: Oscillation Section 4: Pendulums

4. Determine the variables that affect the period of a simple pendulum

II. Instructions KEYWORDS AND CONCEPTS

Mechanics – oldest physical science that deals with both stationary and moving bodies under the influence of forces Statics – branch of mechanics that deals with bodies at rest Dynamics – branch that deals with bodies in motion Fluid mechanics – defined as the science that deals with the behavior of fluids at rest or in motion Fluid statics – the study of fluids at rest Fluid kinematics – study of fluids in motion where pressure forces are not considered Fluid dynamics – branch of science if pressure forces are considered for fluids in motion Hydrodynamics – study of motion of fluids that are practically incompressible Hydraulics – deals with liquid flows in pipes and open channels Fluid – a substance in the liquid or gas phase Mass density – amount of mass per unit volume of that object or substance Weight density – weight of unit volume of that object or substance Pressure – defined as the ratio of force applied per area covered Buoyancy – tendency of a fluid to uplift a submerged body, because of the up-thrust of the fluid Uniform flow – which the velocities of the liquid particles at all sections of a pipe or a channel are equal Non-uniform flow – the velocities of liquid particles at all sections of a pipe or channel are not equal TANAUAN CITY COLLEGE

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Streamline flow – which each liquid particle has a definite path and the paths of individual particles do not cross each other Turbulent flow – It is the parameter that describes the vibration.

Steady flow – the quality of liquid flowing per second is constant Unsteady flow – the quality of liquid flowing per second is not constant Compressible flow – the volume of a fluid and its density changes during the flow Incompressible flow – the volume of a flowing fluid and its density does not change during the flow Rotational flow – the fluid particles also rotate about their own axes while flowing Irrotational flow – the fluid particles do not rotate about their own axes and retain their original orientation One-dimensional flow – the streamlines of its moving particles may be represented by straight line Two-dimensional flow – whose streamlines may be represented by a curve Three-dimensional flow – whose streamlines may be represented in space along three mutually perpendicular directions Harmonic motion – periodic motion in which the displacement is symmetrical about a point Simple harmonic motion – harmonic motion that repeats indefinitely Damped harmonic motion – harmonic motion that gradually diminishes its energy and slows down until it is already at full stop.

Driven harmonic motion – harmonic motion that uses an outside force to continue its movement Period – amount of time taken for the motion to repeat itself Frequency – number of times the motion is repeated per unit of time Amplitude – extreme magnitude of displacement Angular frequency – parameter that describes the vibration Phase angle – amount of shift from the standard sine function Pendulum – object that swings back and forth between two angular positions Simple pendulum – attached to a suspended material with mass negligible compared to that of point mass Physical pendulum – the entire body swings back and forth, that does not pass through its center of mass Torsion pendulum – the suspended material does not swing FLUIDS AND HARMONICS SECTION 1: DENSITY AND FLUID PRESSURE What is A Fluid?

The term "fluid" in everyday language typically refers only to liquids, but in the area of physics, fluid describes any gas or liquid that conforms to the shape of its container. It is any substance which flows because its particles are not rigidly attached to one another.

It can also be defined as a substance that exist in three primary phases: solid, liquid, and gas, where in the liquid or gas phase is referred to as a fluid. The distinction between a solid and a fluid is made on the basis of the substance’s ability to resist an applied shear (or tangential) stress that tends to change its shape. A solid can resist an applied shear stress by deforming, whereas a fluid deforms continuously under the influence of shear stress, no matter how small. In solid stress, it is proportional to strain, but in fluid stress it is proportional to strain rate.

A fluid is also a collection of molecules that are randomly arranged and held together by weak cohesive forces and TANAUAN CITY COLLEGE

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forces exerted by the walls of a container.

Fluid Mechanics is branch of physics that deals with the properties and behavior of fluids either at rest (fluid statics) or in motion (fluid dynamics), and also the interaction of fluids with solids or other fluids at the boundaries.

In our treatment of the mechanics of fluids, we shall see that we do not need to learn any new physical principles to explain such effects as buoyant force acting on submerged object and the dynamic lift acting on an airplane wing.

Areas of Application of Fluid Mechanics  Mechanical Engineering  Design of pumps, turbines, air-conditioning equipment, pollution-control equipment, etc.  Design and analysis of aircraft, boats, submarines, rockets, jet engines, wind turbines, biomedical devices, the cooling of electronic components, and the transportation of water, crude oil, and natural gas. 

Civil Engineering  Transport of river sediments  Pollution of air and water  Design of piping systems  Flood control systems



Chemical Engineering  Design of chemical processing equipment



Turbomachines  Pumps  Turbine  Fan  Blower  Propeller, etc.



Military  Missile  Aircraft  Ship  Underwater vehicle  Dispersion of chemical agents, etc.



Automobile  IC engine  Air conditioning  Fuel flow  External aerodynamics, etc.



Medicine  Heart assist device  Artificial heart valve,  Lab-on-a-chip device  Glucose monitor  Controlled drug delivery, etc.



Electronics  Convective cooling of generated heat



Energy  Combuster  Burner  Boiler  Gas  Hydro and wind turbine, etc.



Oil and Gas  Pipeline  Pump  Valve  Offshore rig  Oil spill cleanup, etc.



Almost everything in our world is either in contact with a fluid or is itself a fluid.

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Properties of Fluids A.

Density A. Mass Density (ρ, Greek letter “rho”) is defined as the average mass density of a substance or an object as the

amount of mass m per unit volume V of that object or substance. It can be computed by simply dividing the mass of the object by its volume.

The SI unit of density is kg/m 3. This unit is awkwardly large for some solids or liquids. Therefore, other units of similar dimension such as g/cm3, lb-m/ft3, kg/L, ton/m3 and g/ml are used more often.

Density is an intensive property and does not depend on the quantity of mass present, for a given material the ratio of mass to volume always remains the same. It can be concluded that volume V is directly proportional to mass m. Density is not only a property of fluids; solids also have densities since they both have mass and volume.

B. Weight Density (D) is defined as the average weight density of a substance or an object as the weight of unit

volume of that object or substance.

Since the weight W = mg, therefore

The typical units of weight density is N/m 3, lb/ft3, and gr-f/cm3.

Example 1 What is the mass of an 80.0 ml acetone (ρ acetone = 0.792 g/mL)?

Solution: To get the mass of acetone, re-arrange the equation first.

Then, substitute the given values: m = ρV = (0.792 g/mL) ( 80.0 mL) m = 63.4 g B. Pressure Pressure (P) is defined as the ratio of force applied per area covered. The force is perpendicular to the surface area.

The units of pressure are Pa or N/m 2 (1 Pa = 1 N/m2 or 1kg/m s2), dyne/cm2, or any similar unit with the same dimension. Another frequently used unit of pressure is atmosphere (atm). It is the same quantity as volume stress but not the same as tensile and shear stress.

Units of Pressure Here is a table of the common units of pressure that is being used in calculations:

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Example 2 1.50 kg water is inside a cylindrical jar 4.00 cm in diameter. What is the pressure at the interior bottom of the jar?

Solution: The pressure experienced at the inside bottom of the jar is due to the weight of the water: F = mg = (1.50 kg) (9.8 m/s2) = 14.7 N Next, obtain the area of the bottom of the cylindrical jar: Since the diameter is 4.00 cm, its 0.0200 m.

radius is 2.00 cm or

A = πr2 = π (0.0200 m)2 = 0.00126 m2 Now, to get the pressure:

SECTION 2: FLUID PRINCIPLES Fluid at Rest For this part of the lecture, it is assumed that the liquid is incompressible. This means that the density is constant and uniform throughout its volume. This is one of the characteristics of an ideal fluid. Real fluids are actually more complicated to deal with. The density changes with varying temperature. However, one can use ideal liquid as an approximation to real liquids granted that the other quantities that make the computation more complicated can be ignored, such as when the temperature is constant or almost constant. Ideal liquids have other characteristics but for a liquid at rest, the only assumption needed is that the density is constant.

Fluid Pressure in Depth The sample problem for pressure demonstrated how the property exerted by a liquid at the base of its container is related to its weight because weight is simply the force F exerted on the base and area A is the area of the base.

There is another equation useful in similar problems, which will be derived. Take a look at the equation of pressure.

Note that the force is the weight of the liquid (F = mg).

The equation for density is

V = Ah where A is the area on the top or bottom of the container and h is the height. Substitute V in the equation for density.

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Re-arrange equations such that both equations are equated to A.

Afterwards, combine both equations.

Since both sides of the equation contain the variable for mass, the variables are therefore cancelled.

Re-arrange the equation to get,

With this equation, one can find the liquid pressure at any depth in the container, not just at the bottom. The only variable here is the height h of the liquid above a particular depth. The constant g = 9.8 m/s 2 or 980 cm/s2 is simply the acceleration due to gravity, while ρ is simply the density of the liquid. This implies that liquid pressure does not depend on mass and volume, only on height.

Consider point A, a particular location inside the partially filled container. The pressure at point A does not depend on the actual mass or volume of the liquid surrounding it. It only depends on the height h of the liquid above point A and the density ρ of the liquid. Hence, the locations deeper than point A experience more pressure and those shallower than point A experience less pressure.

Example 3 A glass container was filled with oil (ρ = 0.80 g/cm 3) up to 9.0 cm in height. What is the pressure (a) at the base of the container? (b) At a depth of 3.0 cm from the surface of the oil? (c) At 4.0 cm above the base?

Solution: a) The height of the oil above the base of the container is 9.0 cm since all of the oil inside the container is above the base.

P=ρgh P = (0.80 g/cm3) (980 cm/s2) (9.0 cm) P = 7,056 dyne/cm2 P = 7.1 × 103 dyne/cm2 b) At depth of 3.0 cm, the height of the liquid above is only 3.0 cm. The other portion of liquid is below this depth and hence, does not exert any pressure.

P=ρgh P = (0.80 g/cm3) (980 cm/s2) (3.0 cm) P = 2,352 dyne/cm2 P = 2.4 × 103 dyne/cm2 c) At 4.0 cm above the base, the height of the liquid above is 5.0 cm (9.0 cm – 4.0 cm = 5.0 cm). The other portion of liquid is below this depth and hence, does not exert any pressure.

P=ρgh P = (0.80 g/cm3) (980 cm/s2) (5.0 cm) P = 3,920 dyne/cm2 P = 3.9 × 103 dyne/cm2

Pascal’s Principle Pascal’s Principle states that pressure applied to an enclosed fluid is transmitted undiminished to every part of the TANAUAN CITY COLLEGE

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fluid, as well as to the walls of the container.

Pascal’s principle can be easily demonstrated using a hydraulic lift similar to the illustration above. In a hydraulic lift, the liquid is confined. Hence, based on Pascal’s Principle, the pressure on the left piston should be equal to the pressure on the right piston.

Since pressure is defined as force per unit area:

Where FL is the force applied on the left piston (also the force exerted by water on the left piston), FR is the force applied on the right piston (also the force exerted by water on the right piston), AL is the cross sectional area of the left piston, and AR is the cross sectional area of the right piston.

Example 4 A certain hydraulic lift has a left piston with diameter of 40.00 cm and a right piston with diameter of 5.00 cm. If a force of 20.0 N is exerted on the right piston, what is the force exerted by the left piston?

Solution: The piston can be visualized such that when a force is exerted to the right piston, the right piston will go downward. This, in turn, will cause the left piston to go upward, exerting force on anything that is placed on top of it. The given quantities are:

The quantity FL, the force exerted by the left piston, is what is needed to be calculated. Manipulate the equation so that FL can be calculated:

To get the areas AL and AR, use the given diameters, dL and dR before using the equation above.

AL = (dL/2)2 = π(0.4000 m/2)2 = 0.1260 m2 AR = π (dR/2)2 = π (0.0500 m/2)2 = 0.00200 m2 Then, substitute the known quantities to the equation:

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Notice that the hydraulic press was able to multiply the input force of 20.0 N with an output force of 1,260 N. The output force is 63 times greater than the input force. The ratio of input and output force varies depending on the cross- sectional area of each piston. A hydraulic lift has many uses. It is used to lift cars in car repair shops, and by dentists and barbers to raise or lower the chairs of their clients.

Archimedes (C. 287-212 B.C.) Archimedes, a Greek mathematician, physicist, astronomer, inventor, and engineer, was perhaps the greatest scientist of antiquity. He was the first to compute accurately the ratio of a circle’s circumference to its diameter, and he showed how to calculate the volume and surface area of sphere, cylinders, and other geometric shapes. He is well known for discovering the nature of buoyant force.

Archimedes’ Principle Archimedes’ principle states that the upward buoyant force pushing an object partially or fully submerged in fluid is equal to the weight of the fluid displaced by that object.

Imagine this scenario. Suppose there is a container that is partially filled with liquid up to a certain height. Mark that height as A. Next, drop something into the liquid. This could be anything; a cork, a piece of rock, a gold nugget, etc. When that object is dropped on the liquid, some of the liquid will rise up. Since something was put into the liquid, expect some of the liquid to be displaced. In the illustration below, some of the liquid rose above A. Based on Archimedes’ principle, the force that is pushing that object up is equal to the weight of the liquid above A.

In the illustration, FB is the magnitude of force pushing the object up, which is also the magnitude of the weight of the liquid above A. FW is the weight of the object.

In most cases, the volume of the liquid displaced is equal to the volume of the object submerged. The exception here is when the object has cavity or when the object can absorb the liquid.

Archimedes’ principle is useful especially in determining which object will float and which object will sink.

Three possible scenarios that could happen in case an object is submerged in a fluid:



The weight of the object is greater than the weight of the fluid displaced (FW > FB). Hence, the object will sink.



The weight of the object is equal to the weight of the liquid displaced (FW = FB). Hence, the object will float.



The weight of the object is less than the weight of the object displaced (FW < FB). Hence, the object will move upward until it is only partly submerged. When the object is only partly submerged, some of the displaced liquid molecules will go back to their original location, causing the buoyant force to decrease until it is equal to the weight of the object. The object will float while being partially submerged.

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Buoyancy Have you ever tried to push a beach ball under water? This is extremely difficult to do because of the large upward force exerted by the water on the ball. The tendency of a fluid to uplift a submerged body, because of the up-thrust of the fluid, is known as force of buoyancy or simply buoyancy. We can determine the magnitude of a buoyant force by applying some logic and Newton’s second law.

The manner in which buoyant forces acts vertically upward through the centroid of the displaced volume is summarized and can be defined mathematically by Archimedes’ principle, which states that the magnitude of the buoyant force always equals the weight of the fluid displaced by the object. The buoyant force acts vertically upward through the point that was the center of gravity of the displaced fluid.

Fd = γ f V d Where Fd = buoyant force γ f = specific weight of fluid Vd displaced volume of fluid Before we proceed with the example, it is instructive for us to compare the forces acting on a body placed over a liquid, either it will submerged or sinks down or floats on the liquid (partly submerged object).

Two forces involved are: 1. Gravitational force (totally submerged object) – when object is totally submerged in a fluid of density ρ f, the magnitude of the upward buoyant force is B = ρ f Vo g, where Vo is the volume of the object. If the object has a mass M and density ρo, its weight is equal to Fg = Mg = ρo Vo g, and the net force on it is B – F g = (ρf – ρo) Vo g.

Hence if the density of the object is less than the buoyant force, and the unconstrained object accelerates upward. If the density of the object is greater than the density of the fluid, then the upward buoyant force is less than the downward force of gravity, and the unsupported object sinks.

2. Up-thrust of the liquid (floating object) – now consider an object of volume V o in static equilibrium floating on a fluid – that is, an object that is only partially submerged. In this case, the upward buoyant force is balanced by the downward gravitational force acting on the object. If V f is the volume of the fluid displaced by the object (this volume is the same as the volume of that part of the object that is beneath the fluid level), the buoyant force has a magnitude B = ρ f Vf g. Because the weight of the object is to Fg = Mg = ρo Vo g and because Fg = B, we see that ρf Vf g = ρo Vo g.

Therefore, if gravitational force is more that up-thrust, the body will sink, while if the up-thrust is more that the gravitational force, the body will float.

Example 6 An ice cube (ρ = 0.917 g/cm3) was placed in a pitcher of drinking water (ρ = 1.00 g/cm 3). What percent of the ice cube’s volume is below the water’s surface?

Solution: The given quantities are: ρ ice = 0.917 g/cm3 ρ water = 1.00 g/cm3 The volume of water displaced must have weight equal to the weight of the ice cube in order for the ice cube to float.

FB = FW ice = FW water displaced V ice = total volume of ice cube V’ = volume of water displaced = volume of ice under water Since, F w ice = Fw water displaced then, ρ ice V ice g = ρ water V’g Cancel g on both sides and re-arrange the equation, to get: TANAUAN CITY COLLEGE

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Note that V’ is the volume of water displaced as well as the volume of ice under the water surface, while Vice is the total volume of ice. Hence, the fraction V’/V ice is the fraction of ice under water.

Substituting the known quantities:

Hence, the fraction of ice under water surface is 0.917 or in percent, its 91.7%.

Fluid in Motion Similar to steady fluids, it will be assumed that the fluids in motion to be dealt with are also ideal fluids.

The motion of real fluids are very complicated and not yet fully understood. An example of very complicated motion is that of smoke that rises up and changes its flow from steady to turbulent.

On the other hand, some fluids, like water flowing through a hose are a close approximation of ideal fluid.

Characteristics of an Ideal Fluid in Motion Here are the assumptions with regard to ideal fluids. a. Uniform Flow – a flow, in which the velocities of the liquid particles at all sections of a pipe or a channel are equal, is called a uniform flow. This term is generally applied to flow in channels. b. Non-Uniform Flow – a low, in which the velocities of liquid particles at all sections of a pipe or channel are not equal, is called a non-uniform flow. c. Streamline Flow – a flow, in which each liquid particle has a definite path and the paths of individual particles do not cross each other, is called a streamline flow. It is also called a laminar flow. d. Turbulent Flow – a flow, in which each liquid particle does not have a definite path and the paths of individual particles also cross each other, is called a turbulent flow. e. Steady Flow – a flow, which the quality of liquid flowing per second is constant, is called a steady flow. A steady flow may be uniform or non-uniform. f.

Unsteady Flow – a flow, in which the quality of liquid flowing per second is not constant, is called an unsteady flow.

g. Compressible Flow – a flow, in which the volume of a fluid and its density changes during the flow, is called compressible flow. All the gases are, generally, considered to have compressible flows. h. Incompressible Flow – a flow, in which the volume of a flowing fluid and its density does not change during the flow, is called an incompressible flow. All the liquids are, generally, considered to have incompressible flows. i.

Rotational Flow – a flow, in which the fluid particles also rotate (i.e., have some angular velocity) about their own axes while flowing, is called a rotational flow. In a rotational flow, if a match stick is thrown on the surface of the moving fluid, it will rotate about its axis.

j.

Irrotational Flow – a flow, in which the fluid particles do not rotate about their own axes and retain their original orientation, is called an irrotational flow. In an irrotational flow, if a match stick is thrown on the surface of the moving fluid, it does not rotate about its axis but its original orientation.

k. One-Dimensional Flow – a flow, in which the streamlines of its moving particles may be represented by straight line is called one-dimensional flow. It is because of the reason that s straight line streamline, being a mathematical line, possesses one dimension only i.e, either x-x or y-y or z-z direction. l.

Two-Dimensional Flow – a flow, whose streamlines may be represented by a curve, is called a twodimensional flow. It is because of the reason that a curved streamline will be along any two mutually perpendicular directions.

m. Three-Dimensional Flow – a flow, whose streamlines may be represented in space i.e., along three mutually perpendicular directions, is called three-dimensional flow.

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The Equation of Continuity Whenever watering plants in the garden or washing a car, one may have noticed that the speed of water flowing out of the hose can be increased by partially closing the hose’s end with one’s thumb.

One can see that the speed of the water is affected by the cross-sectional area of the tube, hose, or any means of its passage. An expression that relates the speed of ideal fluid to the cross-sectional area of the duct shall not be derived.

Look at the figure below. A region within a tube with varying cross-sectional area will be studied. The region is defined by length L. A certain volume of ideal fluid ΔV enters the region from the left within time Δt.

In the figure above, v1 is the velocity of ideal fluid that is entering the region while v 2 is that of exiting the region; A1 is the cross-sectional area of the left portion of the tube while A 2 is that of the right portion.

One can express the volume of ideal fluid entering the region from the left as: ΔV = A1 x1 = A1v1Δt Note that v1 was not labeled as vector. It is because the fluid is steadily flowing in only one direction. Thus, one can ignore the vector properties of the velocity and focus only on its magnitude.

For the volume of an ideal fluid that leaves the region, ΔV = A2 x2 = A2v2Δt Due to the incompressibility of the fluid, the volume of the ideal fluid that enters the region is equal to the volume that exits the region.

A1v1Δt = A2v2Δt Canceling the common factor Δt, one obtains the equation of continuity: A1v1 = A2v2 The equation of continuity, when expressed in words, implies that an ideal fluid flowing through a tube of varying cross- sectional area has a constant volume flow rate. Another way of expressing the equation of continuity is in terms of volume flow rate RV. RV = Av = constant Rm = ρRV = ρAv = constant Example 7 A main water pipe with an inner diameter of 21.0 cm furnishes water, through intermediate pipes, to a faucet pipe with an inner diameter of 1.5 cm. The velocity of water in the faucet is 3.0 cm/s and the density of water is 1.0 g/cm 3. a. What is the volume flow rate? b. What is the mass flow rate? c. What is the velocity of water in the main pipe?

Solution: The given quantities are as follows:

Finding the cross-sectional areas:

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a. Finding the volume flow rate RV = Av = A2V2 = (346 cm2) (3.0 cm/s) = 1,038 cm3/s b. Finding the mass flow rate Rm = ρ RV = (1.0 g/cm3) (5.3 cm3/s) = 5.3 g/s c. Finding the velocity in the main pipe.

RV = Av = A1v1 = A2v2 = constant

Bernoulli’s Equation The equation of continuity provides information with regard to motion of fluid. However, it does not provide information about the pressure of fluid in motion.

Daniel Bernoulli formulated an equation that relates the pressure of a moving fluid to the fluid density, speed and altitude. This equation, called Bernoulli’s equation, is not a new principle, but simply the reformulation of a familiar principle in the form more suitable to fluid mechanics.

Consider a pipe with varying height and cross-sectional area. Similar to the previous example in the equation of continuity, a region will be defined as L. The volume of ideal fluid that leaves the region is equal to the volume that enters the region.

Now apply the equation of work as change in kinetic energy to the volume of fluid inside the region.

W = ΔKE This change in kinetic energy is due to the difference in speed as the fluid enters the region, and as the fluid exits the region. In turn, the change in speed is due to the difference in the cross-sectional area as stated in equation of continuity.

where m is the mass of fluid that enters the region from left and also the mass of the fluid that exits the region to the right during the time interval Δt.

The work done W comes from two sources. One of the sources is work by gravity Wg on the fluid of mass m as it rises from y1 to y2.

Wg = – mg (y2 – y1) Since m = ρV, TANAUAN CITY COLLEGE

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Wg = –ρV g (y2 – y1) This work is negative because the displacement of the fluid is upward and the gravity is downward.

Another source of work done on the system (fluid in region L) is the push of the incoming fluid from the left. In turn, this is countered by the work done by the system as it pushes the fluid just at the right of the region.

Thus, for the second source of work Wp:

Going back to work-kinetic energy theorem:

Rearranging this equation so that all variables with subscript 2 are on the right side and all with subscript 1 are on the left side

The equation above is known as Bernoulli’s Equation. This equation can also be written as:

SECTION 3: OSCILLATION A very special kind of motion occurs when the force acting on a body is proportional to the displacement of the body from some equilibrium position. If this force is always directed toward the equilibrium position, repetitive back-andforth motion occurs about this position. Such motion is called periodic motion, harmonic motion, oscillation, or vibration (the four terms are completely equivalent).

You are most likely familiar with several examples of periodic motion, such as the oscillations of a block attached to a spring, the swinging of a child on a playground swing, the motion of pendulum, and the vibrations of a stringed musical instrument.

Most of the material in this module deals with harmonic motion, in which the displacement is symmetrical about a point. This motion tends to repeat on regular intervals or after every certain length of time. The length of time in this case is called period (T).

Three Types of Harmonic Motion a. Simple harmonic motion – harmonic motion that repeats indefinitely. A true simple harmonic motion does not exist in reality. b. Damped harmonic motion – harmonic motion that gradually diminishes its energy and slows down until it is already at full stop. A damped harmonic motion can be treated as simple harmonic motion when considering a short interval of time. c. Driven harmonic motion – harmonic motion that uses an outside force to continue its movement. For example, one can force a pendulum to swing continuously by attaching it to a machine that will swing it continuously.

Simple Harmonic Motion Simple harmonic motion is the type of motion that repeats intervals.

The period (T) of harmonic motion is the amount of time it takes for the motion to repeat itself. Frequency (f) is the number of times the motion is repeated per unit time. Similar to circular motion, frequency and period are reciprocal TANAUAN CITY COLLEGE

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quantities.

The unit of period (T) is second (s) or any unit of time. The unit of frequency is s –1, /s or any reciprocal unit of time. s–1 is commonly known as hertz (Hz).

1 Hz = 1 s–1 Objects attached to a spring or any elastic object with negligible mass is governed by Hooke’s Law. Note that this is the same Hooke’s Law from where we derived the Elastic Potential Energy: Fr = – kx Where Fr is the restoring force, k is the spring constant and x is the displacement from the spring’s relaxed length.

Restoring force is always opposite the displacement (hence, the negative sign) and also directly proportional to the magnitude of the displacement. This keeps the object in moving back and forth.

Periodic graphs can be represented by sine or cosine functions. If one graphs the position of an object undergoing simple harmonic motion (such as the mass attached to a spring), with time as the independent variable, one will get something like this:

Periodic graphs such as this, no matter how tall or short, how wide or narrow, or whether or not the function value at t = 0 is zero, can be represented by a sine. One can express the position as function of time as: x(t ) = xo sin(ωt + δ).

In the above equation, t is the time (independent variable) and x is the displacement (dependent variable). The parameters xo, ω and δ are constants that respectively determine the extreme values of the function, how wide is the graph and the shift of the graph from the actual sine function. Note that sin(ωt + δ) is unit less and can range only from –1 to 1. Hence, the unit of xo is meter (or any unit of distance). The expression (ωt + δ) is an angle. Therefore, the unit of δ must be π rad and ω must be π rad/s.

The variable xo is called amplitude; the magnitude of the maximum and minimum displacement. The parameter ω is called angular frequency. Angular frequency is a parameter that describes the vibration and does not involve an actual motion or rotation. The period of the harmonic motion depends on angular frequency. Period is related to angular frequency by the formula:

Meaning, if ω is smaller, T is larger and the graph appears wider. If ω is larger, T is smaller and the graph appears narrower.

The variable δ is called phase angle. It represents the shift of the graph from the standard sine function where the function value at 0, π, 2π, 3π, nπ (n = any integer) are 0. If δ is positive, the graph is shifted to the left. If δ is negative, the graph is shifted to the right.

The instantaneous velocity of the object in simple harmonic motion can be computed using the formula: v(t ) = ωxo cos(ωt + δ).

The acceleration of an object in simple harmonic motion also fluctuates between two extremes. In fact, displacement and acceleration are always zero at the same time. However, whenever displacement is at the positive extreme, acceleration is always at the negative extreme, and vice- versa. It is because force and acceleration are always directed to the same direction according to Newton’s second law, while according to Hooke’s law, displacement and force are always towards opposite directions.

The acceleration of an object in simple harmonic motion can be computed using the formula: TANAUAN CITY COLLEGE

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a(t ) = - ω2 x0 sin(ωt + δ) Since x0 from the equation x(t ) = x0 sin(ωt + δ) can be incorporated with the acceleration equation a(t ) = - ω2 x0 sin (ωt + δ), then we can simplify a(t) to: a(t ) = - ω 2 x(t ).

SECTION 4: PENDULUMS While some objects undergoing simple harmonic motion move back and forth as governed by Hooke’s law, there are objects that rotate back and forth between two angular positions. These objects are generally called pendulums.

Take for example, a simple pendulum that moves back and forth between two angles θo and – θo:

The angular position of the pendulum at time t can be computed using the formula: where θo, ω and δ are constants. Similar to the previous equation x(t) = xo sin (ωt + δ), θo is the amplitude of the harmonic motion, ω is the angular frequency and δ is the phase angle. Note that ω and δ, like in equation x(t) = xo sin (ωt + δ), are merely parameters of harmonic motion and not actual angles like θ and θo. Hence, in angular simple harmonic motion, ω is not equal to θ/t or θo/t.

The equation

still holds for angular simple harmonic motion.

Three Types of Pendulum There are three types of pendulum. We will identify and discuss each type.

1. Simple Pendulum A simple pendulum is a system made-up of point mass that swings back and forth, tracing an arc of a circle. The point mass is attached to a string or a thin rod, with a mass negligible compared to that of point mass.

To solve for the angular frequency and period, the following formulas are used.

Where L is the length of the string or rod and g is the acceleration due to gravity (9.8 m/s 2).

2. Physical Pendulum An extended body is pivoted freely about an axis that does not pass through its center of mass. Such an arrangement forms a physical pendulum which functions as a simple harmonic motion for small angular displacements. A practical application of such pendulum is an arm/leg or a bell.

To solve for the angular frequency and period of a physical pendulum, the following formulas are used.

Where m is the mass of the swinging object, r is the distance between the pivot and the object’s center of mass, g is the acceleration due to gravity and I is the object’s moment of inertia.

3. Torsion Pendulum A torsion pendulum consists of an object that is attached to a string or thin rod. The string/rod twists back and forth causing the object to undergo angular harmonic motion. The string/rod itself does not swing.

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To solve for the angular frequency and period of a torsion pendulum, the following formulas are used.

Where I is the object’s moment of inertia and K is the tension constant of the suspending material (string/rod). The torsion constant depends on the length, cross-sectional area and kind of material of the suspending material.

Example 8 A lamp is attached to the ceiling with a string 12 ft long. Find its period of oscillation.

Solution:

Example 9 A pendulum system has a collective torsional constant ofn10 Nm/rad. If the angular frequency of the system is π/6 rad/sec, determine the system’s moment of inertia and period.

Solution: Solving for the system’s moment of inertia I ≠ 0,

The system has a moment of inertia I ≈ 36.48 kg-m 2 Now, solving for the system’s period T ≠ 0,

The system has a period of 12 seconds.

III. Viable and vibrant Activities TANAUAN CITY COLLEGE

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Description of the Learning Activities

Discussion – Participation of students through discussion of Fluids and Harmonics 06 Quiz 1 – The students are required to answer the given quiz in Fluids and Harmonics 06 Laboratory Experiment 1: Properties of Liquids – the students must perform the laboratory experiment using available materials only and they need to answer the questions stated in the said experiment 06 Laboratory Experiment 2: Simple Pendulum – the students must perform the laboratory experiment using available materials only and they need to answer the questions stated in the said experiment IV. Opportunity to reflect and articulate students’ acquired knowledge.

Criteria for Evaluation

06 Quiz 1 – PROBLEM SOLVING: SOLVE THE PROBLEM AND SHOW THE COMPLETE SOLUTION. (15 POINTS)

06 Laboratory Experiment 1: Properties of Liquids – the students must perform the laboratory experiment using available materials only and they need to answer the questions stated in the said experiment

06 Laboratory Experiment 2: Simple Pendulum – the students must perform the laboratory experiment using available materials only and they need to answer the questions stated in the said experiment

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Purpose of Activity:

Discussion/participation of students through discussion of FLUIDS AND HARMONICS 06 Quiz 1 – The students are required to answer the quiz in FLUIDS AND HARMONICS to test their understanding of the lesson 06 Laboratory Experiment 1: Properties of Liquids – the students must perform the laboratory experiment using available materials only and they need to answer the questions stated in the said experiment 06 Laboratory Experiment 2: Simple Pendulum – the students must perform the laboratory experiment using available materials only and they need to answer the questions stated in the said experiment

Summary and Reflection

The students are required to write their learning journal of what they learned in the lesson and what they are expecting to learn in the next topic discussion.

TOPIC/S

I ALREADY KNEW….

I LEARNED……

I STILL WANT TO KNOW…..

REFLECTION

Density and Fluid Pressure Fluid Principles Oscillations Pendulums OVERALL REFLECTION

V. Textbooks and other References

1. Bueche, F.J. & Hecht, E., (1997). Schaum’s Outlines: College Physics (9th ed.). New York, McGraw-Hill.

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2. Fishbane, P.M., Gasiorowicz, S.G. & Thornton, S.T., (2005), Physics for Scientists and Engineers with Modern Physics (3rd ed.), Upper Saddle River, Pearson Educational Inc.

3. Giambattista, A., Richardson, B. M., Richardson, R. C, (2007), College Physics (2nd Ed.). The McGraw-Hill Companies, Inc., New York

4. Halliday, D., Resnick, R. & Walker, J., (2005), Fundamentals of Physics (7th ed.), New York, John Wiley and Sons, Inc.

5. Tayao, A. G., Santos, G.N.C. & Ocampo, J.P. Jr., (1996), Science and Technology IV: Physics, Manila, Rex Book Store.

6. Fluid mechanics for performing work. (2012). Retrieved from http://www.scienceclarified.com/everyday/Real-Life-Chemistry-Vol-3- Physics-Vol-1/Fluid-Mechanics-Reallife-applications.html

7. Hydraulic press. (2012) Retrieved from http://en.wikipedia.org/wiki/Hydraulic_press

8.

Harmonic motion. (2003). Retrieved from http://www.thefreedictionary.com/harmonic+motion

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