Our Universe: GRAVITATION
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From Ancient times many scientists had shown great interest towards the sky. Most of the scientist studied the motion of celestial bodies. One of the most influential greek astronomers and geographers of his time, "Ptolemy" propounded the geocentric theory in a form that prevailed for 1400 years. He made astronomical observations from Alexandria in Egypt during the year. AD 127-41. Infact the first observation which can date exactly was made by Ptolemy on 26 march 127. While last was made on 2nd feb 141. The Almagest is the earliest of ptolemy work and gives in detail the mathematical theory of motions. Of sun, moon and planets. Ptolemy made his most original contribution by presenting details for the motion of each of the planets. The almagest was not super seeded until a century after copernicus presented his heliocentric theory. Ptolemy first of all justifies his descriptions of the universe based on earth - centred system. It is a view of the world based on fixed earth around which the sphere of the fixed stars rotates every day, this carrying with it the sphere of the sun, moon and planets.
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Nicolas Copernicus: In 16th century AD nicolas copernicus, a polish monk was born on 19th feb 1473 and the died on may 24, 1543 Copernicus summerised his heliocentric theory. According to this theory, the earth and other planets moved in perfect circles around the sun located at the centre of these circles. The earth and other planets would also rotate about their own axes while orbiting around the sum in circular orbits of different radii. Johannes Kepler (1571-1630) was a German mathematician, astronomer and astrologer and key figure in the 17th century scientific revolution. He was assistant to astronomer Tycho Brache. He learned both the ptolemic system and copernican system of planetary motion. He believed copernicas at that time. He defended heliocentrism form both theoretical and theological perspective, maintaining that the sun was the principal source of motive power in the Universe. Issac Newton: The English physicist issac newton (1642-1727) introduced the term "Gravity" after he saw an apple falling on to the ground in his garden. "Gravity" is the force of attraction exerted by the earth on an object. The moon orbits around the earth because of gravity too. Newton later proposed that gravity too. Newton later proposed that gravity was fast a particular case of gravitation. Every mass in the universe attracts every other mass. This is the idea of Newton's law of Gravitation. Newton derived the relation in such a way that F is proportional to 'm'. because the force on a body is directly proportional to its mass by Newton's 2nd law of motion F=ma. When the earth exerts a force on the falling body, by the Newton's 3rd law of motion. The falling body exerts an equal and opposite force on the earth. Therefore the gravitational force F is proportion to both the masses of falling body and the earth, i.e., m1 and m2. The inverse
square relationship 1/r2, was justified by observing motion of the moon. Newton also observed that it is not only the earth which attracts other objects, but every object in the universe attracts every other objects and he termed such a general force as www.sakshieducation.com
''gravitational force". One law is known as the Universal law of gravitation". According to this law, the gravitational force of attraction between any two objects is i) Directly proportional to the product of their masses. ii) Inversely proportional to the square of the distance between them. If the masses of any two objects are 'm1' and m2 and the separation between them is r then the gravitation at force of attraction between them is F a m1m2 ..................... (1) F a 1/r2 ..................... (2) combining (1) and (2) mm Fa 1 2 2 r
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or Gm1m2 Fa ...........( 3) r2
Where 'G' is proportionality constant and known as ''Universal Gravitational Constant".
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"Every body in the Universe attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of distance between them. The force acts along the line joining the two bodies" The Universal Gravitational Constant 'G':
Fr 2 Gm1m 2 We know that F = or G = m m ................( 4) 2 1 2 r
Units of G: - In SI system units of 'F' is Newton (N) - Units of Distance (r) is meter (m) - Units of mass (m) is kilogram (kg) Newton × (metre )
\ Unitsof G =
= Nm 2Kg -2
The numerical value of 'G' is experimentally found to be 6.67 × 10–11 Nm2Kg
Acceleration due to gravity: Galileo proved, (from leaning tower of pisa experiment) that objects of different masses and sizes, when dropped simultaneously from the same height, would reach the ground at same time. In other words when an object is dropped from some height, it experiences uniform acceleration by the gravitational pull of the earth and this acceleration does not depend on the mass of the body. The uniform acceleration produced in a freely falling body due to the gravitational pull of www.sakshieducation.com
the earth is known as acceleration due to gravity and denoted by letter 'g' is given by the equation g = GM/r2 .............. (5) where G =Universal Gravitational Constant (6.67 × 10–11 Nm2/kg2) M =Mass of the Earth( 6 × 1024 kg) r =Radius of Earth(6.4 × 106 m) from the equation (5) Value of g is g = 9.8 ms–2
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Variation of 'g' value: At a given place on the earth 'g' is constant. However, it varies due to the fact that the earth is not a perfect sphere and its radius 'r' is not the same at all places on its surface. Due to flattering of the earth at the poles, 'r' is minimum and hence 'g' is maximum at the poles. Since 'r' is maximum at the equator, the value of 'g' becomes minimum at the equator. The value of 'g' decreases as we move upwards from the surface. As we go deep in to the earth (mine), the value of 'g' decreases. Gravity meters: The instrument used to measure small changes in the value of 'g' at a given location is known as gravity meters. The Boldien gravity meter and Gulf gravity meters are two examples of gravity meter. Mass and weight: The mass of a body is the quantity of matter contained in it. Mass is independent of external factors like position and surroundings. Thus the mass of a given body will be same at all place of earth or any where in the University.
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The weight of a body is the force with which it is attracted by the earth towards its centre and is equal to the product of its mass and acceleration due to gravity. Thus w = mg, where 'm' is mass of body and 'g' is the acceleration due to gravity.
Since the value of 'g' changes from place to place, the weight of a given body differs from one place to another. For example: The acceleration due to gravity on moon is 1/6th of the acceleration due to gravity on earth, hence the weight of the body on moon is 1/6th of its weight on earth. Very Short Answer Questions
1. What is Geocentric theory? A- The earth is stationary and is at the centre of the universe, with the sun, the moon, planets and stars revolving the earth.
2. What is Heliocentric theory? A- All the planets revolve in a perfect circle with the stationary sun at the centre of these circles. 3. What is acceleration due to gravity? A- 1. The Uniform acceleration produced in a freely falling body due to gravitational pull of the earth is called acceleration due to gravity. 2. It is denoted by (g) www.sakshieducation.com
4. Define the mass of a body? A- The total quantity of matter contained in a body is defined as its mass. 5. Define the weight of a body. A- The weight of a body is the force with which it is pulled by the earth towards its centre. 6. State Hooke's law? A- According to Hooke's law "Stress is directly proportional to strain when a body is in elastic limit'' 7. Calculate the gravitational force on a stone of mass 10kg?
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A- Acceleration due to gravity (g) = 9.8m/sec2 mass of the stone, m=10kg The gravitational force on the stone, F = mg = 10 Ă— 9.8 = 98 newton. 8. Define 1 kg wt?
The gravitational force acting on a body of mass 1kg is called 1kg wt
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9. What is the relation between 'g' and 'G'? GM g= 2 r A-
SHORT ANSWER QUESTION (2M) 1. State the Universal law of gravitation? and calculate the gravitational force of stone of mass. 10kg. A- Newton's Universal law of gravitation states ''that every body in the universe attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them" Problem: We know that F = mg = F = 10 Ă— 9.8 = 98 N 2. What are the various factors that effect on the value of acceleration due to gravity (g)? A- The various factors that influence the value of 'g' are 1. Shape of the earth a) At poles b) At equator 2. Height (Altitude) 3. Depth 4. Local Conditions
3. Why is the weight of a body not the same at poles and equator? A- The weight of a body W=mg. So it depends on acceleration due to gravity. It is directly proportional to acceleration due to gravity. As the acceleration due to gravity is not same at the poles and equator, the weight of a body is also not the same at the poles and the www.sakshieducation.com
equator. 4. Why does the weight of a body differs from one place to another place? A- The weight of a body w=mg. It means the weight of a body is directly proportional acceleration due to gravity. As the value of acceleration due to gravity 'g' changes from place to place, the weight of a body also changes from place to place. 5. What is gravity meter? Give examples? 1. The value of g at a given location is slightly affected by geological deposits concentration of massive concrete building and topography of the region 2. So,a sensitive instrument is used to measure such small changes in the value of g at a given location. 3. This instrument is called gravity meter. 4. Examples: Gulf gravity meter and Boliden gravity meter. 6. Differential between mass and weight of a body. - The following are the differences between mass and weight. Mass
1. The amount of matter contained in a body. 1. The force with which the earth attracts a body towards its centre. 2. It is constant at all places. 2. It varies from place to place. 3. It is a scalar quantity.
3. It is a vector quantity.
4. It can be measured by using a common 4. It can be measured by using a spring balance. balance. 5. Its units are gm and kilogram
5. Its units are dyne and newton.
Essay Type Questions 1. Write Newton's law of universal Gravitation and derive F = G
m 1m 2 r2
A. Newton's law of universal Gravitation: Everybody in the universe attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The force acts along the line joining the two bodies. Let the masses of any two bodies be m1 and m2 and they are separated by distance ''r''. The gravitational force of attraction between the two bodies is directly proportional to the product of their masses. F a m1m2 .................(1) The Gravitational force of attraction between the bodies is inversely proportional to the square of the distance between them. www.sakshieducation.com
From (1) and (2) mm mm F = a 1 2 2 or F = G 1 2 2 r r Where G is the proportionality constant and is called ''Universal Gravitational Constant"
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2. Explain the change in the value of 'g' as we go from the surface of the earth to deep inside? 1. 'g' value decreases from the surface of the Earth to deep inside. 2. The reason for the decreases in 'g' value is decrease of effective mass of the Earth. 4 3. The effective mass of the earth is given by the equation M = pr 2p 3 Where p is the average density of the Earth. 4. From the above equation, the effective mass M influencing the body decreases inside the mine as effective r decreases. 5. If g is acceleration due to gravity at a depth ''d'' inside the mine and g is acceleration due to gravity on the surface, then the relation between them is given by æ dö g = ggd ç1 - r ÷ d è ø
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6. Thus at the centre of the earth (d=r) so"gd'' value becomes zero at center of earth.
3. Derive the relation between acceleration due to gravity and Gravitational constant? - Let 'G' and 'g' be universal Gravitational constant and acceleration due to gravity. Suppose a stone of mass 'm' is dropped from a distance 'r' from the centre of the Earth of mass M, then the force exerted by the Earth on the stone is given by the Universal law of Gravitation as. Mm F = G 2 ..............( 1) r This force produces acceleration 'g' on the stone but according to Newton's second law of motion. F = mg ............(2) mg = G
M r2 from the above equation, g is independent of the value of the mass of the stone.
Hence g = G
4. Describe an experiment to determine the weight of a body? Aim: To determine the weight of a body. Apparatus: An extensible spring, pan, weights, stand, metre scale, plasticine. www.sakshieducation.com
Method: Suspend the spring to the stand. Attach a pan to the lower end of the spring. Also fix a pin at the lower end of the spring. Also fix a pin at the lower end of the spring at right angles to the spring by means of a little plasticine.
Parts: 1) Spring; 2) Scale; 3) Pan; 4) Pointer; 5) Stand
The horizontal pin serves as a pointer. Fix the metre scale vertically so that the pointer can more over the scale. Record the reading of the pointer on the scale (lo) initially. Add a 10 gram weight of the scale pan. Note the reading of the pointer (l1). Record the readings by adding 10 grams each time.
Balance Readings (m)
Mean Length l (cm)
Extension of the spring (l â€“ l0)
Plot a graph between stretching ((l â€“ l0) against force (F). This graph is a straight and is called Calibration graph for the spring. www.sakshieducation.com
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Now place the body on the pan and note down the extension ((l – l0).
Locate the point of ((l – l0) on y-axis and read the corresponding x-co-ordinate. This gives the weight of the body.
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Multiple Choice Questions 1. According to kepler's law, the planets move around the sun in their orbit. a) Circular Orbit b) Elliptical Orbit c) Rectangular Orbit d) Linear
2. The law of Gravitation a) applies only to large bodies such as planets and stars b) accounts for all known forces c) holds only in the solar system d) holds everywhere in the universe.
3. The weight of 400g stone is a) 0.041 b) 0.4N c) 3.92 N d) 3920 N
4. The distance of the Moon and from the Earth is about? a) 3.85 × 105 km b) 3.85 × 105 m c) 385 × 105 km d) 385 × 105 m
5. The Unit of force in S.I. system is a) metre b) Pascal
c) Newton d) Newton Metre ()
7. The Gravitational force of attraction between any two bodies is mm a) F = 1 2 2 r
c) F = G m1m2 2 r
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6. The units of gravitational constant a) Nm2 kg–2 b) Nmkg – 2 c) Nm2 kg2 d) Nm–2 kg–2
9. As we go from the equator to the poles, the values of 'g' a) remains the same b) decreases c) increases d) decreases up to altitude of 45°
10. Value of 'g' on the Earth is a) 9.8 ms–2 b) 1.67 ms–2 c) 98 ms–2 d) 167 ms–2
11.The relation between g and G is a) G = 9 b) g = GM/r2 c) g = Gr2 d) None of these
12.Acceleration due to gravity at a height 'h' from the surface of the Earth is
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8. All bodies large or small falls on the ground with the same a) Force b) Velocity c) acceleration d) momentum
2h ö æ a) g ç 1 - ÷ r ø è www.sakshieducation.com
r ö æ b) g ç 1 - ÷ 2h ø è 2h ö æ c) ç 1 - ÷ r ø è
2r ö æ d) ç 1 - ÷ hø è
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13.Which of the following instrument used to measure small changes in the value of 'g'? () a) Boliden Gravity Meter b) Ammeter c) Voltmeter d) Galvanometer
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14.1kg Weight = a) 9.8 ergs b) 9.8 joule c) 9.8 kg d) 9.8 newton 15.Which one of the following is changes with change in place? a) mass b) weight c) mass and weight d) None
II. Fill in the blanks 1. Ptolemic theory is also known as _______ 2. Copernicas theory is known as ________ 3. The weight of 500g stone is _______ N 4. Units of Universal Gravitational constant G are _____ 5. Units of acceleration due to gravity are ______ 6. Acceleration due to gravity is independent of _______ 7. Acceleration due to gravity maximum at earth's______ the and minimum at the ______ 8. Mass of a body is ______ any where in the Universe. 9. Mass of a body changes from _____ to place. 10.The distance of the moon from the earth is about ______ 11.According to Newton's second law of Motion F = mass × ______ 12.Acceleration due to gravity is independent of the value of the _____ of the stone. 13.Acceleration due to gravity at the centre of the Earth _____ 14.____ law helps in the determination of the weight of the body. 15.The 'g' value of sun is _____ 16.The weight of the body is the ______ of the earth's gravity acting on it. 17.In formula g =
GM G stands for ______ r2 www.sakshieducation.com
Match the following The Value of 'g' on moon is ( ) The Value of 'g' on sun is () Mass of the earth is () The Value of 'G' is () The Value of 'g' on the earth ( )
Answers. 1. a 2. d 6. a 7. c 11. b 12. a
4. a 9. c 14. d
5. c 10. a 15. b
in the blanks Geocentric Theory Heliocentric Theory 4.9 Nm2 Kg–2 ms–2 mass of the bodies poles, equator same place 3.85 × 105 km acceleration mass zero Hooke's Law 27.4ms-2 Force of attraction Universal Gravitational Constant
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Fill 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.
3. c 8. c 13. a
a) 6.67 × 10–11 Nm2 kg–2 b) 66.7 × 10–11 Nm2 kg2 c) 9.8 ms–2 d) 1.67 ms–2 e) 19.6 ms–2 f) 27.4 ms–2 g) 6.4 × 1020 kg h) 6 × 1024 kg
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III. 1. 2. 3. 4. 5.
Match the following 1. d 2. f