gravitation, newtons law of gravitation satellite motion, critical velocity

1. Gravitation Prof. Mukesh N. Tekwani Department of Physics Ismail Yusuf College, Mumbai mukeshtekwani@hotmail.com 1

2. Isaac Newton Prof. Mukesh N Tekwani 2

3. Newton’s Law of Gravitation Prof. Mukesh N Tekwani 3 Every particle of matter attracts every other particle with a force which is directlyproportional to the product of the masses and inversely proportional to the square of the distance between them.

4. Newton’s Law of Gravitation Prof. Mukesh N Tekwani 4 G is called the Universal Gravitational Constant G = 6.67 x 10-11 N m2/kg2 G is constant throughout the Universe and G does not depend on the medium between the masses

5. Difference between G and g Prof. Mukesh N Tekwani 5

6. Relation between G and g Let M = mass of the Earth m = mass of an object on the surface of the Earth g = acceleration due to gravity on the Earth’s surface R = radius of the Earth Prof. Mukesh N Tekwani 6

7. Relation between G and g Prof. Mukesh N Tekwani 7 m M

8. Relation between G and g Prof. Mukesh N Tekwani 8 Weight of the object is the gravitational force acting on it. m M ………………………….(1)

9. Relation between G and g Prof. Mukesh N Tekwani 9 At height h from the surface of the Earth’s surface, acceleration due to gravity is gh At height h, Weight of object = gravitational force m h M ………………………….(2)

10. Relation between G and g Prof. Mukesh N Tekwani 10 Dividing (2) by (1) we get, m h Thus, g is independent of the mass of the object. M

11. Projection of a Satellite Prof. Mukesh N Tekwani 11

12. Projection of a Satellite Prof. Mukesh N Tekwani 12 Why is it necessary to have at least a two stage rocket to launch a satellite? A rocket with at least two stages is required to launch a satellite because The first stage is used to carry the satellite up to the desired height. In the second stage, rocket is turned horizontally (through 90 degrees) and the satellite is fired with the proper horizontal velocity to perform circular motion around the earth.

13. Critical Velocity of a Satellite Prof. Mukesh N Tekwani 13 The horizontal velocity with which a satellite should be projected from a point above the earth's surface, so that it orbits in a circular path around the earth is called the orbital velocity or critical velocity (Vc) of the satellite.

14. Kepler’s Laws of Motion Prof. Mukesh N Tekwani 14 Born: December 27, 1571 Died: November 15, 1630 German Mathematician, Astronomer Astrologer.

15. Kepler’s First Law – Law of Orbit Prof. Mukesh N Tekwani 15 Every planet revolves in an elliptical orbit around the Sun, with the Sun situated at one focus of the ellipse.

16. Kepler’s Second Lawor Law of Equal Areas Prof. Mukesh N Tekwani 16 The radius vector drawn from the Sun to any planet sweeps out equal areas in equal intervals of time. This law is called the law of areas. The areal velocity of the radius vector is constant.

17. Kepler’s Law of Equal Areas Prof. Mukesh N Tekwani 17

18. Kepler’s Laws Prof. Mukesh N Tekwani 18

19. Kepler’s Third Law - Laws of Period Prof. Mukesh N Tekwani 19 The square of period of revolution of the planet around the Sun is directly proportional to the cube of the semi-major axis of the elliptical orbit. T2 α r3 According to this law, when the planet is closest to the Sun, its speed is maximum and when it is farthest from the Sun, its speed is minimum.

20. Critical Velocity of a Satellite Prof. Mukesh N Tekwani 20 Let M = mass of the Earth R = radius of Earth m = mass of satellite h = height of the satellite above Earth’s r = R + h, where r is the distance of the satellite from the center of the Earth Vc = critical velocity of the satellite vc h R M r

21. Critical Velocity of a Satellite Prof. Mukesh N Tekwani 21 The centripetal force necessary for the circular motion of the satellite around the Earth is provided by the gravitational force of attraction between the Earth and the satellite. Centripetal force = gravitational force

22. Critical Velocity of a Satellite Prof. Mukesh N Tekwani 22 Factors on which Critical Velocity of a satellite depends: Mass of the planet Radius of the planet Height of the satellite Critical velocity is not dependent on the mass of the satellite as m does not appear in the above equation ……………………………………………….(1)

23. Critical Velocity of a Satellite Prof. Mukesh N Tekwani 23 But we know that Substituting this value in eqn (1), we get, …………………………………..(2)

24. Critical Velocity of a Satellite Prof. Mukesh N Tekwani 24 Assignment 1: Modify eqn (2) to find the critical velocity of a satellite orbiting very close to the surface of the Earth (h << R) Assignment 2: How does the critical velocity (or orbital velocity) of a satellite vary with an increase in the height of the satellite above the Earth’s surface?

25. Time Period of a Satellite The time taken by a satellite to complete one revolution around the earth is called its periodic time or time period. Prof. Mukesh N Tekwani 25

26. Time Period of a Satellite Prof. Mukesh N Tekwani 26 Let M = mass of the Earth R = radius of Earth m = mass of satellite h = height of the satellite above Earth’s r = R + h, where r is the distance of the satellite from the center of the Earth Vc = critical velocity of the satellite vc h R M r

27. Time Period of a Satellite Prof. Mukesh N Tekwani 27 Distance covered by the satellite in 1 revolution = Circumference of the circle Time taken to cover this distance is the time period Critical speed Vc = Vc = But

28. Prof. Mukesh N Tekwani 28 Time Period of a Satellite As is a constant, so we get, T2α r3 Thus, the square of the period of revolution is directly proportional to the cube of the radius of its orbit. Squaring both sides, we get

29. Prof. Mukesh N Tekwani 29 Time Period of a Satellite Factors on which Time Period of a satellite depends: Mass of the planet Radius of the planet, and Height of the satellite from the planet’s surface Period of the satellite does not depend on the mass of the satellite. Assignment: Obtain an expression for the time period of a satellite in terms of gh. For a satellite close to the earth, calculate the period of revolution in minutes.

30. Binding Energy of a Satellite Definition: The minimum amount of energy required to remove a satellite from the earth’s gravitational influence is called as bindingenergy of the satellite. A satellite revolving around the Earth has Kinetic energy, and Potential energy Prof. Mukesh N Tekwani 30

31. What is kinetic energy? The energy possessed by a body due to its motion is called its kinetic energy. If m = mass of an object, and v = its velocity K.E = (1/2) mv2 Prof. Mukesh N Tekwani 31

32. What is potential energy? The energy possessed by a body due to its position is called its potential energy. If m = mass of an object, and h = its height above the surface P.E. = mgh Quiz: Does a body in motion have potential energy? Prof. Mukesh N Tekwani 32

33. Binding Energy of a Satellite Prof. Mukesh N Tekwani 33 Let M = mass of the Earth R = radius of Earth m = mass of satellite h = height of the satellite above Earth’s r = R + h, where r is the distance of the satellite from the center of the Earth Vc = critical velocity of the satellite vc h R M r

34. Prof. Mukesh N Tekwani 34 Binding Energy of a Satellite The critical velocity is given by Kinetic energy of motion KE = Substituting (1) in (2), we get, KE = ……………………………………………………...(1) …..……...(2) ..………………………………………………..(3)

35. Prof. Mukesh N Tekwani 35 Binding Energy of a Satellite The gravitational potential at a distance r from the centre of the Earth is given by: GP = Potential energy = gravitational potential x mass of the satellite Therefore, PE = ..…….………………………..(4)

36. Prof. Mukesh N Tekwani 36 Binding Energy of a Satellite The total energy of the satellite is given by TE = KE + PE TE = + TE = The negative sign indicates that the satellite is bound to the Earth due to the gravitational force of the Earth. ..…….…………………………………………..(5)

37. Prof. Mukesh N Tekwani 37 Binding Energy of a Satellite To free the satellite from the Earth’s gravitational influence, an amount of energy equal to its total energy must be supplied. This is called the binding energy of the satellite. Therefore, BE = Where r = R + h Assignment: Calculate the BE of a satellite at rest on the surface of the Earth.

38. Prof. Mukesh N Tekwani 38 Weightlessness in a Satellite The weight of a body is the gravitational force exerted on it by the Earth. When a person stands on a floor, he exerts a force on the floor. The floor in turn exerts a force (normal reaction) on the person. This normal reaction is equal to the weight of the person. The person has a feeling of weight due to this normal reaction.

39. Prof. Mukesh N Tekwani 39 Weightlessness in a Satellite Consider an astronaut of mass m, in a satellite that is moving around the Earth in a circular orbit. There is a centripetal force on the satellite and the astronaut. Thus, both are attracted towards the Earth with the same acceleration, due to the Earth’s gravitational force. So the astronaut is not able to exert a force on the floor of the satellite & the satellite in turn cannot exert normal reaction on the astronaut. This causes the “feeling” of weightlessness.

40. Prof. Mukesh N Tekwani 40 Weightlessness in a Satellite We must remember that the mass of the astronaut DOES NOT become zero. This condition of weightlessness is also known (incorrectly) as zero gravity condition. But, weightlessness does not mean the absence of gravity.

41. Escape Velocity of a Satellite Prof. Mukesh N Tekwani 41 The minimum velocity with which a body should be projected from the surface of the earth so that it escapes the gravitational field of the earth is called the escape velocity of the body.

42. Escape Velocity of a Satellite Prof. Mukesh N Tekwani 42 Consider a satellite of mass m, stationary on the surface of the Earth. The binding energy of the satellite, on the surface of the Earth, is given by BE = To escape from the Earth’s influence, energy must be provided to the satellite in the form of kinetic energy. R

43. Escape Velocity of a Satellite Prof. Mukesh N Tekwani 43 Therefore, KE of satellite = BE KE = Therefore, = R R

44. Numerical Problems Obtain an equation for the escape velocity of a body from the surface of a planet of radius R and mean density ρ. What would be the duration of the year if the distance between the Earth and Sun gets doubled? Prof. Mukesh N Tekwani 44

45. Numerical Problems Calculate the height of a communications satellite from the surface of the Earth. Values of G, M and R are as given in the text book. (These values will also be provided in the question paper) A body weighs 4.5 kgwt on the surface of Earth. How much will it weigh on the surface of a planet whose mass is (1/9)th that of the Earth’s mass and radius is half that of the Earth? Prof. Mukesh N Tekwani 45

46. Variation of g with Altitude Let M = mass of the Earth m = mass of an object on the surface of the Earth g = acceleration due to gravity on the Earth’s surface R = radius of the Earth Prof. Mukesh N Tekwani 46

47. Prof. Mukesh N Tekwani 47 Variation of g with Altitude Weight of the object is the gravitational force acting on it. m M ………………………….(1)

48. Prof. Mukesh N Tekwani 48 Variation of g with Altitude At height h from the surface of the Earth’s surface, acceleration due to gravity is gh At height h, Weight of object = gravitational force m h M ………………………….(2)

49. Prof. Mukesh N Tekwani 49 Variation of g with Altitude Dividing (2) by (1) we get, m h Thus, g is independent of the mass of the object. From this eqn. it is clear that the acceleration due to gravity decreases as altitude of the body from the earth’s surface increases. M

50. Prof. Mukesh N Tekwani 50 Variation of g with Altitude By Binomial expansion: Since the higher powers of h/R are neglected.

51. Variation of g due to depth Prof. Mukesh N Tekwani 51 Let M = mass of the Earth R = radius of Earth g = acceleration due to gravity on the surface of earth ρ = density of earth P R-d

52. Variation of g due to depth The acceleration due to gravity on the surface of the earth is given by: Consider the earth as a sphere of density ρ Mass = Volume x Density Prof. Mukesh N Tekwani 52

53. Variation of g due to depth Prof. Mukesh N Tekwani 53 Mass = Volume x Density Mass = …………………………………………. (1)

54. Variation of g due to depth Consider a point P at the depth d below the surface of the earth. At this point, let the acceleration due to gravity be gd. Distance of point P from centre of Earth is (R – d) Prof. Mukesh N Tekwani 54

55. Variation of g due to depth The acceleration due to gravity at point P due to sphere of radius (R –d) is Here, M’is the mass of inner solid sphere of radius (R - d) Prof. Mukesh N Tekwani 55

56. Variation of g due to depth Mass = volume x density Prof. Mukesh N Tekwani 56 ……………………………………………(2)

57. Prof. Mukesh N Tekwani 57 Variation of g due to depth Dividing (2) by (1), we get, This is the expression for the acceleration due to gravity at a depth d below the surface of the earth. Therefore, acceleration due to gravity decreases with depth.

58. Variation of g due to depth Prof. Mukesh N Tekwani 58 At the centre of the earth, d = R, therefore gd = 0 So if a body of mass m is taken to the centre of the earth, its weight will be equal to zero (since w= mg). But its mass will not become 0.

59. Prof. Mukesh N Tekwani 59 Variation of g due to depth g’ Outside the earth Inside the earth depth altitude

60. What is latitude The latitude of a location on the Earth is the angular distance of that location south or north of the Equator. Latitude of equator is 0o Latitude of North pole is : 90o north (+90o) Latitude of South pole is : 90o south (-90o) Prof. Mukesh N Tekwani 60

61. Variation of g due to latitude The Earth is rotating from west to east and the axis of rotation passes through the poles. Let angular velocity of earth be ω. Every point on the surface of the earth is moving in a circle, i.e. every point is in an accelerated motion Prof. Mukesh N Tekwani 61

62. Variation of g due to latitude Prof. Mukesh N Tekwani 62 NP-North Pole SP-South Pole

63. Variation of g due to latitude Prof. Mukesh N Tekwani 63 Consider a body at point P on the surface of the earth. Let the latitude of point P be  The body at point P moves in a circular path whose center is at Q and radius is PQ. PQ = r XOP = 

64. Variation of g due to latitude Prof. Mukesh N Tekwani 64 Therefore, OPQ =  (alternate angles – transversal cutting two parallel lines) Centripetal acceleration needed for a body at point P is ar = rω2 Consider  OPQ cos =

65. Variation of g due to latitude Prof. Mukesh N Tekwani 65 cos = cos = r = R cos  Therefore, ar = R cos  x ω2 ar = R ω2cos  ………………. (1) a

67. a = R ω2 cos2 ……………… (2)

68. The effective acceleration g’ due to gravity at point P is directed towards the centre of the earth and is given by:g’ = g - a a

70. As latitude  increases, cos  decreases, so g’ will increase.

71. The value of g’ increases as we move from the equator to the pole due to rotation of the earth.a

72. Variation of g due to latitude Prof. Mukesh N Tekwani 68 Assignment 1: Obtain an expression / value for the acceleration due to gravity at the equator. Assignment 2: Obtain an expression / value for the acceleration due to gravity at the poles.