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CHAPTER 7: Gravitation (2 Hours)

CHAPTER 7: Gravitation (2 Hours). Learning Outcome:. 7.1 Gravitational Force and Field Strength(1 hour). At the end of this chapter, students should be able to: State and use the Newton’s law of gravitation, Define gravitational field strength as gravitational force per unit mass

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CHAPTER 7: Gravitation (2 Hours)

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  1. CHAPTER 7: Gravitation(2 Hours)

  2. Learning Outcome: 7.1 Gravitational Force and Field Strength(1 hour) At the end of this chapter, students should be able to: • State and use the Newton’s law of gravitation, • Define gravitational field strength as gravitational force per unit mass • Derive and use gravitational field strength, • Sketch a graph of ag against r and explain the change in ag with altitude and depth from the surface of the earth.

  3. 7.1 Newton’s law of gravitation 7.1.1 Newton’s law of gravitation • States that a magnitude of an …………………..between two point masses is directly ………………....... to the product of their masses and inversely proportional to the square of the …………………between them. OR mathematically, and where

  4. The statement can also be shown by using the Figure 7.1. Figure 7.1 where

  5. Figures 7.2a and 7.2b show the gravitational force, Fg varies with the distance, r. • Notes: • Every spherical object with constant density can be reduced to a point mass at the centre of the sphere. • The gravitational forces always attractive in nature and the forces always act along the line joining the two point masses. Figure 7.2b Figure 7.2a

  6. Example 7.1 : A spaceship of mass 9000 kg travels from the Earth to the Moon along a line that passes through the Earth’s centre and the Moon’s centre. The average distance separating Earth and the Moon is 384,000 km. Determine the distance of the spaceship from the Earth at which the gravitational force due to the Earth twice the magnitude of the gravitational force due to the Moon. (Given the mass of the Earth, mE=6.001024 kg, the mass of the Moon, mM=7.351022 kg and the universal gravitational constant, G=6.671011 N m2 kg2)

  7. Solution : Given

  8. Example 7.2 : Two spheres of masses 3.2 kg and 2.5 kg respectively are fixed at points A and B as shown in Figure 8.3. If a 50 g sphere is placed at point C, determine a. the resultant force acting on it. b. the magnitude of the sphere’s acceleration. (Given G = 6.671011 N m2 kg2) Figure 7.3

  9. Solution : a. The magnitude of the forces on mC,

  10. Solution :

  11. Solution : The magnitude of the nett force is and its direction is

  12. Solution : b. By using the Newton’s second law of motion, thus and the direction of the acceleration in the …………………..of the nett force on the mCi.e………………………………………

  13. Exercise 7.1 : Given G = 6.671011 N m2 kg2 1. Four identical masses of 800 kg each are placed at the corners of a square whose side length is 10.0 cm. Determine the nett gravitational force on one of the masses, due to the other three. ANS. : 8.2103 N; 45 2. Three 5.0 kg spheres are located in the xy plane as shown in Figure 7.4.Calculate the magnitude of the nett gravitational force on the sphere at the origin due to the other two spheres. ANS. : 2.1108 N Figure 7.4

  14. Exercise 7.1 : 3. In Figure 7.5, four spheres form the corners of a square whose side is 2.0 cm long. Calculate the magnitude and direction of the nett gravitational force on a central sphere with mass of m5 = 250 kg. ANS. : 1.68102 N; 45 Figure 7.5

  15. 7.1.2 Gravitational Field • is defined as a ……………………………surrounding a body that has the property of masswherethe …………………….is experienced if a test mass placed in the region. • Field lines are used to show gravitational field around an object with mass. • For spherical objects (such as the Earth) the field is radial as shown in Figure 7.6. Figure 7.6

  16. Note: • The gravitational field in small region near the Earth’s surface are ……………..and can be drawn ……………………..to each other as shown in Figure 7.7. • The field lines indicate two things: • The arrows – the ………………. of the field • The spacing – the ………………. of the field Figure 7.7 The gravitational field is a conservative field in which the work done in moving a body from one point to another is independent of the path taken.

  17. Gravitational field strength, • is defined as the ……………………per ………………of a body (test mass) placed at a point. OR • It is also known as gravitational acceleration (the free-fall acceleration). • It is a ……………………….. • The S.I. unit of the gravitational field strength is ……………..or …………... where

  18. 7.1.3 Gravitational Acceleration • Its direction is in the ……………………..of the gravitational force. • Another formula for the gravitational field strength at a point is given by and where

  19. Figure 7.8 shows the direction of the gravitational field strength on a point S at distance rfrom the centre of the planet. Figure 7.8

  20. The gravitational field in the small region near the Earth’s surface( r R) are uniform where its strength is 9.81 m s2 and its direction can be shown by using the Figure 7.9. Figure 7.9 where

  21. Example 7.3 : Determine the Earth’s gravitational field strength a. on the surface. b. at an altitude of 350 km. (Given G = 6.671011 N m2 kg2, mass of the Earth, M = 6.00  1024 kg and radius of the Earth, R= 6.40  106 m) Solution : a. The gravitational field strength is

  22. Solution : b. The gravitational field strength is given by

  23. Example 7.4 : The gravitational field strength on the Earth’s surface is 9.81 N kg1. Calculate a. the gravitational field strength at a point C at distance 1.5R from the Earth’s surface where R is the radius of the Earth. b. the weight of a rock of mass 2.5 kg at point C. Solution : a. The gravitational field strength on the Earth’s surface is The distance of point C from the Earth’s centre is

  24. Solution : a. Thus the gravitational field strength at point C is given by b. Given The weight of the rock is

  25. Example 7.5 : Figure 7.10 shows an object A at a distance of 5 km from the object B. The mass A is four times of the mass B. Determine the location of a point on the line joining both objects from B at which the nett gravitational field strength is zero. Figure 7.10

  26. Solution : At point C,

  27. 7.1.4 Variation of gravitational field strength on the distance from the centre of the Earth Outside the Earth ( r > R) • Figure 7.11 shows a test mass which is outside the Earth and at a distance r from the centre. • The gravitational field strength outside the Earth is Figure 7.11

  28. On the Earth ( r = R) • Figure 7.12 shows a test mass on the Earth’s surface. • The gravitational field strength on the Earth’s surface is Figure 7.12

  29. Inside the Earth ( r < R) • Figure 7.13 shows a test mass which is inside the Earth and at distance rfrom the centre. • The gravitational field strength inside the Earth is given by where Figure 7.13

  30. By assuming the Earth is a solid sphere and constant density, hence • Therefore the gravitational field strength inside the Earth is

  31. The variation of gravitational field strength, ag as a function of distance from the centre of the Earth, r is shown in Figure 7.14. Figure 7.14

  32. Learning Outcome: 7.2 Gravitational potential (1 hour) At the end of this chapter, students should be able to: • Define gravitational potential in a gravitational field. • Derive and use the formulae, • Sketch the variation of gravitational potential, V with distance, rfrom the centre of the earth.

  33. 7.2.1 Gravitational potential, V • at a point is defined as the ………………...by an external force in bringing a test mass from infinity to a point per unit the test mass. OR mathematically, V is written as: • It is a ……………………... where

  34. The S.I unit for gravitational potential is ……………or ………………. • Another formula for the gravitational potential at a point is given by and and where where

  35. The gravitational potential difference between point A and B (VAB) in the Earth’s gravitational field is defined as the ………………..in bringing a test mass from point B to point A per unit the test mass. OR mathematically, VABis written as: where

  36. A rA B rB M • Figure 7.15 shows two points A and B at a distance rA and rB from the centre of the Earth respectively in the Earth’s gravitational field. • The gravitational potential difference between the points A and B is given by Figure 7.15

  37. V • The gravitational potential difference between point B and A in the Earth’s gravitational field is given by • The variation of gravitational potential, V when the test mass, m move away from the Earth’s surface is illustrated by the graph in Figure 7.16. Figure 7.16 Note: • The Gravitational potential at infinity is zero.

  38. Example 7.6 : When in orbit, a satellite attracts the Earth with a force of 19 kN and the satellite’s gravitational potential due to the Earth is 5.45107 J kg1. a. Calculate the satellite’s distance from the Earth’s surface. b. Determine the satellite’s mass. (Given G = 6.671011 N m2 kg2, mass of the Earth, M = 5.981024 kg and radius of the Earth , R = 6.38106 m) Solution :

  39. Solution : a. By using the formulae of gravitational potential, thus: Therefore the satellite’s distance from the Earth’s surface is:

  40. Solution : b. From the Newton’s law of gravitation, hence:

  41. Example 7.7 : The gravitational potential at the surface of a planet of radius R is 12.8 MJ kg1. Determine the work done in overcoming the gravitational force when a space probe of mass 1000 kg is lifted to a height of 2R from the surface of the planet. Solution : On the surface of the planet, the gravitational potential is

  42. Solution : The final distance of the space probe from the centre of the Earth is The work done required is given by

  43. Example 7.8 : The Moon has a mass of 7.351022 kg and a radius of 1740 km. a. Determine the gravitational potential at its surface. b. A probe of mass 100 kg is dropped from a height 1 km onto the Moon’s surface. Calculate its change in gravitational potential energy. c. If all the gravitational potential energy lost is converted to kinetic energy, calculate the speed at which the probe hits the surface. (Given G = 6.671011 N m2 kg2) Solution : a. The gravitational potential on the moon’s surface is

  44. Solution : b. Given Hence the change in the gravitational potential energy is

  45. Solution : c. Given Gravitational potential energy lost = kinetic energy The speed of the probe when hit the moon’s surface is given by

  46. Learning Outcome: 7.3 Satellite motion in a circular orbit (½ hour) At the end of this chapter, students should be able to: • Explain satellite motion with: • velocity, • period,

  47. 7.3 Satellite motion in a circular orbit 7.3.1 Tangential (linear/orbital) velocity, v • Consider a satellite of mass, m travelling around the Earth of mass, M, radius, R, in a circular orbit of radius, r with …………………………….speed, v as shown in Figure 8.22. Figure 8.22

  48. The centripetal force, Fc is contributed by the gravitational force of attraction, Fg exerted on the satellite by the Earth. Hence the tangential velocity, v is given by where

  49. and • For a satellite close to the Earth’s surface, Therefore • The relationship between tangential velocity and angular velocity is Hence , the period, T of the satellite orbits around the Earth is given by

  50. 7.3.2 Synchronous (Geostationary) Satellite • Figure 7.17 shows a synchronous (geostationary) satellite which stays above the same point on the equator of the Earth. • The satellite have the following characteristics: • It revolves in the ……………………………as the Earth. • It rotates with the …………………….of rotation as that of the Earth (24 hours). • It moves directly above the ……………………... • The centre of a synchronous satellite orbit is at the centre of the Earth. • It is used as a ……………………………... Figure 7.17

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