6 2 Gravitational field and gravitational field strength

6. 2 Gravitational field and gravitational field strength • Gravitational field strength (g) • Relationship between G and g • Variation of g with a distance from the centre of earth • Variation of g with latitude © Manhattan Press (H. K. ) Ltd. 1

6. 2 Gravitational field and gravitational field strength (SB p. 203) Gravitational field A gravitational field is a force field. This field exists around a mass. Another mass introduced into the field will experience a gravitational force. earth’s gravitational field represented by lines of forces Similar to magnetic field © Manhattan Press (H. K. ) Ltd. 2

6. 2 Gravitational field and gravitational field strength (SB p. 203) Gravitational field - only the force of attraction - directed towards centre of earth - stronger as closer to earth’s surface Go to More to Know 3 More to Know 4 © Manhattan Press (H. K. ) Ltd. 3

6. 2 Gravitational field and gravitational field strength (SB p. 203) Gravitational field strength (g) The gravitational field strength (g) at a point in a gravitational field is defined as the gravitational force of attraction per unit mass at that point. Unit – N kg-1 © Manhattan Press (H. K. ) Ltd. vector 4

6. 2 Gravitational field and gravitational field strength (SB p. 204) Gravitational field strength (g) By Newton’s 2 nd Law which is equal to g (acceleration due to gravity) Unit of gravitational field strength = Unit for acceleration Gravitational field strength on earth’s surface = 9. 81 N kg-1 (produces acceleration of 9. 81 m s-2) © Manhattan Press (H. K. ) Ltd. 5

6. 2 Gravitational field and gravitational field strength (SB p. 204) Relationship between G and g Go to More to Know 5 Go to More to Know 6 1. Mass of earth (ME) = 2. Average density of earth ( ) = Go to Example 2 Example 3 Example 4 © Manhattan Press (H. K. ) Ltd. 6

6. 2 Gravitational field and gravitational field strength (SB p. 208) Variation of g with a distance from the centre of earth 1. For particles on or above the earth’s surface (a) At r (r RE) © Manhattan Press (H. K. ) Ltd. 7

6. 2 Gravitational field and gravitational field strength (SB p. 208) Variation of g with a distance from the centre of earth 1. For particles on or above the earth’s surface (b) r = RE Go to Example 5 © Manhattan Press (H. K. ) Ltd. 8

6. 2 Gravitational field and gravitational field strength (SB p. 209) Variation of g with a distance from the centre of earth 2. For particles below the earth’s surface Gravitational force acted on particle is only due to part of earth inside it Go to More to Know 7 © Manhattan Press (H. K. ) Ltd. 9

6. 2 Gravitational field and gravitational field strength (SB p. 210) Variation of g with a distance from the centre of earth 2. For particles below the earth’s surface Go to More to Know 8 © Manhattan Press (H. K. ) Ltd. 10

6. 2 Gravitational field and gravitational field strength (SB p. 210) Variation of g with latitude Two main reasons why g varies with latitude • Shape of earth - earth is ellipsoid, R 1 R 2 • Self-rotation of earth - all objects on earth’s surface (except at the poles) need centripetal forces © Manhattan Press (H. K. ) Ltd. 11

6. 2 Gravitational field and gravitational field strength (SB p. 211) Variation of g with latitude 1. Gravitational field strength at the poles gp © Manhattan Press (H. K. ) Ltd. 12

6. 2 Gravitational field and gravitational field strength (SB p. 211) Variation of g with latitude 2. Gravitational field strength at the equator ge ge < gp © Manhattan Press (H. K. ) Ltd. 13

6. 2 Gravitational field and gravitational field strength (SB p. 212) Variation of g with latitude 3. Gravitational field strength at latitude (g ) Centripetal force (towards Q) = m(RE cos ) 2 which is provided by components of gravitational force (towards O) and other force from object Go to More to Know 9 © Manhattan Press (H. K. ) Ltd. 14

6. 2 Gravitational field and gravitational field strength (SB p. 212) Variation of g with latitude 3. Gravitational field strength at latitude (g ) If = 45 o g = 9. 79 N kg-1 Go to More to Know 10 More to Know 11 © Manhattan Press (H. K. ) Ltd. 15

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6. 2 Gravitational field and gravitational field strength (SB p. 203) Radial and uniform gravitational fields On the earth or other planets, the gravitational field lines are radially inward (radial field). The parallel field lines indicate that it is a uniform field. Return to © Manhattan Press (H. K. ) Ltd. Text 17

6. 2 Gravitational field and gravitational field strength (SB p. 203) Uniform gravitational field If a body is located in a uniform gravitational field, the gravitational force acted on the body is the same for all positions. Return to © Manhattan Press (H. K. ) Ltd. Text 18

6. 2 Gravitational field and gravitational field strength (SB p. 204) Uniform gravitational field near the earth's surface Since the radius of the earth is very great, the gravitational field near the earth's surface is assumed as uniform. Return to © Manhattan Press (H. K. ) Ltd. Text 19

6. 2 Gravitational field and gravitational field strength (SB p. 204) g for an isolated sphere Other than the earth, the gravitational field strength (g) of a point at a distance r from the centre of an isolated sphere of mass M is also equal to: Return to © Manhattan Press (H. K. ) Ltd. Text 20

6. 2 Gravitational field and gravitational field strength (SB p. 205) Q: The mass of the earth is approximately 80 times that of the moon and the earth’s radius is 3. 7 times that of the moon. If the acceleration of free fall on the earth is 9. 81 m s− 2, estimate the gravitational field strength (g) on the moon’s surface. Solution © Manhattan Press (H. K. ) Ltd. 21

6. 2 Gravitational field and gravitational field strength (SB p. 205) Solution: If M = mass of moon, then mass of earth = 80 M ; if r = radius of moon, then radius of earth = 3. 7 r. Return to Text © Manhattan Press (H. K. ) Ltd. 22

6. 2 Gravitational field and gravitational field strength (SB p. 205) Q: A space capsule travels from the earth towards the moon. Calculate its distance from the earth where it is under zero gravity. (Consider only the gravity due to the earth and the moon. ) (Mass of earth ME = 5. 98 × 1024 kg; mass of moon MM = 7. 35 × 1022 kg; distance between centre of earth and centre of moon = 3. 84 × 108 m. ) Solution © Manhattan Press (H. K. ) Ltd. 23

6. 2 Gravitational field and gravitational field strength (SB p. 206) Solution: Let x be the distance of space capsule from the earth when it is under zero gravity. This occurs if Return to Text © Manhattan Press (H. K. ) Ltd. 24

6. 2 Gravitational field and gravitational field strength (SB p. 206) Q: A certain star of mass M and radius r rotates so rapidly that materials at its equator only just remain on its surface. Given that the gravitational constant is G, find the period of self-rotation of the star in terms of G, M and r. Solution © Manhattan Press (H. K. ) Ltd. 25

6. 2 Gravitational field and gravitational field strength (SB p. 206) Solution: For a mass m on the equator, the gravitational force: As the star rotates, the mass moves in a circle of radius r. The centripetal force is provided by the gravitational force. Return to Text © Manhattan Press (H. K. ) Ltd. 26

6. 2 Gravitational field and gravitational field strength (SB p. 209) Q: The radius of the earth (RE) is 6. 38 × 106 m and the acceleration due to gravity at the earth’s surface is 9. 81 m s− 2. Find the acceleration of free fall at a height of 6. 38 × 104 m from the earth’s surface. Solution © Manhattan Press (H. K. ) Ltd. 27

6. 2 Gravitational field and gravitational field strength (SB p. 209) Solution: The distance from the earth’s centre (r) = 6. 38 × 104 + 6. 38 × 106 = 6. 44 × 106 m = 1. 01 RE Acceleration of free fall at the height ( g’) Return to Text © Manhattan Press (H. K. ) Ltd. 28

6. 2 Gravitational field and gravitational field strength (SB p. 209) 1. In the case of a shell and a particle outside the shell, the shell attracts the particle as if all the mass of the shell were concentrated at its centre. 2. If the particle is inside the shell, the net force acted on the particle is zero. Return to © Manhattan Press (H. K. ) Ltd. Text 29

6. 2 Gravitational field and gravitational field strength (SB p. 210) g at earth’s centre At the earth’s centre, r = 0. Hence, g = 0. Return to Text © Manhattan Press (H. K. ) Ltd. 30

6. 2 Gravitational field and gravitational field strength (SB p. 212) Centripetal force In the calculation, the gravitational force is assumed to be the same everywhere on the earth. Therefore, the centripetal force is affected by latitude only. Return to Text © Manhattan Press (H. K. ) Ltd. 31

6. 2 Gravitational field and gravitational field strength (SB p. 212) If a plumb-line is hung at P, it will not be exactly normal to the earth’s surface but will be along the direction PB. Return to Text © Manhattan Press (H. K. ) Ltd. 32

6. 2 Gravitational field and gravitational field strength (SB p. 213) g on the earth Generally, the gravitational field strength varies from place to place on the earth's surface. It increases from the equator to the poles. Return to Text © Manhattan Press (H. K. ) Ltd. 33