1. 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
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2. earth’s gravitational field represented by lines of forces
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
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3. 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
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More to Know 4
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4. Gravitational field strength (g)
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.
vector
Unit – N kg-1
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5. Gravitational field strength (g)
6.2 Gravitational field and gravitational field strength (SB p. 204)
Gravitational field strength (g)
By Newton’s 2nd 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)
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6. Relationship between G and g
6.2 Gravitational field and gravitational field strength (SB p. 204)
Relationship between G and g
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Mass of earth (ME) =
2. Average density of earth () =
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Example 2
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Example 3
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Example 4
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7. Variation of g with a distance from the centre of earth
6.2 Gravitational field and gravitational field strength (SB p. 208)
Variation of g with a distance from the centre of earth
For particles on or above the earth’s surface
(a) At r (r  RE)
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8. Variation of g with a distance from the centre of earth
6.2 Gravitational field and gravitational field strength (SB p. 208)
Variation of g with a distance from the centre of earth
For particles on or above the earth’s surface
(b) r = RE
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Example 5
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9. Variation of g with a distance from the centre of earth
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
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10. Variation of g with a distance from the centre of earth
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
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11. Variation of g with latitude
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, R1  R2
Self-rotation of earth - all objects on earth’s surface (except at the poles) need centripetal forces
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12. Variation of g with latitude
6.2 Gravitational field and gravitational field strength (SB p. 211)
Variation of g with latitude
1. Gravitational field strength at the poles gp
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13. Variation of g with latitude
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
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14. Variation of g with latitude
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
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More to Know 9
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15. Variation of g with latitude
6.2 Gravitational field and gravitational field strength (SB p. 212)
Variation of g with latitude
3. Gravitational field strength at latitude  (g)
If  = 45o
g = 9.79 N kg-1
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More to Know 11
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17. More to Know 3 Text Radial and uniform gravitational fields
6.2 Gravitational field and gravitational field strength (SB p. 203)
More to Know 3
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.
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18. More to Know 4 Text Uniform gravitational field
6.2 Gravitational field and gravitational field strength (SB p. 203)
More to Know 4
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.
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19. 6.2 Gravitational field and gravitational field strength (SB p. 204)
More to Know 5
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.
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20. More to Know 6 Text g for an isolated sphere
6.2 Gravitational field and gravitational field strength (SB p. 204)
More to Know 6
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:
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21. 6.2 Gravitational field and gravitational field strength (SB p. 205)
Example 2
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 m s−2, estimate the gravitational field strength (g) on the moon’s surface.
Solution
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22. Example 2 Text Solution:
6.2 Gravitational field and gravitational field strength (SB p. 205)
Example 2
Solution:
If M = mass of moon, then mass of earth = 80M ; if r = radius of moon, then radius of earth = 3.7r.
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23. 6.2 Gravitational field and gravitational field strength (SB p. 205)
Example 3
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
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24. Example 3 Text Solution:
6.2 Gravitational field and gravitational field strength (SB p. 206)
Example 3
Solution:
Let x be the distance of space capsule from the earth when it is under zero gravity. This occurs if
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25. 6.2 Gravitational field and gravitational field strength (SB p. 206)
Example 4
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
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26. Example 4 Text Solution:
6.2 Gravitational field and gravitational field strength (SB p. 206)
Example 4
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.
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27. 6.2 Gravitational field and gravitational field strength (SB p. 209)
Example 5
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 × 104 m from the earth’s surface.
Solution
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28. Example 5 Text Solution:
6.2 Gravitational field and gravitational field strength (SB p. 209)
Example 5
Solution:
The distance from the earth’s centre (r) = 6.38 × ×106 = 6.44 ×106 m = 1.01RE
Acceleration of free fall at the height ( g’)
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29. 6.2 Gravitational field and gravitational field strength (SB p. 209)
More to Know 7
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.
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30. More to Know 8 Text g at earth’s centre At the earth’s centre, r = 0.
6.2 Gravitational field and gravitational field strength (SB p. 210)
More to Know 8
g at earth’s centre
At the earth’s centre, r = 0.
Hence, g = 0.
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31. More to Know 9 Text Centripetal force
6.2 Gravitational field and gravitational field strength (SB p. 212)
More to Know 9
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.
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32. 6.2 Gravitational field and gravitational field strength (SB p. 212)
More to Know 10
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.
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33. More to Know 11 Text g on the earth
6.2 Gravitational field and gravitational field strength (SB p. 213)
More to Know 11
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.
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