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5 Gravitation
Spherical Shell
The gravitational intensity due to a spherical shell of radius a.
g(r ) = 0 (r < a)
= −G M/r
2
(r > a)
(5.6)
where r is measured from the centre of the shell. The potential
V (r ) = −G M/a (r < a)
= −G M/r (r > a)
(5.7)
Uniform Solid Sphere
g(r ) = −G Mr/a (r ≤ a)
= −G M/r
2
(r ≥ a)
(5.8a)
V (r ) = −
G M
2a
3 −
r 2
a 2
(r ≤ a)
= −G M/r (r ≥ a)
(5.8b)
Potential energy of a uniform sphere
U = −3G M
2
/5a
(5.9)
Variation of g on Earth
(a) Altitude : g = g 0 /(1 + h/R)
2
(5.10)
g = g 0
1 −
2h
R
(h << R)
( 5.10a)
(b) Latitude (λ) (at sea level)
g 0 = 9.83215 − 0.05178 cos
2
λ
(5.11)
Formula (5.11) is accurate to better than two parts in a million.
(c) Rotation of earth:
g
= g − Rω
2 cos
2
λ
(5.12)
where ω = 7.27 × 10 −5 /s and R = 6.4 × 10 6 m
(d) Depth (d) (constant density model)
g = g 0 (1 − d/R)
(5.13)