Gravity - The Solid Earth - page 195

This acceleration is the same as the acceleration for the situation when the entire
mass of the shell 4 b 2

t is concentrated at O, the centre of the shell.
(ii) When the point P is inside the spherical shell, the limits D min and D max in
Eq. (5.10) are b-r and b

π ρ

+ r , respectively. In this case, the potential at P is

=− G ρ t 2 π b 2 D
br

b + r

V

b − r

=− G ρ t 4 π b (5.13)

This potential is a constant, which is independent of the position of point P
inside the shell. The gravitational acceleration, being the negative gradient of the
potential, is therefore zero inside the shell.
Using these results for the gravitational potential and acceleration of a spher-
ical shell, we can immediately see that, at any external point, the gravitational
potential and the acceleration due to a sphere are the same as those values due to
an equal mass placed at the centre of the sphere. In addition, at any point within
a sphere the gravitational acceleration is that due to all the material closer to the
centre than the point itself. The contribution from all the material outside the
point is zero. Imagine the sphere being made of a series of thin uniform spherical
shells; then, from Eq. (5.13), we can see that none of the shells surrounding the
point makes any contribution to the acceleration. The gravitational acceleration
a at a distance r from a sphere of radius b ( b < r ) and density ρ is, therefore,

3 π b 3
r 2

4

G ρ GM

r 2

a =− =− (5.14)

where M is the mass of the sphere. The minus sign in Eqs. (5.12) and (5.14)
arises because gravitational acceleration is positive inwards, whereas r is positive
outwards. A radial variation of density within the sphere does not affect these
results, but any lateral variations within each spherical shell render them invalid.

5.3 Gravity of the Earth
5.3.1 The reference gravity formula
We can apply Eq. (5.14)tothe Earth if we assume it to be perfectly spherical.
The gravitational acceleration towards the Earth is then given by

GM E
r 2

− a = (5.15)

where M E is the mass of the Earth. The value of the gravitational acceleration at
the surface (denoted by g ,which is taken to be positive inwards) of a spherical
Earth is, therefore, GM E / r 2 ,where R is the radius of the Earth. At the Earth's

surface, the acceleration due to gravity has a value of about 9.81 m s − 2 .
The first person to measure the Earth's gravity was Galileo. A celebrated
legend is that he conducted his experiments by dropping objects from the top of
the leaning tower in Pisa and timing their falls to the ground. (In fact, he slid