Question

The acceleration due to gravity, at a depth d below the surface of the earth, equals to gd . Obtain the formula for the ratio (gd/ge) where ge is the acceleration due to gravity on the surface of the earth.

Answer 1

we know, acceleration due to gravity on the surface of the earth is given by,

$G_e=\frac{GM}{R^2}$ .....(1)

where, G is gravitational constant, M is mass of the earth and R is radius of the earth.

now let body be taken to the depth ‘d’ below the surface of the earth.

then acceleration due to gravity at depth ‘d’ below the surface of the earth is given by, $g_d=\frac{GM}{R^2}\left(1-\frac{d}{R}\right)$.......(2)

from equations (1), and (2),

$\frac{G_d}{G_e}=\left(1-\frac{d}{R}\right)$this is the required expression.