Question

If $V$ is the gravitational potential on the surface of the earth, then what is its value at the centre of the earth?
(Assume that mass distribution is uniform)

• A. $2V$
• B. $3V$
• C. $\frac{3}{2}V$
• D. $\frac{2}{3}V$

Answer 1

The correct option is C $\frac{3}{2}V$

Let the mass of the earth is $M$ and radius is $R$.

The gravitational potential due to a uniform solid sphere, at a distance $r$ from centre is given by
$V=-\frac{GM}{2R^3}(3R^2-r^2)\dots \left(i\right)$

Gravitational potential at the surface of the earth is
Substitute $r=R$ in equation $\left(i\right)$

$\Rightarrow V_s=-\frac{GM}{R}=V\dots \left(ii\right)$

Gravitational potential at the centre of the earth is

Substitute $r=0$ in equation $\left(i\right)$

$\Rightarrow V_c=-\frac{3GM}{2R}$
$\Rightarrow V_c=\frac{3}{2}Vs=\frac{3}{2}V$ (By equation $\left(ii\right)$

Hence, option (c) is correct.