Question

A particle of mass $2m$ is placed at the centre of a uniform spherical shell of mass $m$ and radius $R$. Find the gravitational potential at a distance$\frac{R}{4}$ from the centre.

Answer 1

The gravitational potential due to shell will be constant inside the shell since gravitational field is 0 inside the shell. Hence, gravitational potential due to shell is same at all points as that at the center.

$V_{shell}=-\frac{Gm}{R}$

Gravitational potential due to mass at the center will be

$V_{mass}=-\frac{G2m}{R/4}=-\frac{8Gm}{R}$

Thus, net potential,

$V=V_{shell}+V_{mass}=-\frac{8Gm}{R}-\frac{Gm}{R}=-9\frac{Gm}{R}$

Answer is $-9\frac{Gm}{R}$.