Question

A particle of mass $M$ is placed at the centre of a uniform spherical shell of equal mass and radius $a$. Find the gravitational potential at a point $\text{P}$ at a distance $a/2$ from the centre.

• A. $-\frac{3GM}{a}$
• B. $-\frac{5GM}{a}$
• C. $-\frac{5GM}{2a}$
• D. $-\frac{2GM}{3a}$

Answer 1

The correct option is A $-\frac{3GM}{a}$

The gravitational potential at the point $\text{P}$ due to the particle at centre is,

$V_1=-\frac{GM}{r}$

$\Rightarrow V_1=-\frac{GM}{\left(a/2\right)}=-\frac{2GM}{a}(\because r=a/2)$

And the potential at point $\text{P}$ due to shell will be

$V_2=-\frac{GM}{a}$

So, the net potential at $\text{P}$ is given by

$V_p=V_1+V_2$

$\Rightarrow V_p=-\frac{2GM}{a}+(-\frac{GM}{a})$

$\therefore V_p=-\frac{3GM}{a}$

Hence, option $\left(a\right)$ is correct.

Why this question ? Tip: The net potential at any point $\text{P}$ will be obtained by superposition of potential due to all interacting masses at that point.