Two concentric shells have mass M and m and their radii are R and r respectively, where R > r . What is the gravitational potential at their common centre

The correct option is -G[(M/R) + (m/r)].

Step-by-Step Explanation:

1. Gravitational Potential of a Spherical Shell:
The gravitational potential ($V$) at any point inside or on the surface of a thin, uniform spherical shell of mass $M$ and radius $R$ is constant and equal to its value on the surface:
$V=-\frac{GM}{R}$
where $G$ is the universal gravitational constant.

2. Potential due to the Outer Shell:
The common center of the concentric shells lies inside the outer shell of mass $M$ and radius $R$. Therefore, the gravitational potential ($V_1$) at the center due to this outer shell is:
$V_1=-\frac{GM}{R}$

3. Potential due to the Inner Shell:
Similarly, the common center lies inside the inner shell of mass $m$ and radius $r$. The gravitational potential ($V_2$) at the center due to this inner shell is:
$V_2=-\frac{Gm}{r}$

4. Superposition Principle:
The total gravitational potential ($V$) at the common center is the sum of the potentials produced by each shell individually:
$V=V_1+V_2$
Substituting the values of $V_1$ and $V_2$:
$V=-\frac{GM}{R}-\frac{Gm}{r}$
Factoring out $-G$ gives:
$V=-G\left[\frac{M}{R}+\frac{m}{r}\right]$