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

Options

A. `-(GM)/R`
B. `-(GM)/r`
C. `-G[M/R-m/r]`
D. `-G[M/R+m/r]`

Explanation

To find the gravitational potential at the common center of two concentric shells with masses \( M \) and \( m \) and radii \( R \) and \( r \) respectively, we can follow these steps:

### Step 1: Understanding Gravitational Potential
The gravitational potential \( V \) at a distance \( d \) from a mass \( M \) is given by the formula:
\[ V = -\frac{G M}{d} \]
where \( G \) is the gravitational constant.

### Step 2: Gravitational Potential Inside a Shell
According to the shell theorem, the gravitational potential inside a uniform spherical shell is constant and equal to the potential at its surface. Therefore, the gravitational potential inside each shell at the center (which is also at a distance less than the radius of the shell) can be calculated using the radius of the shell.

### Step 3: Calculate the Potential Due to Each Shell
1. **For the larger shell (mass \( M \) and radius \( R \))**:
The gravitational potential at the center due to this shell is:
\[ V_M = -\frac{G M}{R} \]

2. **For the smaller shell (mass \( m \) and radius \( r \))**:
The gravitational potential at the center due to this shell is:
\[ V_m = -\frac{G m}{r} \]

### Step 4: Total Gravitational Potential at the Center
The total gravitational potential \( V \) at the center is the sum of the potentials due to both shells:
\[ V = V_M + V_m = -\frac{G M}{R} - \frac{G m}{r} \]
This can be simplified to:
\[ V = -G \left( \frac{M}{R} + \frac{m}{r} \right) \]

### Final Answer
Thus, the gravitational potential at the common center of the two concentric shells is:
\[ V = -G \left( \frac{M}{R} + \frac{m}{r} \right) \]

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