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: Understand the Gravitational Potential Formula
The gravitational potential \( V \) at a distance \( r \) from a mass \( m \) is given by the formula:
\[ V = -\frac{Gm}{r} \]
where \( G \) is the gravitational constant.

### Step 2: Determine the Potential Due to Each Shell
1. **Inner Shell**: For the inner shell with mass \( m \) and radius \( r \):
- Since we are at the center, the potential due to the inner shell is:
\[ V_1 = -\frac{Gm}{r} \]

2. **Outer Shell**: For the outer shell with mass \( M \) and radius \( R \):
- Similarly, the potential due to the outer shell at the center is:
\[ V_2 = -\frac{GM}{R} \]

### Step 3: Calculate the Total Gravitational Potential
The total gravitational potential \( V \) at the common center is the sum of the potentials due to both shells:
\[ V = V_1 + V_2 \]
Substituting the expressions for \( V_1 \) and \( V_2 \):
\[ V = -\frac{Gm}{r} - \frac{GM}{R} \]

### Step 4: Combine the Terms
Factoring out \( -G \):
\[ V = -G \left( \frac{m}{r} + \frac{M}{R} \right) \]

### 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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