A particle of mass m is placed at the centre of a unifrom spherical shell of mass 3 m and radius R The gravitational potential on the surface of the shell is .

Options

A. `-(Gm)/R`
B. `-(3Gm)/R`
C. `-(4Gm)/R`
D. `-(2Gm)/R`

Explanation

To find the gravitational potential on the surface of a uniform spherical shell with a mass of 3m and radius R, while a particle of mass m is placed at the center, 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: Calculate the Potential Due to the Shell
The shell has a mass of \( 3m \) and the distance from the center of the shell to its surface is \( R \). Therefore, the gravitational potential \( V_{\text{shell}} \) at the surface of the shell is:
\[ V_{\text{shell}} = -\frac{G(3m)}{R} \]

### Step 3: Calculate the Potential Due to the Particle at the Center
The particle of mass \( m \) is located at the center of the shell. The distance from the particle to the surface of the shell is also \( R \). Thus, the gravitational potential \( V_{\text{particle}} \) due to the particle at the center is:
\[ V_{\text{particle}} = -\frac{Gm}{R} \]

### Step 4: Combine the Potentials
The total gravitational potential \( V_{\text{total}} \) at the surface of the shell is the sum of the potentials due to the shell and the particle:
\[ V_{\text{total}} = V_{\text{shell}} + V_{\text{particle}} \]
Substituting the values we calculated:
\[ V_{\text{total}} = -\frac{G(3m)}{R} - \frac{Gm}{R} \]
\[ V_{\text{total}} = -\frac{3Gm}{R} - \frac{Gm}{R} \]
\[ V_{\text{total}} = -\frac{4Gm}{R} \]

### Final Answer
Thus, the gravitational potential on the surface of the shell is:
\[ V = -\frac{4Gm}{R} \]