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. `-(GM)/ ( 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 Gravitational Potential
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: Gravitational Potential Due to the Shell
For a uniform spherical shell, the gravitational potential inside the shell (at the center) is constant and equal to the potential on the surface of the shell. Therefore, we can calculate the potential at the surface of the shell.

### Step 3: Calculate the Potential Due to the Shell
The mass of the shell is \(3m\) and the radius is \(R\). The potential at the surface of the shell (at distance \(R\) from the center) due to the shell is:
\[ V_{\text{shell}} = -\frac{G(3m)}{R} \]

### Step 4: Calculate the Potential Due to the Particle at the Center
The particle of mass \(m\) located at the center contributes to the gravitational potential at the surface of the shell. However, since the potential due to a mass at the center is also calculated at the surface, we have:
\[ V_{\text{particle}} = -\frac{Gm}{R} \]

### Step 5: Total Gravitational Potential at the Surface
Now, we can find the total gravitational potential \(V_{\text{total}}\) at the surface of the shell by adding the contributions from both the shell and the particle:
\[ V_{\text{total}} = V_{\text{shell}} + V_{\text{particle}} = -\frac{G(3m)}{R} - \frac{Gm}{R} \]
\[ V_{\text{total}} = -\frac{3Gm}{R} - \frac{Gm}{R} = -\frac{4Gm}{R} \]

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