A body of mass m is placed at the centre of the spherical shell of radius R and mass M. The gravitation potential on the surface of the shell is

Options

A. `-G/R (M+m)`
B. `-G/R (M-m)`
C. `-G/R ((mM)/(M+m))`
D. `-G/R ((mM)/(M-m))`

Answer

To find the gravitational potential on the surface of a spherical shell with a mass \( M \) and radius \( R \) when a body of mass \( m \) is placed at the center, we can follow these steps:

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

### Step 2: Consider the Spherical Shell
For a spherical shell of mass \( M \), the gravitational potential inside the shell (at the center) is constant and equal to the potential on the surface of the shell. This is a key property of spherical shells in gravitational fields.

### Step 3: Calculate the Potential on the Surface
At the surface of the shell (where \( r = R \)), the potential due to the mass \( M \) of the shell is:
\[ V = -\frac{G M}{R} \]

### Step 4: Add the Contribution from the Mass \( m \)
Since the mass \( m \) is at the center, it does not contribute to the potential at the surface of the shell. The potential due to mass \( m \) at the surface of the shell is zero because the gravitational potential is defined relative to the mass creating it.

### Step 5: Final Expression for Potential
Thus, the total gravitational potential \( V \) at the surface of the shell is simply:
\[ V = -\frac{G M}{R} \]

### Conclusion
The gravitational potential on the surface of the shell is:
\[ V = -\frac{G M}{R} \]