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

Answer

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

### Step 1: Understand the Concept of Gravitational Potential
Gravitational potential \( V \) at a point in a gravitational field is defined as the work done in bringing a unit mass from infinity to that point without any acceleration. The formula for gravitational potential due to a mass \( M \) at a distance \( r \) is given by:

\[ V = -\frac{G M}{r} \]

where \( G \) is the universal gravitational constant.

### Step 2: Analyze the Spherical Shell
According to the shell theorem, a uniform spherical shell of mass \( M \) exerts no net gravitational force on a mass located inside it. Therefore, the gravitational potential inside the shell (including at the center) is constant and equal to the potential at the surface of the shell.

### Step 3: Calculate the Gravitational Potential at the Surface
To find the gravitational potential at the surface of the spherical shell, we need to use the formula for gravitational potential at a distance \( R \) from the center (which is also the radius of the shell):

\[ V_{\text{surface}} = -\frac{G M}{R} \]

### Step 4: Consider the Mass \( m \) at the Center
The mass \( m \) at the center does not affect the potential on the surface of the shell due to the shell theorem. Thus, we only consider the mass \( M \) of the shell for calculating the potential at the surface.

### Step 5: Final Expression for Gravitational Potential
The gravitational potential on the surface of the shell is given by:

\[ V = -\frac{G M}{R} \]

This is the final answer for the gravitational potential on the surface of the spherical shell.

### Summary
The gravitational potential on the surface of the spherical shell is:

\[ V = -\frac{G M}{R} \]