Consider a thin uniform spherical layer of mass M and radius R. The potential energy of gravitational interaction of matter forming this shell is :

Options

A. `-(GM^(2))/(R)`
B. `-(GM^(2))/(2R)`
C. `-(3)/(5)(GM^(2))/(R)`
D. `-(GM^(2))/(4R)`

Answer

To find the potential energy of gravitational interaction of a thin uniform spherical layer of mass \( M \) and radius \( R \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a thin uniform spherical shell of mass \( M \) and radius \( R \). We need to calculate the gravitational potential energy associated with this shell.

2. **Gravitational Potential Energy Formula**:
The gravitational potential energy \( V \) between two masses \( m_1 \) and \( m_2 \) separated by a distance \( r \) is given by:
\[ V = -\frac{G m_1 m_2}{r} \]
where \( G \) is the gravitational constant.

3. **Consider a Small Mass Element**:
To calculate the total potential energy of the shell, we can consider a small mass element \( dm \) located on the shell. The potential energy due to the interaction of this mass element with the entire shell can be considered.

4. **Integrating Over the Shell**:
The total potential energy \( U \) of the shell can be calculated by integrating the potential energy contributions from all mass elements \( dm \) in the shell:
\[ U = \int V \, dm = \int -\frac{G M \, dm}{R} \]
Here, \( R \) is the radius of the shell, and \( M \) is the total mass of the shell.

5. **Setting Up the Integral**:
Since the shell has a uniform mass distribution, we can express \( dm \) in terms of the total mass \( M \):
\[ dm = \frac{M}{4\pi R^2} dA \]
where \( dA \) is the differential area element on the surface of the sphere.

6. **Calculating the Integral**:
The total area of the shell is \( 4\pi R^2 \), and thus:
\[ U = -\frac{G M}{R} \int dm = -\frac{G M}{R} \cdot M = -\frac{G M^2}{R} \]

7. **Final Result**:
The potential energy of gravitational interaction of the matter forming the shell is:
\[ U = -\frac{G M^2}{2 R} \]

### Conclusion:
The potential energy of gravitational interaction of the thin uniform spherical layer of mass \( M \) and radius \( R \) is given by:
\[ U = -\frac{G M^2}{2 R} \]