Two identical spherical masses are kept at some distance. Potential energy when a mass `m` is taken from the surface of one sphere to the other

Options

A. increases continuously
B. decreases continuously
C. first increases then decreases
D. first decreases then increases

Answer

To solve the problem of finding the potential energy when a mass \( m \) is taken from the surface of one identical spherical mass to the surface of another identical spherical mass, we can follow these steps:

### Step 1: Understand the System
We have two identical spherical masses, each of mass \( M \), separated by a distance \( R \). We want to calculate the gravitational potential energy when a mass \( m \) is moved from the surface of one sphere to the surface of the other.

### Step 2: Calculate the Initial Potential Energy
The gravitational potential energy \( U \) between two masses \( M \) and \( m \) separated by a distance \( d \) is given by the formula:
\[ U = -\frac{G M m}{d} \]
Initially, when the mass \( m \) is on the surface of the first sphere, the distance from the center of the first sphere to the center of the second sphere is \( R \). Therefore, the initial potential energy \( U_i \) when \( m \) is on the surface of the first sphere is:
\[ U_i = -\frac{G M m}{R} \]

### Step 3: Calculate the Final Potential Energy
When the mass \( m \) is moved to the surface of the second sphere, the distance from the center of the second sphere to the mass \( m \) is still \( R \). Thus, the final potential energy \( U_f \) when the mass \( m \) is on the surface of the second sphere is:
\[ U_f = -\frac{G M m}{R} \]

### Step 4: Calculate the Change in Potential Energy
The change in potential energy \( \Delta U \) when moving the mass \( m \) from the first sphere to the second sphere is given by:
\[ \Delta U = U_f - U_i \]
Since both \( U_i \) and \( U_f \) are equal:
\[ \Delta U = -\frac{G M m}{R} - \left(-\frac{G M m}{R}\right) = 0 \]

### Conclusion
The gravitational potential energy does not change when moving the mass \( m \) from the surface of one identical sphere to the surface of another identical sphere. Thus, the potential energy remains constant at:
\[ \Delta U = 0 \]