A mass m on the surface of the Earth is shifted to a target equal to the radius of the Earth. If R is the radius and M is the mass of the Earth, then work done in this process is

Options

A. `(mgR)/(2)`
B. mgR
C. 2 mgR
D. `(mgR)/( 4)`

Answer

To solve the problem of calculating the work done in moving a mass \( m \) from the surface of the Earth to a distance equal to the radius of the Earth \( R \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Initial and Final Positions**:
- The initial position of the mass \( m \) is at the surface of the Earth, which is at a distance \( R \) from the center of the Earth.
- The final position is at a distance \( 2R \) from the center of the Earth (which is the radius of the Earth plus the radius of the Earth).

2. **Understand the Gravitational Force**:
- The gravitational force acting on the mass \( m \) at a distance \( r \) from the center of the Earth is given by:
\[ F = \frac{G M m}{r^2} \]
- Here, \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the distance from the center of the Earth.

3. **Calculate the Work Done**:
- The work done \( W \) in moving the mass \( m \) from \( R \) to \( 2R \) against the gravitational force can be calculated using the integral of the force over the distance:
\[ W = \int_{R}^{2R} F \, dr = \int_{R}^{2R} \frac{G M m}{r^2} \, dr \]

4. **Evaluate the Integral**:
- The integral can be evaluated as follows:
\[ W = G M m \left[ -\frac{1}{r} \right]_{R}^{2R} = G M m \left( -\frac{1}{2R} + \frac{1}{R} \right) \]
- Simplifying this gives:
\[ W = G M m \left( \frac{1}{R} - \frac{1}{2R} \right) = G M m \left( \frac{1}{2R} \right) \]

5. **Relate to Gravitational Acceleration**:
- We know that the gravitational acceleration at the surface of the Earth \( g \) is given by:
\[ g = \frac{G M}{R^2} \]
- Therefore, we can express \( G M \) as \( g R^2 \):
\[ W = \frac{g R^2 m}{2R} = \frac{g m R}{2} \]

### Final Answer:
The work done in moving the mass \( m \) from the surface of the Earth to a distance equal to the radius of the Earth is:
\[ W = \frac{g m R}{2} \]