The work done to raise a mass m from the surface of the earth to a height h, which is equal to the radius of the earth, is :

Options

A. `(3)/(2)mgR`
B. `mgR`
C. `2 mgR`
D. `(1)/(2)mgR`

Explanation

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

### Step-by-Step Solution:

1. **Understanding the Work Done**:
The work done \( W \) in raising an object in a gravitational field is given by the change in gravitational potential energy (PE). The formula for gravitational potential energy is:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass being raised, and \( r \) is the distance from the center of the Earth.

2. **Initial Potential Energy**:
When the mass \( m \) is at the surface of the Earth, the distance \( r \) is equal to the radius of the Earth \( R \):
\[ U_i = -\frac{G M m}{R} \]

3. **Final Potential Energy**:
When the mass \( m \) is raised to a height \( h \) equal to the radius of the Earth, the new distance from the center of the Earth is \( R + h = R + R = 2R \):
\[ U_f = -\frac{G M m}{2R} \]

4. **Calculating Work Done**:
The work done \( W \) is the difference between the final and initial potential energies:
\[ W = U_f - U_i \]
Substituting the values we found:
\[ W = \left(-\frac{G M m}{2R}\right) - \left(-\frac{G M m}{R}\right) \]
Simplifying this expression:
\[ W = -\frac{G M m}{2R} + \frac{G M m}{R} \]
\[ W = \frac{G M m}{R} - \frac{G M m}{2R} \]
\[ W = \frac{G M m}{2R} \]

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

6. **Final Result**:
Thus, the work done to raise the mass \( m \) to a height \( h \) equal to the radius of the Earth is:
\[ W = \frac{1}{2} g m R \]

### Conclusion:
The work done to raise a mass \( m \) from the surface of the Earth to a height equal to the radius of the Earth is \( \frac{1}{2} g m R \).