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 find 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 1: Understand the Initial and Final Potential Energy
The gravitational potential energy \( U \) at a distance \( r \) from the center of the Earth is given by the formula:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( m \) is the mass being raised.

### Step 2: Calculate Initial Potential Energy \( U_i \)
At the surface of the Earth, where \( r = R \):
\[ U_i = -\frac{G M m}{R} \]

### Step 3: Calculate Final Potential Energy \( U_f \)
When the mass is raised to a height \( h = R \), the distance from the center of the Earth becomes \( r = R + R = 2R \):
\[ U_f = -\frac{G M m}{2R} \]

### Step 4: Calculate the Work Done \( W \)
The work done in raising the mass is equal to the change in potential energy:
\[ W = U_f - U_i \]
Substituting the values we found:
\[ W = \left(-\frac{G M m}{2R}\right) - \left(-\frac{G M m}{R}\right) \]
This simplifies to:
\[ W = -\frac{G M m}{2R} + \frac{G M m}{R} \]
\[ W = \frac{G M m}{R} - \frac{G M m}{2R} = \frac{G M m}{2R} \]

### Step 5: Relate \( G M \) to \( g \)
We know that the acceleration due to gravity \( g \) at the surface of the Earth is given by:
\[ g = \frac{G M}{R^2} \]
Thus, we can express \( G M \) as:
\[ G M = g R^2 \]

### Step 6: Substitute \( G M \) into the Work Done Equation
Now substituting \( G M \) into the work done equation:
\[ W = \frac{g R^2 m}{2R} \]
This simplifies to:
\[ W = \frac{1}{2} g m R \]

### Final Answer
Thus, the work done to raise the mass \( m \) from the surface of the Earth to a height equal to the radius of the Earth is:
\[ W = \frac{1}{2} g m R \]