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. `mgR`
B. `2 mgR`
C. `(1)/(2) m gR`
D. `(3)/(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 1: Understand the situation
We are raising a mass \( m \) from the surface of the Earth to a height \( h = R \). The initial position is at the surface of the Earth, and the final position is at a height equal to the radius of the Earth.

### Step 2: Identify the gravitational potential energy formula
The gravitational potential energy \( U \) at a distance \( r \) from the center of the Earth is given by:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( m \) is the mass being raised.

### Step 3: Calculate the initial potential energy
At the surface of the Earth (where \( r = R \)):
\[ U_i = -\frac{G M m}{R} \]

### Step 4: Calculate the final potential energy
At a height \( h = R \) (where the distance from the center of the Earth is \( r = 2R \)):
\[ U_f = -\frac{G M m}{2R} \]

### Step 5: Calculate the change in potential energy
The work done \( W \) in raising the mass is equal to the change in potential energy:
\[ W = U_f - U_i \]
Substituting the values we found:
\[ W = \left(-\frac{G M m}{2R}\right) - \left(-\frac{G M m}{R}\right) \]
\[ W = -\frac{G M m}{2R} + \frac{G M m}{R} \]
\[ W = \frac{G M m}{R} - \frac{G M m}{2R} \]
\[ W = \frac{G M m}{2R} \]

### Step 6: Express \( G M \) in terms of \( g \)
We know that the acceleration due to gravity \( g \) at the surface of the Earth is given by:
\[ g = \frac{G M}{R^2} \]
From this, we can express \( G M \) as:
\[ G M = g R^2 \]

### Step 7: Substitute \( G M \) into the work done formula
Substituting \( G M \) into our work done equation:
\[ W = \frac{g R^2 m}{2R} \]
\[ W = \frac{g m R}{2} \]

### Conclusion
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:
\[ W = \frac{m g R}{2} \]

### Final Answer
The correct option is \( \frac{m g R}{2} \).

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