A mass m is raised from a distance 2 R from the surface of the earth to 3R. Work done to do so against gravity will-

Options

A. `(mgR)/10`
B. `(mgR)/11`
C. `(mgR)/12`
D. `(mgR)/14`

Answer

To solve the problem of calculating the work done in raising a mass \( m \) from a distance \( 2R \) from the surface of the Earth to \( 3R \), we will follow these steps:

### Step 1: Understand the Initial and Final Positions
- The initial position (A) of the mass \( m \) is at a distance of \( 2R \) from the Earth's surface.
- The final position (B) of the mass \( m \) is at a distance of \( 3R \) from the Earth's surface.
- The distance from the center of the Earth to point A is \( R + 2R = 3R \).
- The distance from the center of the Earth to point B is \( R + 3R = 4R \).

### Step 2: Calculate the Potential Energy at Both Positions
- The gravitational potential energy \( U \) at a distance \( r \) from the center of the Earth is given by:
\[ U = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( m \) is the mass of the object.

- **Potential Energy at Point A (Initial Position)**:
\[ U_A = -\frac{GMm}{3R} \]

- **Potential Energy at Point B (Final Position)**:
\[ U_B = -\frac{GMm}{4R} \]

### Step 3: Calculate the Work Done
- The work done \( W \) against gravity is the change in potential energy:
\[ W = U_B - U_A \]

- Substituting the values of \( U_A \) and \( U_B \):
\[ W = \left(-\frac{GMm}{4R}\right) - \left(-\frac{GMm}{3R}\right) \]
\[ W = -\frac{GMm}{4R} + \frac{GMm}{3R} \]

### Step 4: Simplify the Expression
- To combine the fractions, we need a common denominator, which is \( 12R \):
\[ W = \left(-\frac{3GMm}{12R} + \frac{4GMm}{12R}\right) \]
\[ W = \frac{(4 - 3)GMm}{12R} = \frac{GMm}{12R} \]

### Step 5: Substitute \( GM \) with \( gR^2 \)
- We know that \( g = \frac{GM}{R^2} \), therefore \( GM = gR^2 \).
- Substitute \( GM \) into the work done equation:
\[ W = \frac{gR^2 m}{12R} = \frac{g m R}{12} \]

### Final Answer
The work done to raise the mass \( m \) from a distance \( 2R \) to \( 3R \) against gravity is:
\[ W = \frac{g m R}{12} \]