Energy required in moving a body of mass `m` from a distance 2R to 3R from centre of earth of mass M is

Options

A. `(GMm)/(12 R^(2))`
B. `(GMm)/(2R^(2))`
C. `(GMm)/(8R)`
D. `(GMm)/(6R)`

Explanation

To find the energy required to move a body of mass \( m \) from a distance \( 2R \) to \( 3R \) from the center of the Earth (mass \( M \)), we can use the concept of gravitational potential energy.

### Step-by-Step Solution:

1. **Understanding Gravitational Potential Energy**:
The gravitational potential energy \( U \) of a mass \( m \) at a distance \( r \) from the center of the Earth is given by the formula:
\[ U = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

2. **Calculate Potential Energy at Distance \( 2R \)**:
Using the formula for gravitational potential energy, we can find the potential energy when the body is at a distance \( 2R \):
\[ U_{2R} = -\frac{GMm}{2R} \]

3. **Calculate Potential Energy at Distance \( 3R \)**:
Similarly, we calculate the potential energy when the body is at a distance \( 3R \):
\[ U_{3R} = -\frac{GMm}{3R} \]

4. **Calculate the Work Done (Energy Required)**:
The work done (or energy required) to move the body from \( 2R \) to \( 3R \) is the difference in potential energy:
\[ W = U_{3R} - U_{2R} \]
Substituting the values we calculated:
\[ W = \left(-\frac{GMm}{3R}\right) - \left(-\frac{GMm}{2R}\right) \]
\[ W = -\frac{GMm}{3R} + \frac{GMm}{2R} \]

5. **Finding a Common Denominator**:
To simplify the expression, we need a common denominator, which is \( 6R \):
\[ W = \left(-\frac{2GMm}{6R}\right) + \left(\frac{3GMm}{6R}\right) \]
\[ W = \frac{GMm}{6R} \]

6. **Final Result**:
Thus, the energy required to move the body from a distance \( 2R \) to \( 3R \) from the center of the Earth is:
\[ W = \frac{GMm}{6R} \]