Energy required to move a body of mass m from an orbit of radius 2R to 3R is

Options

A. `(GMm)/(12R^2)`
B. `(GMm)/(3R^2)`
C. `(GMm)/(8R)`
D. `(GMm)/(6R)`

Explanation

To find the energy required to move a body of mass \( m \) from an orbit of radius \( 2R \) to \( 3R \), we need to calculate the change in gravitational potential energy between these two orbits.

### Step-by-Step Solution:

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

2. **Calculate Potential Energy at Radius \( 2R \)**:
For the orbit at radius \( 2R \):
\[ U_1 = -\frac{G M m}{2R} \]

3. **Calculate Potential Energy at Radius \( 3R \)**:
For the orbit at radius \( 3R \):
\[ U_2 = -\frac{G M m}{3R} \]

4. **Calculate the Change in Potential Energy**:
The energy required to move the body from radius \( 2R \) to \( 3R \) is the change in potential energy, which can be calculated as:
\[ \Delta U = U_2 - U_1 \]
Substituting the values we found:
\[ \Delta U = \left(-\frac{G M m}{3R}\right) - \left(-\frac{G M m}{2R}\right) \]
This simplifies to:
\[ \Delta U = -\frac{G M m}{3R} + \frac{G M m}{2R} \]

5. **Finding a Common Denominator**:
The common denominator for \( 3R \) and \( 2R \) is \( 6R \). Rewriting the terms:
\[ \Delta U = \left(-\frac{2G M m}{6R} + \frac{3G M m}{6R}\right) \]
This simplifies to:
\[ \Delta U = \frac{(3 - 2)G M m}{6R} = \frac{G M m}{6R} \]

6. **Final Answer**:
Therefore, the energy required to move the body from an orbit of radius \( 2R \) to \( 3R \) is:
\[ \Delta U = \frac{G M m}{6R} \]