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 an orbit of radius \( 3R \), we can follow these steps:

### Step 1: Write the expression for gravitational potential energy
The gravitational potential energy \( U \) of a body of mass \( m \) in an orbit of radius \( r \) is given by the formula:
\[ U = -\frac{GMm}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the central body (e.g., a planet), and \( r \) is the radius of the orbit.

### Step 2: Calculate the initial potential energy at radius \( 2R \)
For the initial orbit at radius \( 2R \):
\[ U_i = -\frac{GMm}{2R} \]

### Step 3: Calculate the final potential energy at radius \( 3R \)
For the final orbit at radius \( 3R \):
\[ U_f = -\frac{GMm}{3R} \]

### Step 4: Find the change in potential energy
The energy required to move the body from the initial orbit to the final orbit is the change in potential energy, which is given by:
\[ \Delta U = U_f - U_i \]
Substituting the values from steps 2 and 3:
\[ \Delta U = \left(-\frac{GMm}{3R}\right) - \left(-\frac{GMm}{2R}\right) \]
\[ \Delta U = -\frac{GMm}{3R} + \frac{GMm}{2R} \]

### Step 5: Simplify the expression
To simplify \( \Delta U \), we need a common denominator, which is \( 6R \):
\[ \Delta U = \left(-\frac{2GMm}{6R}\right) + \left(\frac{3GMm}{6R}\right) \]
\[ \Delta U = \frac{3GMm}{6R} - \frac{2GMm}{6R} = \frac{GMm}{6R} \]

### Conclusion
The energy required to move the body from an orbit of radius \( 2R \) to \( 3R \) is:
\[ \Delta U = \frac{GMm}{6R} \]

### Final Answer
Thus, the answer is \( \frac{GMm}{6R} \).

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