The energy required to move a body of mass`m` from an orbit of radius `3R` to `4R` is

Options

A. `(GMm)/(2R)`
B. `(GMm)/(6R)`
C. `(GMm)/(12R)`
D. `(GMm)/(24R)`

Explanation

To solve the problem of finding the energy required to move a body of mass `m` from an orbit of radius `3R` to `4R`, we will follow these steps:

### Step 1: Understand the Total Energy in Orbit
The total energy \( E \) of a body in a circular orbit of radius \( R \) is given by the formula:
\[ E = -\frac{GMm}{2R} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the central body, and \( m \) is the mass of the orbiting body.

### Step 2: Calculate Total Energy at Radius \( 3R \)
Using the formula for total energy, we calculate the energy at the radius \( 3R \):
\[ E_{3R} = -\frac{GMm}{2 \cdot 3R} = -\frac{GMm}{6R} \]

### Step 3: Calculate Total Energy at Radius \( 4R \)
Next, we calculate the energy at the radius \( 4R \):
\[ E_{4R} = -\frac{GMm}{2 \cdot 4R} = -\frac{GMm}{8R} \]

### Step 4: Find the Change in Energy
The energy required to move the body from the orbit of radius \( 3R \) to \( 4R \) is the difference in total energy:
\[ \Delta E = E_{4R} - E_{3R} \]
Substituting the values we calculated:
\[ \Delta E = \left(-\frac{GMm}{8R}\right) - \left(-\frac{GMm}{6R}\right) \]
This simplifies to:
\[ \Delta E = -\frac{GMm}{8R} + \frac{GMm}{6R} \]

### Step 5: Simplify the Expression
To combine the fractions, we need a common denominator, which is \( 24R \):
\[ \Delta E = \left(-\frac{3GMm}{24R} + \frac{4GMm}{24R}\right) \]
This results in:
\[ \Delta E = \frac{1GMm}{24R} \]

### Final Answer
Thus, the energy required to move the body from an orbit of radius \( 3R \) to \( 4R \) is:
\[ \Delta E = \frac{GMm}{24R} \]