The work done to increases the radius of orbit of a satellite of mass 'm' revolving around a planet of mass `M` from orbit of radius `R` into another orbit of radius `3R` is

Options

A. `(2GMm)/(3R)`
B. `(GMm)/(3R)`
C. `(GMm)/(6R)`
D. `(GMMm)/(24R)`

Explanation

To solve the problem of finding the work done to increase the radius of orbit of a satellite from radius \( R \) to \( 3R \), we can follow these steps:

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

### Step 2: Calculate the Total Energy at Radius \( R \)
Using the formula for total energy, we can calculate the energy when the satellite is at radius \( R \):
\[ E(R) = -\frac{GMm}{2R} \]

### Step 3: Calculate the Total Energy at Radius \( 3R \)
Next, we calculate the total energy when the satellite is at radius \( 3R \):
\[ E(3R) = -\frac{GMm}{2(3R)} = -\frac{GMm}{6R} \]

### Step 4: Find the Change in Total Energy
The work done \( W \) to move the satellite from radius \( R \) to radius \( 3R \) is equal to the change in total energy:
\[ W = E(3R) - E(R) \]
Substituting the values we calculated:
\[ W = \left(-\frac{GMm}{6R}\right) - \left(-\frac{GMm}{2R}\right) \]
This simplifies to:
\[ W = -\frac{GMm}{6R} + \frac{GMm}{2R} \]

### Step 5: Simplify the Expression
To combine the terms, we need a common denominator:
\[ W = \frac{GMm}{2R} - \frac{GMm}{6R} = \frac{GMm \cdot 3}{6R} - \frac{GMm}{6R} = \frac{(3 - 1)GMm}{6R} = \frac{2GMm}{6R} \]
Thus, we have:
\[ W = \frac{GMm}{3R} \]

### Final Answer
The work done to increase the radius of orbit of the satellite from \( R \) to \( 3R \) is:
\[ W = \frac{GMm}{3R} \]

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