A sayellite of mass m revolves in a circular orbit of radius R a round a planet of mass M. Its total energy E is :-

Options

A. `- (GMm)/(2R)`
B. `+ (GMm)/(3R)`
C. `- (GMm)/R`
D. `+ (GMm)/R`

Answer

To find the total energy \( E \) of a satellite of mass \( m \) revolving in a circular orbit of radius \( R \) around a planet of mass \( M \), we need to consider both the gravitational potential energy and the kinetic energy of the satellite.

### Step-by-Step Solution:

1. **Identify the Potential Energy**:
The gravitational potential energy \( U \) of the satellite in orbit is given by the formula:
\[ U = -\frac{G M m}{R} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the planet, \( m \) is the mass of the satellite, and \( R \) is the distance from the center of the planet to the satellite.

2. **Identify the Kinetic Energy**:
The kinetic energy \( K \) of the satellite can be expressed as:
\[ K = \frac{1}{2} m v^2 \]
where \( v \) is the orbital speed of the satellite.

3. **Relate Kinetic Energy to Gravitational Force**:
For a satellite in circular motion, the gravitational force provides the necessary centripetal force. Thus, we can set up the equation:
\[ \frac{m v^2}{R} = \frac{G M m}{R^2} \]
Here, the left side is the centripetal force, and the right side is the gravitational force. We can simplify this by canceling \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{v^2}{R} = \frac{G M}{R^2} \]
Rearranging gives:
\[ v^2 = \frac{G M}{R} \]

4. **Substitute \( v^2 \) into the Kinetic Energy Equation**:
Now, substituting \( v^2 \) back into the kinetic energy formula:
\[ K = \frac{1}{2} m \left(\frac{G M}{R}\right) = \frac{G M m}{2R} \]

5. **Calculate Total Energy**:
The total energy \( E \) of the satellite is the sum of its kinetic and potential energy:
\[ E = K + U \]
Substituting the expressions we derived:
\[ E = \frac{G M m}{2R} - \frac{G M m}{R} \]
Combining these terms:
\[ E = \frac{G M m}{2R} - \frac{2G M m}{2R} = -\frac{G M m}{2R} \]

6. **Final Result**:
Thus, the total energy \( E \) of the satellite is:
\[ E = -\frac{G M m}{2R} \]