For a satellite orbiting very close to earth's surface, total energy is

Options

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

Explanation

To find the total energy of a satellite orbiting very close to the Earth's surface, we need to consider both its kinetic energy (KE) and potential energy (PE).

### Step-by-Step Solution:

1. **Identify the Kinetic Energy (KE)**:
The kinetic energy of a satellite in orbit is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the satellite and \( v \) is its orbital velocity.

2. **Determine the Orbital Velocity (v)**:
For a satellite orbiting close to the Earth's surface, the orbital velocity can be derived from the gravitational force acting as the centripetal force. The gravitational force is given by:
\[ F = \frac{GMm}{R^2} \]
where \( G \) is the universal gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth. This force provides the necessary centripetal force:
\[ F = \frac{mv^2}{R} \]
Setting these equal gives:
\[ \frac{GMm}{R^2} = \frac{mv^2}{R} \]
Simplifying this equation, we find:
\[ v^2 = \frac{GM}{R} \]

3. **Substituting for Kinetic Energy**:
Now substituting \( v^2 \) back into the kinetic energy formula:
\[ KE = \frac{1}{2} m \left(\frac{GM}{R}\right) = \frac{GMm}{2R} \]

4. **Identify the Potential Energy (PE)**:
The gravitational potential energy of the satellite is given by:
\[ PE = -\frac{GMm}{R} \]

5. **Calculate Total Energy (E)**:
The total energy \( E \) of the satellite is the sum of its kinetic and potential energies:
\[ E = KE + PE \]
Substituting the expressions for KE and PE:
\[ E = \frac{GMm}{2R} - \frac{GMm}{R} \]
This simplifies to:
\[ E = \frac{GMm}{2R} - \frac{2GMm}{2R} = -\frac{GMm}{2R} \]

6. **Conclusion**:
Thus, the total energy of a satellite orbiting very close to the Earth's surface is:
\[ E = -\frac{GMm}{2R} \]