A satellite of mass `m` is orbiting the earth (of radius `R`) at a height `h` from its surface. The total energy of the satellite in terms of `g_(0)`, the value of acceleration due to gravity at the earth's surface,

Options

A. `(2mg_(0)R^(2))/(R+h)`
B. `-(2mg_(0)R^(2))/(R+h)`
C. `(mg_(0)R^(2))/(2(R+h))`
D. `-(mg_(0)R^(2))/(2(R+h))`

Explanation

To find the total energy of a satellite of mass `m` orbiting the Earth at a height `h` from its surface in terms of `g₀`, the acceleration due to gravity at the Earth's surface, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Radius of Orbit**:
The radius of the orbit `r` of the satellite is the sum of the Earth's radius `R` and the height `h` above the Earth's surface:
\[ r = R + h \]

2. **Total Energy of the Satellite**:
The total mechanical energy `E` of a satellite in orbit is given by the formula:
\[ E = -\frac{G M m}{2r} \]
where:
- `G` is the gravitational constant,
- `M` is the mass of the Earth,
- `m` is the mass of the satellite,
- `r` is the distance from the center of the Earth to the satellite.

3. **Relate Gravitational Constant to g₀**:
The acceleration due to gravity at the surface of the Earth `g₀` is given by:
\[ g₀ = \frac{G M}{R^2} \]
We can express `G M` in terms of `g₀`:
\[ G M = g₀ R^2 \]

4. **Substitute G M in Total Energy Formula**:
Substitute `G M` in the total energy formula:
\[ E = -\frac{g₀ R^2 m}{2r} \]

5. **Substitute r**:
Now substitute `r = R + h` into the equation:
\[ E = -\frac{g₀ R^2 m}{2(R + h)} \]

6. **Final Expression**:
Thus, the total energy of the satellite in terms of `g₀` is:
\[ E = -\frac{g₀ R^2 m}{2(R + h)} \]

### Final Answer:
The total energy of the satellite in terms of `g₀` is:
\[ E = -\frac{g₀ R^2 m}{2(R + h)} \]