At a height H from the surface of earth, the total energy of a satellite is equal to the potential energy of a body of equal mass at a height 3R from the surface of the earth (R = radius of the earth). The value of H is

Options

A. `R`
B. `(4R)/(3)`
C. `3R`
D. `(R)/(3)`

Explanation

To solve the problem, we need to find the height \( H \) from the surface of the Earth where the total energy of a satellite is equal to the potential energy of a body of equal mass at a height \( 3R \) from the surface of the Earth.

### Step-by-Step Solution:

1. **Understanding Total Energy of a Satellite**:
The total energy \( E \) of a satellite in orbit at a height \( H \) above the Earth's surface can be expressed as:
\[ E = -\frac{GMm}{2(R + H)} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( R \) is the radius of the Earth.

2. **Understanding Potential Energy of a Body**:
The potential energy \( U \) of a body of mass \( m \) at a height \( 3R \) from the surface of the Earth (which means at a distance \( R + 3R = 4R \) from the center of the Earth) is given by:
\[ U = -\frac{GMm}{4R} \]

3. **Setting the Energies Equal**:
According to the problem, the total energy of the satellite is equal to the potential energy of the body:
\[ -\frac{GMm}{2(R + H)} = -\frac{GMm}{4R} \]

4. **Cancelling Common Terms**:
Since \( GMm \) is common on both sides, we can cancel it out (assuming \( m \neq 0 \)):
\[ \frac{1}{2(R + H)} = \frac{1}{4R} \]

5. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ 4R = 2(R + H) \]

6. **Expanding and Rearranging**:
Expanding the right side:
\[ 4R = 2R + 2H \]
Rearranging this equation:
\[ 4R - 2R = 2H \]
\[ 2R = 2H \]

7. **Solving for \( H \)**:
Dividing both sides by 2:
\[ R = H \]

Thus, the value of \( H \) is equal to the radius of the Earth \( R \).

### Final Answer:
\[ H = R \]