Potential energy of a satellite having mass ‘m’ and rotating at a height of `6.4 xx 10^6` m from the earth surface is

Options

A. `-0.5 mgR_e`
B. `-mgR_e`
C. `-2mgR_e`
D. `4mgR_e`

Explanation

To find the potential energy of a satellite of mass 'm' rotating at a height of \(6.4 \times 10^6\) m from the Earth's surface, we can use the formula for gravitational potential energy.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Mass of the satellite, \( m \)
- Height of the satellite from the Earth's surface, \( h = 6.4 \times 10^6 \) m
- Radius of the Earth, \( R \approx 6.4 \times 10^6 \) m (approximately)

2. **Calculate the Distance from the Center of the Earth**:
The distance \( r \) from the center of the Earth to the satellite is given by:
\[ r = R + h = 6.4 \times 10^6 \, \text{m} + 6.4 \times 10^6 \, \text{m} = 12.8 \times 10^6 \, \text{m} \]

3. **Use the Formula for Gravitational Potential Energy**:
The gravitational potential energy \( U \) of a satellite at a distance \( r \) from the center of the Earth is given by:
\[ U = -\frac{G M m}{r} \]
where:
- \( G \) is the gravitational constant, \( G \approx 6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)
- \( M \) is the mass of the Earth, \( M \approx 5.972 \times 10^{24} \, \text{kg} \)

4. **Substituting the Values**:
Substitute \( r = 12.8 \times 10^6 \, \text{m} \) into the potential energy formula:
\[ U = -\frac{(6.674 \times 10^{-11}) (5.972 \times 10^{24}) m}{12.8 \times 10^6} \]

5. **Calculate the Value**:
Calculate the numerical value:
\[ U = -\frac{(6.674 \times 10^{-11}) (5.972 \times 10^{24}) m}{12.8 \times 10^6} \approx -3.14 \times 10^{14} m \, \text{J} \]

### Final Answer:
The potential energy of the satellite is approximately:
\[ U \approx -3.14 \times 10^{14} m \, \text{J} \]