An artificial satellite moving in a circular orbit around the earth has a total energy `E_(0)`. Its potential energy is

Options

A. `-E_(0)`
B. `E_(0)`
C. `2E_(0)`
D. `-2E_(0)`

Explanation

To find the potential energy of an artificial satellite moving in a circular orbit around the Earth, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the total energy of the satellite**:
The total energy \( E_0 \) of a satellite in a circular orbit is given by the formula:
\[ E_0 = -\frac{G M m}{2r} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the Earth,
- \( m \) is the mass of the satellite,
- \( r \) is the orbital radius (distance from the center of the Earth to the satellite).

2. **Write the formula for potential energy**:
The gravitational potential energy \( U \) of the satellite in orbit is given by:
\[ U = -\frac{G M m}{r} \]

3. **Relate total energy to potential energy**:
For a satellite in a circular orbit, the total energy \( E_0 \) is related to the potential energy \( U \) by the equation:
\[ E_0 = U + K \]
where \( K \) is the kinetic energy. For circular motion, the kinetic energy can be expressed as:
\[ K = \frac{1}{2} mv^2 \]
and it can be shown that \( K = -\frac{1}{2} U \). Therefore, we can write:
\[ E_0 = U - \frac{1}{2} U = -\frac{1}{2} U \]

4. **Express potential energy in terms of total energy**:
Rearranging the equation \( E_0 = -\frac{1}{2} U \) gives us:
\[ U = -2E_0 \]

5. **Final result**:
Thus, the potential energy \( U \) of the satellite is:
\[ U = -2E_0 \]

### Conclusion:
The potential energy of the artificial satellite is \( U = -2E_0 \).

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