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

Options

A. `-2E`
B. 2E
C. `(2E)/(3)`
D. `-(2E)/(3)`

Explanation

To find the potential energy of an artificial satellite moving in a circular orbit around the Earth in terms of its total energy \( E_0 \), we can follow these steps:

### Step 1: Understand the relationship between total energy and potential energy
The total energy \( E \) of a satellite in a circular orbit is given by the sum of its kinetic energy \( K \) and potential energy \( U \):
\[ E = K + U \]

### Step 2: Write the expressions for kinetic and potential energy
The gravitational potential energy \( U \) of the satellite is given by:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, \( m \) is the mass of the satellite, and \( r \) is the distance from the center of the Earth to the satellite.

The kinetic energy \( K \) of the satellite in a circular orbit can be expressed as:
\[ K = \frac{1}{2} m v^2 \]
Using the formula for centripetal force, we know that the gravitational force provides the necessary centripetal force:
\[ \frac{G M m}{r^2} = \frac{m v^2}{r} \]
From this, we can derive the expression for \( v^2 \):
\[ v^2 = \frac{G M}{r} \]
Substituting this back into the kinetic energy equation gives:
\[ K = \frac{1}{2} m \left(\frac{G M}{r}\right) = \frac{G M m}{2r} \]

### Step 3: Substitute the expressions into the total energy equation
Now, substituting \( K \) and \( U \) into the total energy equation:
\[ E = K + U = \frac{G M m}{2r} - \frac{G M m}{r} \]
Combining these terms gives:
\[ E = \frac{G M m}{2r} - \frac{2G M m}{2r} = -\frac{G M m}{2r} \]

### Step 4: Relate total energy to potential energy
From the total energy expression, we can see that:
\[ E = -\frac{G M m}{2r} \]
This means that the potential energy \( U \) can be expressed in terms of the total energy \( E_0 \):
\[ U = -\frac{G M m}{r} \]
Since we have \( E_0 = -\frac{G M m}{2r} \), we can relate \( U \) to \( E_0 \):
\[ U = -2E_0 \]

### Final Result
Thus, the potential energy \( U \) of the satellite in terms of the total energy \( E_0 \) is:
\[ U = -2E_0 \]