A satellite of mass m moves around the Earth in a circular orbit with speed v. The potential energy of the satellite is

Options

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

Answer

To find the potential energy of a satellite of mass \( m \) moving in a circular orbit around the Earth, we can follow these steps:

### Step 1: Understand the Formula for Gravitational Potential Energy
The gravitational potential energy \( U \) of a two-body system (like the Earth and the satellite) is given by the formula:
\[ U = -\frac{G M m}{r} \]
where:
- \( G \) is the universal 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.

### Step 2: Determine the Distance \( r \)
In this scenario, since the satellite is assumed to be very close to the Earth, we can take \( r \) to be equal to the radius of the Earth \( R \). Therefore, we can replace \( r \) with \( R \):
\[ U = -\frac{G M m}{R} \]

### Step 3: Relate Gravitational Constant to Orbital Velocity
The orbital velocity \( v \) of a satellite in a circular orbit can be expressed as:
\[ v = \sqrt{\frac{G M}{R}} \]
Squaring both sides gives:
\[ v^2 = \frac{G M}{R} \]
From this, we can express \( G M \) in terms of \( v^2 \):
\[ G M = v^2 R \]

### Step 4: Substitute \( G M \) into the Potential Energy Formula
Now, we can substitute \( G M \) back into the potential energy formula:
\[ U = -\frac{G M m}{R} = -\frac{v^2 R m}{R} \]
This simplifies to:
\[ U = -v^2 m \]

### Conclusion
Thus, the potential energy of the satellite is:
\[ U = -v^2 m \]