A satellite of mass m is revolving at a height R above theh sruface of Earth. Here R is the radius of the Earth. The gravitational potential energy of this satellite

Options

A. `-mgR`
B. `-(mgR)/2`
C. `-(mgR)/3`
D. `-(mgR)/7`

Answer

To find the gravitational potential energy (U) of a satellite of mass \( m \) revolving at a height \( R \) above the surface of the Earth, where \( R \) is the radius of the Earth, we can follow these steps:

### Step 1: Understand the formula for gravitational potential energy
The gravitational potential energy \( U \) of a mass \( m \) at a distance \( r \) from the center of a planet of mass \( M \) is given by the formula:
\[ U = -\frac{G M m}{r} \]
where \( G \) is the universal gravitational constant.

### Step 2: Determine the distance from the center of the Earth
In this case, the satellite is at a height \( R \) above the Earth's surface. Therefore, the distance \( r \) from the center of the Earth to the satellite is:
\[ r = R + R = 2R \]

### Step 3: Substitute the value of \( r \) into the potential energy formula
Now, substituting \( r = 2R \) into the potential energy formula:
\[ U = -\frac{G M m}{2R} \]

### Step 4: Relate \( G \) and \( g \)
We can relate the gravitational constant \( G \) to the acceleration due to gravity \( g \) at the surface of the Earth. The relationship is given by:
\[ g = \frac{G M}{R^2} \]
From this, we can express \( G M \) as:
\[ G M = g R^2 \]

### Step 5: Substitute \( G M \) into the potential energy formula
Now, substituting \( G M = g R^2 \) into the potential energy expression:
\[ U = -\frac{g R^2 m}{2R} \]

### Step 6: Simplify the expression
Simplifying the expression gives:
\[ U = -\frac{g R m}{2} \]

### Final Answer
Thus, the gravitational potential energy of the satellite is:
\[ U = -\frac{g R m}{2} \]

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