The binding of a satellite of mass m in a orbit of radius r is

Options

A. `(mgR^(2))/(r)`
B. `(mgR^(2))/(2r)`
C. `(-mgR^(3))/(r)`
D. `(- m gR^(3))/(2 r)`

Explanation

To find the binding energy of a satellite of mass \( m \) in an orbit of radius \( r \), we can follow these steps:

### Step 1: Understand Binding Energy
Binding energy is defined as the amount of energy required to free a satellite from its gravitational influence. When the satellite is free from gravitational influence, its total energy becomes zero.

### Step 2: Calculate Total Energy of the Satellite
The total energy \( E \) of a satellite in orbit is given by the formula:
\[ E = -\frac{G M m}{2r} \]
where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet,
- \( m \) is the mass of the satellite,
- \( r \) is the radius of the orbit.

### Step 3: Relate Binding Energy to Total Energy
Since the binding energy \( BE \) is the energy required to make the total energy zero, we can express it as:
\[ BE = -E \]
Substituting the expression for total energy:
\[ BE = -\left(-\frac{G M m}{2r}\right) = \frac{G M m}{2r} \]

### Step 4: Express Binding Energy in Terms of \( g \)
We know that the acceleration due to gravity \( g \) at the surface of the planet is given by:
\[ g = \frac{G M}{r^2} \]
To express binding energy in terms of \( g \), we can rearrange this equation to find \( G M \):
\[ G M = g r^2 \]
Now substituting this into the binding energy equation:
\[ BE = \frac{g r^2 m}{2r} \]
This simplifies to:
\[ BE = \frac{g m r}{2} \]

### Final Expression
Thus, the binding energy of the satellite is:
\[ BE = \frac{g m r}{2} \]

### Summary
The binding energy of a satellite of mass \( m \) in an orbit of radius \( r \) is given by:
\[ BE = \frac{g m r}{2} \]