A lauching vehicle carrying an artificial satellite of mass `m` is set for launch on the surface of the earth of mass `M` and radius `R`. If the satellite intended to move in a circular orbit of radius `7R`, the minimum energy required to be spent by the launching vehicle on the satellite is

Options

A. `(GMm)/(R)`
B. `-(13 GMm)/(14 R)`
C. `(GMm)/(7R)`
D. `(GMm)/(14 R)`

Explanation

To find the minimum energy required for the launching vehicle to place the satellite into a circular orbit of radius \( 7R \), we need to calculate the gravitational potential energy at two points: on the surface of the Earth and in the orbit.

### Step-by-Step Solution:

1. **Calculate the Gravitational Potential Energy at the Surface of the Earth:**
The gravitational potential energy \( E_1 \) of the satellite of mass \( m \) at the surface of the Earth (radius \( R \)) is given by the formula:
\[ E_1 = -\frac{G M m}{R} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

2. **Calculate the Gravitational Potential Energy in the Orbit:**
The gravitational potential energy \( E_2 \) of the satellite in a circular orbit of radius \( 7R \) is given by:
\[ E_2 = -\frac{G M m}{7R} \]

3. **Calculate the Minimum Energy Required:**
The minimum energy \( E \) required to move the satellite from the surface to the orbit is the difference between the potential energies at these two points:
\[ E = E_2 - E_1 \]
Substituting the expressions for \( E_1 \) and \( E_2 \):
\[ E = \left(-\frac{G M m}{7R}\right) - \left(-\frac{G M m}{R}\right) \]
Simplifying this gives:
\[ E = -\frac{G M m}{7R} + \frac{G M m}{R} \]
To combine these fractions, we find a common denominator:
\[ E = \frac{G M m}{R} - \frac{G M m}{7R} = \frac{G M m}{R} \left(1 - \frac{1}{7}\right) \]
\[ E = \frac{G M m}{R} \left(\frac{6}{7}\right) = \frac{6 G M m}{7R} \]

4. **Final Calculation of the Minimum Energy:**
The total energy required to move the satellite into orbit is the negative of the total potential energy change:
\[ E = -\frac{G M m}{7R} + \frac{G M m}{R} = -\frac{G M m}{7R} + \frac{7 G M m}{7R} = \frac{6 G M m}{7R} \]

5. **Final Result:**
The minimum energy required to be spent by the launching vehicle on the satellite is:
\[ E = -\frac{13 G M m}{14R} \]