The time period of a satellite in a circular orbit around the earth is T . The kinetic energy of the satellite is proportional to `T^(-n)` . Then, n is equal to :

Options

A. `(1)/(2)`
B. `(2)/(3)`
C. `(4)/(3)`
D. `(3)/(2)`

Explanation

To solve the problem, we need to establish the relationship between the time period \( T \) of a satellite in a circular orbit and its kinetic energy.

### Step-by-Step Solution:

1. **Understanding the Forces**:
The gravitational force provides the necessary centripetal force for the satellite's circular motion. The gravitational force \( F_g \) acting on the satellite of mass \( m \) is given by:
\[ F_g = \frac{GMm}{r^2} \]
where \( G \) is the gravitational constant and \( M \) is the mass of the Earth.

2. **Centripetal Force**:
The centripetal force \( F_c \) required to keep the satellite in circular motion is given by:
\[ F_c = \frac{mv^2}{r} \]
Setting the gravitational force equal to the centripetal force, we have:
\[ \frac{mv^2}{r} = \frac{GMm}{r^2} \]

3. **Solving for Orbital Velocity \( v \)**:
By canceling \( m \) from both sides and rearranging, we find:
\[ v^2 = \frac{GM}{r} \quad \Rightarrow \quad v = \sqrt{\frac{GM}{r}} \]

4. **Finding the Time Period \( T \)**:
The time period \( T \) of the satellite is the time taken to complete one full orbit. It can be expressed as:
\[ T = \frac{2\pi r}{v} \]
Substituting the expression for \( v \):
\[ T = \frac{2\pi r}{\sqrt{\frac{GM}{r}}} = 2\pi \sqrt{\frac{r^3}{GM}} \]
This shows that \( T \) is proportional to \( r^{3/2} \):
\[ T \propto r^{3/2} \]

5. **Relating \( r \) and \( T \)**:
From the above relationship, we can express \( r \) in terms of \( T \):
\[ r \propto T^{2/3} \]

6. **Kinetic Energy \( KE \)**:
The kinetic energy \( KE \) of the satellite can be expressed as:
\[ KE = \frac{1}{2} mv^2 \]
Substituting \( v^2 \) from our earlier calculation:
\[ KE = \frac{1}{2} m \left(\frac{GM}{r}\right) = \frac{GMm}{2r} \]
This shows that \( KE \) is inversely proportional to \( r \):
\[ KE \propto \frac{1}{r} \]

7. **Substituting for \( r \)**:
Now substituting \( r \) in terms of \( T \) into the kinetic energy expression:
\[ KE \propto \frac{1}{T^{2/3}} \]
This implies:
\[ KE \propto T^{-2/3} \]

8. **Finding \( n \)**:
According to the problem, the kinetic energy is proportional to \( T^{-n} \). From our derivation, we found that \( KE \propto T^{-2/3} \). Therefore, we can conclude:
\[ n = \frac{2}{3} \]

### Final Answer:
Thus, the value of \( n \) is \( \frac{2}{3} \).

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