Suppose the force of gravitation is inversely proportional to the cube of the radius of circular orbit in which satellite is revolving then its time period is proportional to

Options

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

Answer

To solve the problem, we need to analyze the relationship between the gravitational force acting on a satellite in a circular orbit and its time period when the gravitational force is inversely proportional to the cube of the radius of the orbit.

### Step-by-Step Solution:

1. **Understanding the Forces**:
- The gravitational force \( F_g \) acting on the satellite provides the necessary centripetal force \( F_c \) for circular motion.
- We can express this relationship as:
\[ F_g = F_c \]

2. **Expressing Gravitational Force**:
- According to the problem, the gravitational force is inversely proportional to the cube of the radius \( r \):
\[ F_g \propto \frac{1}{r^3} \]
- We can express this as:
\[ F_g = \frac{k}{r^3} \]
where \( k \) is a constant of proportionality.

3. **Centripetal Force Expression**:
- The centripetal force can be expressed in terms of the satellite's mass \( m \) and angular velocity \( \omega \):
\[ F_c = m \omega^2 r \]

4. **Setting the Forces Equal**:
- Since \( F_g = F_c \), we have:
\[ \frac{k}{r^3} = m \omega^2 r \]

5. **Rearranging the Equation**:
- Rearranging gives:
\[ \omega^2 = \frac{k}{m r^4} \]

6. **Relating Angular Velocity to Time Period**:
- We know that angular velocity \( \omega \) is related to the time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]
- Therefore, we can substitute \( \omega \) in the equation:
\[ \left(\frac{2\pi}{T}\right)^2 = \frac{k}{m r^4} \]

7. **Solving for Time Period \( T \)**:
- Rearranging this equation gives:
\[ T^2 = \frac{(2\pi)^2 m r^4}{k} \]
- Taking the square root:
\[ T = 2\pi \sqrt{\frac{m}{k}} r^2 \]

8. **Determining the Proportionality**:
- From the final expression, we can see that the time period \( T \) is proportional to the square of the radius \( r \):
\[ T \propto r^2 \]

### Conclusion:
Thus, the time period \( T \) of the satellite is proportional to the square of the radius of its orbit.