If the force acting on the body moving in uniform circular motion is inversely proportional to `r^3`, then the time period of its revolution is proportional to ?

Answer

To solve the problem, we need to analyze the relationship between the force acting on a body in uniform circular motion and the time period of its revolution when the force is inversely proportional to \( r^3 \).

### Step-by-Step Solution:

1. **Understanding Uniform Circular Motion**:
- In uniform circular motion, the speed of the body is constant, but its direction changes continuously. The only acceleration present is centripetal acceleration, which is directed towards the center of the circular path.

2. **Centripetal Force**:
- The centripetal force \( F_c \) required to keep a body of mass \( m \) moving in a circle of radius \( r \) with speed \( v \) is given by:
\[ F_c = \frac{mv^2}{r} \]

3. **Relationship Between Speed and Angular Velocity**:
- The linear speed \( v \) can be expressed in terms of angular velocity \( \omega \):
\[ v = \omega r \]
- Substituting this into the centripetal force equation gives:
\[ F_c = \frac{m(\omega r)^2}{r} = m\omega^2 r \]

4. **Given Condition**:
- According to the problem, the force acting on the body is inversely proportional to \( r^3 \):
\[ F_c \propto \frac{1}{r^3} \]
- We can express this as:
\[ F_c = \frac{c}{r^3} \]
- where \( c \) is a constant of proportionality.

5. **Equating the Forces**:
- From the centripetal force equation, we have:
\[ m\omega^2 r = \frac{c}{r^3} \]
- Rearranging gives:
\[ \omega^2 = \frac{c}{m} \cdot \frac{1}{r^4} \]

6. **Finding the Time Period**:
- The angular velocity \( \omega \) is related to the time period \( T \) by:
\[ \omega = \frac{2\pi}{T} \]
- Substituting this into the equation for \( \omega^2 \):
\[ \left(\frac{2\pi}{T}\right)^2 = \frac{c}{m} \cdot \frac{1}{r^4} \]
- Rearranging gives:
\[ T^2 = \frac{4\pi^2 m}{c} \cdot r^4 \]

7. **Proportionality of Time Period**:
- From the equation \( T^2 \propto r^4 \), we can deduce:
\[ T \propto r^2 \]

### Conclusion:
The time period \( T \) of the revolution is proportional to \( r^2 \).