A particle is revolving in a circle of radius R. If the force acting on it is inversely proportional to R, then the time period is proportional to

Options

A. R
B. `R^(2)`
C. `1//R`
D. `1//R^(2)`

Explanation

To solve the problem, we need to analyze the relationship between the force acting on a particle revolving in a circular path and the time period of its revolution.

### Step-by-Step Solution:

1. **Understanding the Force**:
We know that the force acting on the particle is inversely proportional to the radius \( R \). Therefore, we can express this relationship mathematically as:
\[ F = \frac{C}{R} \]
where \( C \) is a constant.

2. **Centripetal Force**:
For a particle of mass \( m \) moving in a circle of radius \( R \) with angular velocity \( \omega \), the centripetal force required to keep the particle in circular motion is given by:
\[ F = m R \omega^2 \]

3. **Equating Forces**:
Since the force \( F \) acting on the particle is also equal to the centripetal force, we can set the two expressions for \( F \) equal to each other:
\[ \frac{C}{R} = m R \omega^2 \]

4. **Rearranging the Equation**:
Rearranging the equation gives us:
\[ C = m R^2 \omega^2 \]

5. **Solving for Angular Velocity**:
From the equation \( C = m R^2 \omega^2 \), we can express \( \omega^2 \) as:
\[ \omega^2 = \frac{C}{m R^2} \]

6. **Finding Angular Velocity**:
Taking the square root gives us:
\[ \omega = \sqrt{\frac{C}{m}} \cdot \frac{1}{R} \]
This shows that \( \omega \) is inversely proportional to \( R \):
\[ \omega \propto \frac{1}{R} \]

7. **Relating Angular Velocity to Time Period**:
The relationship between angular velocity \( \omega \) and the time period \( T \) is given by:
\[ \omega = \frac{2\pi}{T} \]
Therefore, substituting this into our previous relationship gives:
\[ \frac{2\pi}{T} \propto \frac{1}{R} \]

8. **Finding the Time Period**:
Rearranging this relationship leads to:
\[ T \propto R \]
This means that the time period \( T \) is directly proportional to the radius \( R \).

### Conclusion:
The time period \( T \) is proportional to the radius \( R \).