A particle performing uniform circular motion has angular momentum `L`. If its angular frequency is double and its kinetic energy halved, then the new angular momentum is :

Options

A. 2L
B. 4L
C. `(L)/(2)`
D. `(L)/(4)`

Explanation

To solve the problem, we need to analyze the relationships between angular momentum, kinetic energy, and angular frequency for a particle in uniform circular motion.

### Step-by-Step Solution:

1. **Understand the Relationships**:
- The angular momentum \( L \) of a particle is given by the formula:
\[ L = I \omega \]
where \( I \) is the moment of inertia and \( \omega \) is the angular frequency.
- The kinetic energy \( K \) of a rotating object is given by:
\[ K = \frac{1}{2} I \omega^2 \]

2. **Express Kinetic Energy in Terms of Angular Momentum**:
- We can express kinetic energy in terms of angular momentum:
\[ K = \frac{1}{2} I \omega^2 = \frac{1}{2} \left( \frac{L}{\omega} \right) \omega^2 = \frac{L \omega}{2} \]
- Rearranging gives:
\[ L = 2K / \omega \]

3. **Initial Conditions**:
- Let the initial angular frequency be \( \omega \) and the initial kinetic energy be \( K \).
- Thus, the initial angular momentum is:
\[ L = 2K / \omega \]

4. **New Conditions**:
- The problem states that the angular frequency is doubled:
\[ \omega' = 2\omega \]
- The kinetic energy is halved:
\[ K' = \frac{K}{2} \]

5. **Calculate New Angular Momentum**:
- Using the new kinetic energy and new angular frequency, we can find the new angular momentum \( L' \):
\[ L' = \frac{2K'}{\omega'} = \frac{2 \left( \frac{K}{2} \right)}{2\omega} \]
- Simplifying this gives:
\[ L' = \frac{K}{\omega} \]

6. **Relate New Angular Momentum to Initial Angular Momentum**:
- We know from our earlier expression that:
\[ L = \frac{2K}{\omega} \]
- Therefore, we can express \( K/\omega \) in terms of \( L \):
\[ L' = \frac{K}{\omega} = \frac{L}{4} \]

7. **Final Result**:
- The new angular momentum \( L' \) is:
\[ L' = \frac{L}{4} \]

### Conclusion:
The new angular momentum is \( \frac{L}{4} \).