A particle is confined to rotate in a circular path decreasing linear speed, then which of the following is correct?

Options

A. `vecL` (angular momentum) is conserved about the centre
B. Only direction of angular momentum `vecL` is conserved
C. It spirals towards the centre
D. Its acceleration is towards the centre

Explanation

To solve the problem of a particle confined to rotate in a circular path with decreasing linear speed, we need to analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The particle is moving in a circular path, which means it has a tangential velocity (v) and is subject to centripetal acceleration directed towards the center of the circle.

2. **Identifying the Variables**:
- Let \( r \) be the radius of the circular path.
- The angular velocity \( \omega \) is related to the linear speed by the equation \( v = \omega r \).
- Since the linear speed is decreasing, we can conclude that \( \omega \) must also be decreasing.

3. **Angular Momentum Calculation**:
- The angular momentum \( L \) of the particle about the center is given by \( L = r \times mv \).
- As both \( v \) and \( \omega \) are decreasing, we need to determine if \( L \) is conserved.
- Since \( v \) is decreasing, \( L \) will also decrease, indicating that angular momentum is not conserved.

4. **Direction of Angular Momentum**:
- The direction of angular momentum is given by the right-hand rule. As long as the radius \( r \) and the direction of \( v \) remain constant, the direction of \( L \) will remain constant even if its magnitude decreases.

5. **Spiraling Towards the Center**:
- The statement that the particle spirals towards the center is incorrect. Although both \( v \) and \( \omega \) are decreasing, they do so in a way that keeps the radius \( r \) constant. Therefore, the particle does not spiral inward.

6. **Acceleration Analysis**:
- The particle experiences centripetal acceleration directed towards the center, but since the speed is decreasing, there is also a tangential acceleration acting in the opposite direction of motion.
- The net acceleration is not directed solely towards the center; it is the vector sum of centripetal acceleration and tangential acceleration.

### Conclusion:
From the analysis, we conclude that:
- The correct statement is that **only the direction of angular momentum is conserved**.