A particle is moving along a circular path with uniform speed. Through what angle does its angular velocity change when it completes half of the circular path ?

Options

A. `0^@`
B. `45^@`
C. `180^@`
D. `360^@`

Explanation

To solve the problem, we need to understand the motion of a particle moving along a circular path with uniform speed and how its angular velocity behaves during this motion.

### Step-by-Step Solution:

1. **Understanding Circular Motion**:
A particle moving in a circular path with uniform speed means that while the speed (magnitude of velocity) remains constant, the direction of the velocity vector is continuously changing.

2. **Defining Angular Velocity**:
Angular velocity (\( \omega \)) is defined as the rate of change of angular displacement with respect to time. It is directed along the axis of rotation and is always perpendicular to the plane of the circular path.

3. **Initial and Final Position**:
When the particle completes half of the circular path, it moves from one point on the circle (let's say point A) to the point directly opposite (point B).

4. **Direction of Angular Velocity**:
At point A, the angular velocity vector points in a certain direction (let's say out of the plane of the circle). When the particle reaches point B, the angular velocity vector will still be perpendicular to the plane of the circular path but will point in the opposite direction (into the plane of the circle).

5. **Change in Angular Velocity**:
The change in angular velocity is determined by the difference in direction of the angular velocity vector from point A to point B. Since the particle has moved half the circle, the angular velocity vector has changed direction by 180 degrees.

6. **Conclusion**:
Therefore, the angle through which the angular velocity changes when the particle completes half of the circular path is 180 degrees.

### Final Answer:
The angular velocity changes through an angle of 180 degrees when the particle completes half of the circular path.

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