A particle is moving along a circular path. The angular velocity, linear velocity, Angular acceleration, and centripetal acceleration of the particle at any instant. Respectively are `vec omega, vec v, vec prop, and vec a_c`. Which of the following relations is not correct ?

Options

A. `vec omega _|_ vec v`
B. `vec omega _|_ vec prop`
C. `vec omega _|_ vec a_c`
D. `vec v _|_ vec a_c`.

Explanation

To solve the problem, we need to analyze the relationships between the angular velocity (ω), linear velocity (v), angular acceleration (α), and centripetal acceleration (a_c) of a particle moving along a circular path.

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- **Angular Velocity (ω)**: This is a vector quantity that represents the rate of rotation of the particle around the center of the circular path. It is directed along the axis of rotation.
- **Linear Velocity (v)**: This is the tangential velocity of the particle at any point on the circular path. It is always directed tangent to the path.
- **Angular Acceleration (α)**: This represents the rate of change of angular velocity. It can be in the same direction as ω (if ω is increasing) or in the opposite direction (if ω is decreasing).
- **Centripetal Acceleration (a_c)**: This is the acceleration directed towards the center of the circular path, responsible for keeping the particle in circular motion.

2. **Analyzing the Relationships**:
- **ω is perpendicular to v**:
- Since v is tangential to the circular path and ω is directed along the axis of rotation (perpendicular to the plane of motion), this relationship is correct.
- **ω is perpendicular to α**:
- If ω is increasing, α is in the same direction as ω. If ω is decreasing, α is in the opposite direction. Therefore, ω and α are not necessarily perpendicular. This relationship is incorrect.
- **ω is perpendicular to a_c**:
- Since a_c is directed towards the center of the circular path and ω is directed along the axis of rotation, they are perpendicular. This relationship is correct.
- **v is perpendicular to a_c**:
- The linear velocity v is tangential to the circular path while a_c is directed towards the center. Hence, they are also perpendicular. This relationship is correct.

3. **Conclusion**:
- The only incorrect relationship among the given options is **ω is perpendicular to α**.

### Final Answer:
The relation that is not correct is **ω is perpendicular to α**.