A particle 'P' is moving in a circle of radius 'a' with a uniform speed 'u' 'C' is the centre of the circle and AB is a diameter. The angular velocity of P about A and C are in the ratio

Options

A. `1 : 1`
B. `1 : 2`
C. `2 : 1`
D. `4 : 1 `

Explanation

To solve the problem, we need to find the ratio of the angular velocities of a particle \( P \) moving in a circle about two points: point \( A \) (one end of the diameter) and point \( C \) (the center of the circle).

### Step-by-Step Solution:

1. **Understand the Setup**:
- The particle \( P \) is moving in a circle of radius \( a \) with a uniform speed \( u \).
- Point \( C \) is the center of the circle, and \( A \) and \( B \) are the endpoints of the diameter.

2. **Define Angular Velocity**:
- Angular velocity \( \omega \) is defined as the rate of change of angular displacement with respect to time.
- For point \( A \), angular velocity \( \omega_A = \frac{d\theta_1}{dt} \).
- For point \( C \), angular velocity \( \omega_C = \frac{d\theta_2}{dt} \).

3. **Relate Angles**:
- The angle \( \theta_2 \) at the center \( C \) is related to the angle \( \theta_1 \) at point \( A \) by the relationship:
\[ \theta_2 = 2\theta_1 \]
- This is due to the fact that the angle subtended at the center of the circle is twice the angle subtended at any point on the circumference.

4. **Differentiate the Relationship**:
- Taking the derivative of both sides with respect to time gives:
\[ \frac{d\theta_2}{dt} = 2 \frac{d\theta_1}{dt} \]
- This implies:
\[ \omega_C = 2 \omega_A \]

5. **Find the Ratio of Angular Velocities**:
- From the relationship \( \omega_C = 2 \omega_A \), we can express the ratio of the angular velocities:
\[ \frac{\omega_C}{\omega_A} = 2 \]
- Therefore, the ratio of angular velocities \( \omega_C : \omega_A = 2 : 1 \).

### Final Answer:
The ratio of the angular velocities of particle \( P \) about points \( A \) and \( C \) is \( 2:1 \).