A particle P is moving in a circle of radius 'a' with a uniform speed v. C is the centre of the circle and AB is a diameter. When passing through B 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`

Answer

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: A (one end of the diameter) and C (the center of the circle).

### Step-by-Step Solution:

1. **Identify the Parameters**:
- Let the radius of the circle be \( a \).
- The diameter \( AB \) is \( 2a \).
- The speed of the particle \( P \) is \( v \).

2. **Determine the Angular Velocity about Point C**:
- The angular velocity \( \omega_C \) about the center \( C \) can be calculated using the formula:
\[ \omega_C = \frac{v}{r} \]
- Here, \( r \) is the radius of the circle, which is \( a \).
- Therefore, we have:
\[ \omega_C = \frac{v}{a} \]

3. **Determine the Angular Velocity about Point A**:
- The distance from point \( A \) to point \( P \) when \( P \) is at point \( B \) is the diameter of the circle, which is \( 2a \).
- The angular velocity \( \omega_A \) about point \( A \) is given by:
\[ \omega_A = \frac{v}{R} \]
- Here, \( R \) is the distance from \( A \) to \( P \), which is \( 2a \).
- Thus, we have:
\[ \omega_A = \frac{v}{2a} \]

4. **Calculate the Ratio of Angular Velocities**:
- Now, we can find the ratio of the angular velocities \( \frac{\omega_A}{\omega_C} \):
\[ \frac{\omega_A}{\omega_C} = \frac{\frac{v}{2a}}{\frac{v}{a}} = \frac{v}{2a} \cdot \frac{a}{v} = \frac{1}{2} \]

5. **Final Result**:
- Therefore, the ratio of the angular velocities of particle \( P \) about points \( A \) and \( C \) is:
\[ \omega_A : \omega_C = 1 : 2 \]

### Conclusion:
The answer is \( 1 : 2 \).

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