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

Options

A. `1:2`
B. `2:1`
C. `1:3`
D. `3:1`

Answer

To solve the problem, we need to find the ratio of the angular velocities of the particle P about points A and C when it is moving in a circle of radius r with a uniform speed u.

### Step-by-Step Solution:

1. **Understanding Angular Velocity**:
Angular velocity (ω) is defined as the rate of change of angular displacement and can be calculated using the formula:
\[ \omega = \frac{v}{r} \]
where \( v \) is the linear speed and \( r \) is the radius of the circular path.

2. **Angular Velocity about Point A**:
- The distance from point A to the particle P when it is at the position V (on the circumference) is equal to the radius \( r \) of the circle.
- Therefore, the angular velocity of P about point A (ω_A) can be calculated as:
\[ \omega_A = \frac{u}{r} \]

3. **Angular Velocity about Point C**:
- The distance from point C (the center of the circle) to the particle P is also the radius \( r \) of the circle.
- Thus, the angular velocity of P about point C (ω_C) is given by:
\[ \omega_C = \frac{u}{r} \]

4. **Finding the Ratio of Angular Velocities**:
- Now, we need to find the ratio of the angular velocities about points A and C:
\[ \text{Ratio} = \frac{\omega_A}{\omega_C} = \frac{\frac{u}{r}}{\frac{u}{r}} = 1 \]

5. **Conclusion**:
- The ratio of the angular velocities of the particle P about points A and C is:
\[ \text{Ratio} = 1:1 \]