A particle P is moving with uniform speed in a circular path . C is the centre of circle and AB is a diameter of circle. Let `omega_(A)` be the angular speed is particle about C and A. Then value of `omega_(C)//omega_(A)` is

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to find the ratio of the angular speeds \( \omega_C \) and \( \omega_A \) of a particle moving along a circular path. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the setup
The particle \( P \) is moving with uniform speed in a circular path with center \( C \) and a diameter \( AB \). The angular speeds \( \omega_C \) and \( \omega_A \) refer to the rates of change of the angle \( \theta \) at points \( C \) and \( A \) respectively.

### Step 2: Define angular speeds
- **Angular speed about point \( C \)**: This is defined as the rate of change of the angle \( \theta \) at the center \( C \). Mathematically, it can be expressed as:
\[ \omega_C = \frac{d(2\theta)}{dt} \]
since the angle subtended at the center \( C \) is twice the angle subtended at the circumference.

- **Angular speed about point \( A \)**: This is defined as the rate of change of the angle \( \theta \) at point \( A \). Mathematically, it is expressed as:
\[ \omega_A = \frac{d\theta}{dt} \]

### Step 3: Calculate the ratio \( \frac{\omega_C}{\omega_A} \)
Now, we can find the ratio of the angular speeds:
\[ \frac{\omega_C}{\omega_A} = \frac{\frac{d(2\theta)}{dt}}{\frac{d\theta}{dt}} \]
This simplifies to:
\[ \frac{\omega_C}{\omega_A} = \frac{2 \frac{d\theta}{dt}}{\frac{d\theta}{dt}} = 2 \]

### Conclusion
Thus, the value of the ratio \( \frac{\omega_C}{\omega_A} \) is:
\[ \frac{\omega_C}{\omega_A} = 2 \]