A particle moves in a circle with constant angular velocity `omega` about a point P on its circumference. The angular velocity of the particle about the centre C of the circle is

Options

A. `2omega`
B. `(omega)/(2)`
C. `omega`
D. Not constant

Answer

To solve the problem, we need to find the angular velocity of a particle moving in a circle with constant angular velocity \( \omega \) about a point \( P \) on its circumference, with respect to the center \( C \) of the circle.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The particle is moving in a circular path around point \( P \) on the circumference of the circle.
- Let's denote the angle covered by the particle as \( \theta \) when it moves from position \( A \) to position \( B \) around point \( P \).

2. **Relating Angles**:
- According to the properties of circular motion, if the angle covered by the point \( P \) (the center of rotation for the particle) is \( \theta \), then the angle \( \phi \) covered by the particle with respect to the center \( C \) of the circle will be \( 2\theta \).
- This is because the angle subtended at the center of a circle is always twice the angle subtended at any point on the circumference.

3. **Angular Velocity Definition**:
- Angular velocity \( \omega \) is defined as the rate of change of angular displacement with respect to time. Mathematically, it is given by:
\[ \omega = \frac{d\theta}{dt} \]

4. **Calculating Angular Velocity about Center \( C \)**:
- Since the angle \( \phi \) covered by the particle with respect to center \( C \) is \( 2\theta \), we can express the angular velocity with respect to center \( C \) as:
\[ \omega_C = \frac{d\phi}{dt} = \frac{d(2\theta)}{dt} = 2\frac{d\theta}{dt} \]
- Substituting \( \frac{d\theta}{dt} \) with \( \omega \):
\[ \omega_C = 2\omega \]

5. **Conclusion**:
- Therefore, the angular velocity of the particle about the center \( C \) of the circle is \( 2\omega \).

### Final Answer:
The angular velocity of the particle about the center \( C \) of the circle is \( 2\omega \).

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