Angular Velocity of P about A and C in a Circular Motion

Angular velocity is defined as the rate of change of angular displacement with respect to time. In a circular motion, the angular velocity of a particle can be determined by considering the relationship between linear velocity, radius, and angular velocity.

1. Definition of Angular Velocity:
Angular velocity (ω) is the rate at which an object rotates or revolves around an axis. It is measured in radians per second (rad/s) and is denoted by ω. The angular velocity is given by the equation:

ω = Δθ/Δt

where Δθ is the change in angular displacement and Δt is the change in time.

2. Particle P in Circular Motion:
In the given scenario, particle P is moving in a circle of radius "r" with a uniform speed v. Let AB be the diameter of the circle, and C be the center. We need to determine the ratio of the angular velocity of P about points A and C.

3. Angular Velocity of P about A:
The angular velocity of P about A can be calculated by considering the relationship between linear velocity, radius, and angular velocity. The linear velocity of P is given by v = ωA * r, where ωA is the angular velocity of P about A.

4. Angular Velocity of P about C:
Similarly, the angular velocity of P about C can be determined by considering the relationship between linear velocity, radius, and angular velocity. The linear velocity of P is given by v = ωC * r, where ωC is the angular velocity of P about C.

5. Ratio of Angular Velocities:
To find the ratio of the angular velocities of P about A and C, we can divide the equation for the angular velocity of P about A by the equation for the angular velocity of P about C:

ωA/ωC = (v/r)/(v/r)
= 1

Therefore, the ratio of the angular velocities of P about A and C is 1.

In conclusion, the angular velocities of P about points A and C in a circular motion are equal. This implies that the particle P takes the same amount of time to complete one revolution around points A and C.

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