Question

A particle p is moving in a circle of radius r with a uniform speed v, C is the centre of the circle and AB is the diameter. The angular velocity of p about A and C is in the ratio:

• A. \(1:1\)
• B. \(1:2\)
• C. \(2:1\)
• D. \(4:1\)

Hint:

Angular velocity depends on the reference point. For points on diameter, distance becomes \(2r\), reducing angular velocity.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Angular velocity \( \omega \) is the rate of change of angle subtended at a point.
A geometric property of circles states that the angle subtended by an arc at the center is double the angle subtended by it at any point on the circumference.
: Key Formula or Approach:
1. Angular velocity about center \( C \): \( \omega_C = \frac{v}{r} \).
2. Geometric relation: \( \theta_C = 2 \theta_A \).
Step 2: Detailed Explanation:
Let the particle move along a small arc \( ds \).
The angle change at center \( C \) is \( d\theta_C = \frac{ds}{r} \).
The angle change at point \( A \) (on the circumference) for the same arc \( ds \) is \( d\theta_A = \frac{d\theta_C}{2} = \frac{ds}{2r} \).
Calculating angular velocities:
\[ \omega_C = \frac{d\theta_C}{dt} = \frac{v}{r} \]
\[ \omega_A = \frac{d\theta_A}{dt} = \frac{1}{2} \frac{d\theta_C}{dt} = \frac{v}{2r} \]
Taking the ratio \( \omega_A : \omega_C \):
\[ \frac{\omega_A}{\omega_C} = \frac{v/2r}{v/r} = \frac{1}{2} \]
Step 3: Final Answer:
The ratio of angular velocity about A and C is 1:2.