Question

Two particles A and B are moving in uniform circular motion in concentric circles of radii \(r_{A}\) and \(r_{B}\) with speeds \(v_{A}\) and \(v_{B}\) respectively. \(T_{A}\) and \(T_{B}\) are the time periods of rotation of the particles respectively and their relation is \(T_{A}=3T_{B}\). The ratio of angular speed of B to that of A will be

• A. \(r_{B}:3r_{A}\)
• B. \(3r_{B}:r_{A}\)
• C. \(1:3\)
• D. \(3:1\)

Hint:

Angular speed is inversely proportional to the time period, so the ratio of angular speeds is the inverse ratio of periods.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Angular speed of a particle in uniform circular motion is \(\omega=\dfrac{2\pi}{T}\).

Step 2: Ratio
\(\dfrac{\omega_B}{\omega_A}=\dfrac{2\pi/T_B}{2\pi/T_A}=\dfrac{T_A}{T_B}\)

Step 3: Substitute
Given \(T_A=3T_B\), so \(\dfrac{\omega_B}{\omega_A}=3\).
The ratio is \(3:1\).

Final Answer:
The ratio of angular speed of B to A is 3 to 1. \[ \boxed{\text{(D) }3:1} \]