Question

A thin circular ring of mass 'M' and radius 'r' is rotating about its axis with a constant angular velocity \(\omega\). Four objects, each of mass 'M / 2', are kept gently to the opposite ends of two perpendicular diameters of ring. The angular velocity of the ring will be

• A. \(\omega /3\)
• B. \(\omega /2\)
• C. \(3\omega\)
• D. \(\omega\)

Hint:

No external torque acts, so angular momentum is conserved while the moment of inertia increases.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
No external torque acts on the system, so angular momentum \(I\omega\) is conserved.

Step 2: Calculation
Initial moment of inertia of the ring: \(I_{1}=Mr^{2}\). Four masses of \(\frac{M}{2}\) each on the rim add \(4\times\frac{M}{2}\times r^{2}=2Mr^{2}\). So \(I_{2}=3Mr^{2}\).

Step 3: Apply conservation
\(Mr^{2}\omega=3Mr^{2}\omega'\), so \(\omega'=\frac{\omega}{3}\).

Final Answer:
The ring's angular velocity becomes one third of the original. \[ \boxed{\text{(A) }\dfrac{\omega}{3}} \]