Question

A thin circular ring of mass M and radius R is rotating about its axis with a constant angular velocity \(\omega\). Two objects each of mass m are attached gently to the ring on either side. The ring now rotates with an angular velocity:

• A. \(\frac{M}{M + m} \omega\)
• B. \(\frac{M - 2m}{M + 2m} \omega\)
• C. \(\frac{M + 2m}{M - 2m} \omega\)
• D. \(\frac{M}{M + 2m} \omega\)

Hint:

When adding mass to a rotating body without external torque, use angular momentum conservation: \(I_i \omega_i = I_f \omega_f\).

Answer 1

The Correct Option is D

Step 1: Conservation of angular momentum.
No external torque acts, so L_i = L_f.

Step 2: Initial angular momentum.
L_i = I_ring ω = M R² ω.

Step 3: Final moment of inertia.
After attaching two masses at the rim, I_f = M R² + 2 m R² = (M + 2m) R².

Step 4: Equating and solving.
M R² ω = (M + 2m) R² ω_f → ω_f = [M/(M + 2m)] ω.

Step 5: Conclusion.
The new angular velocity is M/(M+2m) times ω.