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 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:
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When adding mass to a rotating body without external torque, use angular momentum conservation: \(I_i \omega_i = I_f \omega_f\).
Updated On: Jun 19, 2026
- \(\frac{M}{M + m} \omega\)
- \(\frac{M - 2m}{M + 2m} \omega\)
- \(\frac{M + 2m}{M - 2m} \omega\)
- \(\frac{M}{M + 2m} \omega\)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the problem.
The ring is rotating about its axis with angular velocity \(\omega\). Two small masses \(m\) are attached gently. No external torque acts on the system. This is a conservation of angular momentum problem.
Step 2: Initial angular momentum.
\[ L_i = I_\text{ring} \omega = M R^2 \omega \] (Thin ring moment of inertia about axis: \(I = MR^2\))
Step 3: Final moment of inertia.
After attaching two masses at the periphery: \[ I_f = I_\text{ring} + 2 m R^2 = MR^2 + 2 m R^2 = (M + 2m) R^2 \]
Step 4: Conservation of angular momentum.
\[ L_i = L_f \Rightarrow I_i \omega = I_f \omega_f \] \[ MR^2 \omega = (M + 2m) R^2 \omega_f \] \[ \omega_f = \frac{M}{M + 2m} \omega \]
Step 5: Conclusion.
The ring rotates with angular velocity \(\frac{M}{M + 2m} \omega\).
The ring is rotating about its axis with angular velocity \(\omega\). Two small masses \(m\) are attached gently. No external torque acts on the system. This is a conservation of angular momentum problem.
Step 2: Initial angular momentum.
\[ L_i = I_\text{ring} \omega = M R^2 \omega \] (Thin ring moment of inertia about axis: \(I = MR^2\))
Step 3: Final moment of inertia.
After attaching two masses at the periphery: \[ I_f = I_\text{ring} + 2 m R^2 = MR^2 + 2 m R^2 = (M + 2m) R^2 \]
Step 4: Conservation of angular momentum.
\[ L_i = L_f \Rightarrow I_i \omega = I_f \omega_f \] \[ MR^2 \omega = (M + 2m) R^2 \omega_f \] \[ \omega_f = \frac{M}{M + 2m} \omega \]
Step 5: Conclusion.
The ring rotates with angular velocity \(\frac{M}{M + 2m} \omega\).
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