JEE Mains · Physics · STD 11 - 6. system of particles and rotational motion

A thin circular ring of mass \(M\) and radius \(R\) is rotating with a constant angular velocity \(2 \; rads ^{-1}\) in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass \(m\) be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in \(rads ^{-1}\) )

A \(\frac{M}{(M+m)}\)
B \(\frac{( M +2 m )}{2 M }\)
C \(\frac{2 M }{( M +2 m )}\)
D \(\frac{2( M +2 m )}{ M }\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{2 M }{( M +2 m )}\)

Step-by-step Solution

Detailed explanation

Applying conservation of angular momentum \(M R^{2} \omega=\left(M R^{2}+2 m R^{2}\right) \omega^{\prime}\) \(\omega^{\prime}=\frac{2 M }{ M +2 m }\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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