A thin uniform rod of mass \(M\) and length \( L\) is rotating about a perpendicular axis passing through its centre with a constant angular velocity \(\omega\). Two objects each of mass \(\frac{ M }{3}\) are attached gently to the two ends of the rod. The rod will now rotate with an angular velocity of

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.