A thin uniform bar of length \(L\) and mass \(8\,m\) lies on a smooth horizontal table. Two point masses \(m\) and \(2\,m\) moving in the same horizontal plane from opposite sides of the bar with speeds \(2v\) and \(v\) respectively. The masses stick to the bar after collision at a distance \(\frac{L}{3}\) and \(\frac{L}{6}\) respectively from the centre of the bar. If the bar starts rotating about its center of mass as a result of collision, the angular speed of the bar will be

A \(\frac{v}{{6L}}\)
B \(\frac{6v}{{5L}}\)
C \(\frac{3v}{{5L}}\)
D \(\frac{v}{{5L}}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{v}{{6L}}\)

Step-by-step Solution

Detailed explanation

Linear momentum is conserved \(2 m v-2 m v=(2 m+m+M) V_{c m}\) \(\Rightarrow V_{c m}=0\) Angular momentum is conserve \(2 m v \frac{L}{6}+2 m v \times \frac{L}{3}+0=I w\) \(...(i)\) \(I=\mathrm{M.I.}\) of rod \(+\mathrm{M.I.}\) of mass \(2 \mathrm{m}+\mathrm{M.I}\) of mass…

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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