A solid cylinder of mass \(2 \mathrm{~kg}\) and radius \(0.2 \mathrm{~m}\) is rotating about its own axis without friction with angular velocity \(5 \mathrm{~rad} \mathrm{s}^{-1}\). A particle of mass \(1 \mathrm{~kg}\) moving with a velocity of \(5 \mathrm{~ms}^{-1}\) strikes the cylinder and sticks to it as shown in figure.

The angular velocity of the system after the particle sticks to it will be

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.