MHT CET · Physics · Rotational Motion

A solid sphere of mass ' \(M\) ' and radius ' \(R\) ' is rotating about its diameter. A solid cylinder of same mass and radius is also rotating about its geometrical \(1: 3\) axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation ( \(\mathrm{K}_{\text {sphere }}\) to \(\mathrm{K}_{\text {cylinder }}\) ) will be

A \(1: 8\)
B \(1: 6\)
C \(1: 3\)
D \(1: 5\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(1: 5\)

Step-by-step Solution

Detailed explanation

Kinetic energy of sphere is given by:
\(E_{\text {sphere }}=\frac{1}{2} I_{\text {sphere }} \omega_{\text {sphere }}^2\)
\(E_{\text {sphere }}=\frac{1}{2} \times \frac{2}{5} m R^2 \omega_{\text {sphere }}^2\)
Kinetic energy of cylinder is given by:
\(\begin{aligned} & E_{\text {cylinder }}=\frac{1}{2} I_{\text {cylinder }} \omega_{\text {cylinder }}^2 \\ & E_{\text {cylinder }}=\frac{1}{2} \times \frac{1}{2} m R^2 \omega_{\text {cylinder }}^2 \\ & \text { Given, } \omega_{\text {cylinder }}=2 \omega_{\text {sphere }} \\ & \frac{E_{\text {sphere }}}{E_{\text {cylinder }}}=\frac{1}{5}\end{aligned}\)

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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