MHT CET · Physics · Rotational Motion

A solid cylinder of mass \(M\) and radius \(R\) is rotating about its geometrical axis. A solid sphere of the same mass and same radius is also rotating about its diameter with an angular speed half that of the cylinder. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder will be

A \(1: 4\)
B \(1: 5\)
C \(2: 3\)
D \(3: 2\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(1: 5\)

Step-by-step Solution

Detailed explanation

\(I_{\text {sphere }}=I_S=\frac{2}{5} m R^2 \text { and } I_{\text {cylinder }}=I_c=\frac{1}{2} m R^2\)
Let \(\omega_{\mathrm{S}}=\) Angular speed of sphere,
\(\omega_c=\) Angular speed of cylinder.
It is given that, \(\omega_{\mathrm{s}}=\frac{\omega_{\mathrm{c}}}{2}\)
\(\begin{aligned}
\therefore \quad & \frac{K \cdot E_{\text {sphere }}}{K \cdot E_{\text {cylinder }}}=\frac{I_s \omega_s^2}{I_c^2 \omega_c^2}=\frac{\frac{2}{5} m R^2 \times\left(\frac{\omega_c}{2}\right)^2}{\frac{1}{2} m R^2 \times \omega_c{ }^2} \\
& \frac{K \cdot E_{\text {sphere }}}{K \cdot E_{\text {cylinder }}}=\frac{1}{5}
\end{aligned}\)

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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