MHT CET · Physics · Rotational Motion

Two spheres of equal masses, one of which is a thin spherical shell and the other a solid, have the same moment of inertia about their respective diameters. The ratio of their radii will be

A \(5: 7\)
B \(3: 5\)
C \(\sqrt{3}: \sqrt{5}\)
D \(\sqrt{3}: \sqrt{7}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\sqrt{3}: \sqrt{5}\)

Step-by-step Solution

Detailed explanation

Let the radii of the thin spherical shell and the solid sphere are \(R_{1}\) and \(R_{2}\) respectively. Then the moment of inertia of the spherical shell about their diameter
\(I=\frac{2}{3} M R_{1}^{2}\) ...(i)
and the moment of inertia of the solid sphere is given by
\(I=\frac{2}{5} M R_{2}^{2}\) ...(ii)
Given that the masses and moment of inertia for both the bodies are equal, then from Eqs. (i) and (ii)
\(\frac{2}{3} M R_{1}^{2}=\frac{2}{5} M R_{2}^{2} \Rightarrow \frac{R_{1}^{2}}{R_{2}^{2}}=\frac{3}{5}\)
\(\Rightarrow \frac{R_{1}}{R_{2}}=\sqrt{\frac{3}{5}} \Rightarrow R_{1}: R_{2}=\sqrt{3}: \sqrt{5}\)

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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