MHT CET · Physics · Rotational Motion

A solid metallic sphere of radius ' \(R\) ' having moment of inertia ' I ' about diameter is melted and recast into a solid disc of radius ' \(r\) ' of a uniform thickness. The moment of inertia of a disc about an axis passing through its edge and perpendicular to its plane is also equal to ' I '. The ratio \(\frac{r}{R}\) is

A \(\frac{1}{\sqrt{2}}\)
B \(\frac{2}{\sqrt{5}}\)
C \(\frac{2}{\sqrt{10}}\)
D \(\frac{2}{\sqrt{15}}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{2}{\sqrt{15}}\)

Step-by-step Solution

Detailed explanation

M.I. of the solid sphere about a diameter
\(\mathrm{I}=\frac{2}{5} \mathrm{MR}^2...(i)\)
M.I. of the disc about an axis through its edge and perpendicular to its plane is
\(\begin{array}{ll}
& I=\frac{3}{2} \mathrm{Mr}^2 ...(ii)\\
\therefore \quad & \frac{2}{5} \mathrm{MR}^2=\frac{3}{2} \mathrm{Mr}^2 \\
\therefore \quad & \mathrm{r}=\frac{2}{\sqrt{15}} \mathrm{R}
\end{array}\)
...[From(i) and (ii)]

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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