MHT CET · Physics · Rotational Motion

The moment of inertia of a uniform square plate about an axis perpendicular to its plane and passing through the centre is \(\frac{\mathrm{Ma}^2}{6}\), where ' \(\mathrm{M}\) ' is the mass and ' \(a\) ' is the side of square plate. Moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

A \(\frac{\mathrm{Ma}^2}{6}\)
B \(\frac{2 \mathrm{Ma}^2}{3}\)
C \(\frac{\mathrm{Ma}^2}{3}\)
D \(\frac{2 \mathrm{Ma}^2}{5}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{2 \mathrm{Ma}^2}{3}\)

Step-by-step Solution

Detailed explanation

\(\begin{aligned} \mathrm{I}_0 & =\frac{\mathrm{Ma}^2}{6} \\ \mathrm{AB} & =\sqrt{2 \mathrm{a}^2}=\sqrt{2} \mathrm{a} \\ \therefore \quad \mathrm{AO} & =\frac{\mathrm{a}}{\sqrt{2}}=\mathrm{h} \\ \therefore \quad \mathrm{I}_{\mathrm{A}} & =\mathrm{I}_0+\mathrm{Mh}^2 \\ & =\frac{\mathrm{Ma}^2}{6}+\frac{\mathrm{Ma}^2}{2} \\ & =\frac{8 \mathrm{Ma}^2}{12}=\frac{2}{3} \mathrm{Ma}^2\end{aligned}\)

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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