MHT CET · Physics · Rotational Motion

A thin uniform rod of length 'L' and mass 'M' is bent at the middle point ' 0 ' at an angle of \(45^{\circ}\) as shown in the figure. The moment of inertia of the system about an axis passing through '0' and perpendicular to the plane of the bent rod, is

A \(\frac{\mathrm{ML}^{2}}{12}\)
B \(\frac{M L^{2}}{24}\)
C \(\frac{\mathrm{M} \mathrm{L}^{2}}{3}\)
D \(\frac{\mathrm{M} L^{2}}{6}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{\mathrm{ML}^{2}}{12}\)

Step-by-step Solution

Detailed explanation

Moment of inertia of each half of the rod about the mid point is given by
\(I_{1}=\frac{\frac{M}{2}\left(\frac{L}{2}\right)^{2}}{3}=\frac{M L^{2}}{24}\)
Total moment of inertia \(=\mathrm{I}=2 \mathrm{I}_{1}=\frac{\mathrm{ML}^{2}}{12}\)

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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