A thin uniform rod of length L and mass M is | vedclass

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(A) The rod is bent at the middle point $O$ into two segments,each of length $l = \frac{L}{2}$ and mass $m = \frac{M}{2}$.Each segment acts as a rod o

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$\frac{ML^{2}}{12}$

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(A) The rod is bent at the middle point $O$ into two segments,each of length $l = \frac{L}{2}$ and mass $m = \frac{M}{2}$.Each segment acts as a rod of length $l$ rotating about one of its ends.The moment of inertia of a uniform rod of mass $m$ and length $l$ about an axis passing through one of its ends and perpendicular to its length is given by $I = \frac{ml^2}{3}$.For each segment,$m = \frac{M}{2}$ and $l = \frac{L}{2}$.Therefore,the moment of inertia of one segment about point $O$ is:$I_1 = \frac{(\frac{M}{2})(\frac{L}{2})^2}{3} = \frac{(\frac{M}{2})(\frac{L^2}{4})}{3} = \frac{ML^2}{24}$.Since the system consists of two such segments,the total moment of inertia $I$ about the axis passing through $O$ and perpendicular to the plane is:$I = I_1 + I_1 = 2 	imes \frac{ML^2}{24} = \frac{ML^2}{12}$.

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

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