Three thin rods,each of mass M and length L,a | vedclass

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(C) Let the three rods be $1$,$2$,and $3$ placed along the $X$,$Y$,and $Z$ axes respectively.$1$. The moment of inertia of rod $1$ (along $X$ axis) ab

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$\frac{2 ML^2}{3}$

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(C) Let the three rods be $1$,$2$,and $3$ placed along the $X$,$Y$,and $Z$ axes respectively.$1$. The moment of inertia of rod $1$ (along $X$ axis) about the $Z$ axis: Since the rod lies along the $X$ axis,its distance from the $Z$ axis varies from $0$ to $L$. The moment of inertia of a rod of mass $M$ and length $L$ about an axis passing through one end and perpendicular to the rod is $I = \frac{ML^2}{3}$. Thus,$I_1 = \frac{ML^2}{3}$.$2$. The moment of inertia of rod $2$ (along $Y$ axis) about the $Z$ axis: Similarly,the rod lies along the $Y$ axis,and the $Z$ axis is perpendicular to it at the origin. Thus,$I_2 = \frac{ML^2}{3}$.$3$. The moment of inertia of rod $3$ (along $Z$ axis) about the $Z$ axis: The rod lies along the $Z$ axis itself. Therefore,every mass element of the rod is at a distance of $0$ from the $Z$ axis. Thus,$I_3 = 0$.Total moment of inertia $I = I_1 + I_2 + I_3 = \frac{ML^2}{3} + \frac{ML^2}{3} + 0 = \frac{2 ML^2}{3}$.

Three thin rods,each of mass $M$ and length $L$,are placed along the $X$,$Y$,and $Z$ axes,which are mutually perpendicular. One end of each rod is at the origin. The moment of inertia of the system about the $Z$ axis is:

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