A thin uniform rod has mass M and length L. T | vedclass

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(C) To find the moment of inertia about an axis passing through a point at a distance $\frac{L}{3}$ from one end,we use the parallel axis theorem: $I

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$\frac{ML^{2}}{9}$

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(C) To find the moment of inertia about an axis passing through a point at a distance $\frac{L}{3}$ from one end,we use the parallel axis theorem: $I = I_{cm} + Mh^2$.Here,$I_{cm}$ is the moment of inertia about the center of mass,which is $\frac{ML^2}{12}$.The distance $h$ between the center of mass (at $\frac{L}{2}$ from the end) and the given axis (at $\frac{L}{3}$ from the end) is $h = |\frac{L}{2} - \frac{L}{3}| = \frac{L}{6}$.Substituting these values into the theorem:$I = \frac{ML^2}{12} + M(\frac{L}{6})^2$$I = \frac{ML^2}{12} + \frac{ML^2}{36}$$I = \frac{3ML^2 + ML^2}{36} = \frac{4ML^2}{36} = \frac{ML^2}{9}$.

$A$ thin uniform rod has mass $M$ and length $L$. The moment of inertia about an axis perpendicular to it and passing through the point at a distance $\frac{L}{3}$ from one of its ends,will be

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