The moment of inertia of a rod of mass M and | vedclass

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(D) To find the moment of inertia about an axis passing through a point midway between the center and the end,we use the parallel axis theorem: $I = I

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$\frac{7 M L^{2}}{48}$

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(D) To find the moment of inertia about an axis passing through a point midway between the center and the end,we use the parallel axis theorem: $I = I_{CM} + M d^2$.Here,$I_{CM} = \frac{M L^2}{12}$ is the moment of inertia about the center of mass.The distance $d$ between the center of mass and the new axis is $d = \frac{L}{4}$.Substituting these values into the theorem:$I = \frac{M L^2}{12} + M \left( \frac{L}{4} \right)^2$$I = \frac{M L^2}{12} + \frac{M L^2}{16}$Taking the least common multiple $(LCM)$ of $12$ and $16$,which is $48$:$I = \frac{4 M L^2 + 3 M L^2}{48} = \frac{7 M L^2}{48}$.

The moment of inertia of a rod of mass $M$ and length $L$ about an axis passing through a point midway between the center and the end is:

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