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(D) The moment of inertia $I$ of a body about an axis is given by $I = \int r^2 dm$,where $r$ is the perpendicular distance of the mass element $dm$ f

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$HF$

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(D) The moment of inertia $I$ of a body about an axis is given by $I = \int r^2 dm$,where $r$ is the perpendicular distance of the mass element $dm$ from the axis. For a given mass distribution,the moment of inertia is minimum about an axis passing through the center of mass that is closest to the bulk of the mass. In a rectangular plate,the moment of inertia is minimum about the axis passing through the center of mass that is parallel to the longer side,as this axis keeps the mass elements at the smallest average distance $r$. Let the length of the plate be $L$ and width be $W$,where $L > W$. The axis $HF$ passes through the center of mass and is parallel to the longer side $AB$ (or $CD$). The axis $EG$ passes through the center of mass and is parallel to the shorter side $AD$ (or $BC$). Since $HF$ is parallel to the longer side,the mass is distributed closer to this axis compared to $EG$. Therefore,the moment of inertia is minimum about the axis $HF$.

Column $I$	Column $II$
$(A)$ 'k' for a solid sphere rotating about its tangent	$(P)$ $\sqrt{2}R$
$(B)$ 'k' for a ring rotating about its tangent perpendicular to its plane	$(Q)$ $\frac{R}{2}$
$(C)$ 'k' for a uniform solid right circular cone rotating about its central axis	$(R)$ $\frac{\sqrt{7}}{\sqrt{5}}R$
$(D)$ 'k' for a uniform disc rotating about its diameter	$(S)$ $\frac{\sqrt{3}}{\sqrt{10}}R$

We have a rectangular plate of uniform thickness. $E, F, G, H$ are the midpoints of $AB, BC, CD,$ and $AD$ respectively. About which axis will the moment of inertia be minimum?

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