The moment of inertia of a thin square plate | vedclass

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Question

(D) Let $I$ be the moment of inertia of the square plate about an axis passing through the centre '$O$' and perpendicular to the plane of the plate.Ac

Vedclass · Accepted Answer

$I_1+I_3$

Vedclass · Answer

(D) Let $I$ be the moment of inertia of the square plate about an axis passing through the centre '$O$' and perpendicular to the plane of the plate.According to the perpendicular axis theorem,for any two mutually perpendicular axes in the plane of the plate intersecting at the centre,the sum of the moments of inertia about these axes equals the moment of inertia about the axis perpendicular to the plane.For axes $1$ and $2$ (diagonals),$I = I_1 + I_2$.For axes $3$ and $4$ (mid-lines),$I = I_3 + I_4$.Since the plate is a square,by symmetry,$I_1 = I_2$ and $I_3 = I_4$.Thus,$I = 2I_1$ and $I = 2I_3$,which implies $I_1 = I_3$.Therefore,the moment of inertia about the perpendicular axis is $I = I_1 + I_3$ (or $I_2 + I_4$,or $I_1 + I_4$,etc.).Comparing with the given options,the correct expression is $I_1 + I_3$.

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