Let I be the moment of inertia of a uniform s | vedclass

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(A) For a uniform square plate,the moment of inertia about any axis passing through its center and lying in its plane is constant,provided the axis is

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$I$

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(A) For a uniform square plate,the moment of inertia about any axis passing through its center and lying in its plane is constant,provided the axis is in the plane of the plate.Let $I_{AB}$ be the moment of inertia about axis $AB$ (parallel to sides),so $I_{AB} = I$.Let $I_{EF}$ be the moment of inertia about axis $EF$ (also parallel to sides and perpendicular to $AB$). By symmetry,$I_{EF} = I_{AB} = I$.According to the perpendicular axis theorem,the moment of inertia about an axis perpendicular to the plane and passing through the center $(I_z)$ is $I_z = I_{AB} + I_{EF} = I + I = 2I$.Now,consider any axis $CD$ in the plane of the square passing through the center. Let its moment of inertia be $I_{CD}$. Let $IJ$ be an axis in the plane,perpendicular to $CD$ and passing through the center. By symmetry,$I_{IJ} = I_{CD}$.Applying the perpendicular axis theorem again: $I_z = I_{CD} + I_{IJ} = 2I_{CD}$.Since $I_z = 2I$,we have $2I_{CD} = 2I$,which implies $I_{CD} = I$.Thus,the moment of inertia is independent of the angle $	heta$ and remains $I$.

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