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 the centre and lying in the plane of the plate is the same,provide

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$I$

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(A) For a uniform square plate,the moment of inertia about any axis passing through the centre and lying in the plane of the plate is the same,provided the axis is symmetric with respect to the square's geometry.Let $I_Z$ be the moment of inertia about an axis passing through the centre and perpendicular to the plane of the square.According to the perpendicular axis theorem,$I_Z = I_{AB} + I_{A'B'}$,where $AB$ and $A'B'$ are two mutually perpendicular axes in the plane of the plate.Since the square is symmetric,$I_{AB} = I_{A'B'} = I$. Thus,$I_Z = 2I$.Similarly,for any two mutually perpendicular axes $CD$ and $C'D'$ in the plane of the plate passing through the centre,$I_Z = I_{CD} + I_{C'D'}$.Due to the rotational symmetry of the square,the moment of inertia about any axis in the plane passing through the centre is constant and equal to $I$. Therefore,$I_{CD} = I$.

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