A uniform square plate of side a and mass m i | vedclass

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Question

(A) The moment of inertia of a square plate about an axis passing through its center $O$ and perpendicular to its plane is given by the perpendicular

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$\frac{2}{3}ma^2$

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(A) The moment of inertia of a square plate about an axis passing through its center $O$ and perpendicular to its plane is given by the perpendicular axis theorem:$I_O = I_x + I_y$Since $I_x = I_y = \frac{ma^2}{12}$ for a square plate,$I_O = \frac{ma^2}{12} + \frac{ma^2}{12} = \frac{ma^2}{6}$Now,we use the parallel axis theorem to find the moment of inertia about an axis passing through a corner (e.g.,point $C$):$I = I_O + md^2$Here,the distance $d$ from the center $O$ to the corner $C$ is half the diagonal of the square:$d = \frac{\sqrt{a^2 + a^2}}{2} = \frac{a\sqrt{2}}{2} = \frac{a}{\sqrt{2}}$Substituting the values:$I = \frac{ma^2}{6} + m\left(\frac{a}{\sqrt{2}}\right)^2$$I = \frac{ma^2}{6} + \frac{ma^2}{2} = \frac{ma^2 + 3ma^2}{6} = \frac{4ma^2}{6} = \frac{2}{3}ma^2$

$A$ uniform square plate of side $a$ and mass $m$ is given. The moment of inertia of this plate about an axis perpendicular to the plate and passing through one of its corners is .......

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