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(D) For a thin uniform square sheet of side $a$ and mass $m$:$1$. The moment of inertia about an axis passing through the center and perpendicular to

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$I = \frac{ma^2}{12}$

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(D) For a thin uniform square sheet of side $a$ and mass $m$:$1$. The moment of inertia about an axis passing through the center and perpendicular to the plane of the sheet is $I_z = \frac{ma^2}{6}$.$2$. By the perpendicular axis theorem,$I_z = I_x + I_y$,where $I_x$ and $I_y$ are the moments of inertia about the axes passing through the center and parallel to the sides.$3$. Due to symmetry,$I_x = I_y = \frac{ma^2}{12}$.$4$. Let $I_d$ be the moment of inertia about a diagonal. By the perpendicular axis theorem in the plane of the sheet,if we consider two perpendicular axes along the diagonals,their sum must equal $I_z$.$5$. Since the square is symmetric about both diagonals,the moment of inertia about each diagonal is the same,say $I_d$.$6$. Thus,$I_d + I_d = I_z$,which means $2I_d = \frac{ma^2}{6}$.$7$. Therefore,$I_d = \frac{ma^2}{12}$.

Consider a thin uniform square sheet made of a rigid material. If its side is $a$,mass is $m$,and moment of inertia about one of its diagonals is $I$,then:

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