Rotational Motion Part 2

Let the length of side of square plate is l. The diagram of square plate having an axis passing from centre as $X-X'$ and an axis passing through one of the corner and perpendicular to plane of square plate $Z-Z'$ is shown below. The distance between the centre of square plate and the corner of square plate is $d=\sqrt{\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}}=\frac{1}{\sqrt{2}}$ We know that, moment of inertia of square plate about centre, $I_{XX'}=\frac{MI^{2}}{6}$ By parallel axis theorem, moment of inertia of square plate about axis $Z-Z'$ will be   $I_{ZZ'}=I_{XX'}+M\left(\frac{1}{\sqrt{2}}\right)^{2}$ $I_{ZZ'}=\frac{MI^{2}}{6}+\frac{MI^{2}}{2}$ $=MI^{2}\left[\frac{1}{6}+\frac{1}{2}\right]$ $I_{ZZ'}=\frac{2MI^{2}}{3}$ Thus, the moment of inertia about an axis passing through corner of square plate and perpendicular to plane of plate is $\frac{2(MI)^{2}}{3}.$