Physics - Rotation Question with Solution | TestHub

For a uniform square plate, the moment of inertia about any axis passing through its center and lying in its plane is the same due to symmetry.

Therefore, the moments of inertia about the given axes are equal:
${\mathrm{I}}_1={\mathrm{I}}_2={\mathrm{I}}_3={\mathrm{I}}_4=\mathrm{I}$

According to the perpendicular axis theorem, the moment of inertia about an axis passing through the center and perpendicular to the plane of the plate (${\mathrm{I}}_z$) is equal to the sum of the moments of inertia about any two mutually perpendicular axes in the plane of the plate.

Taking axes 3 and 4 (which are mutually perpendicular):
${\mathrm{I}}_z={\mathrm{I}}_3+{\mathrm{I}}_4=\mathrm{I}+\mathrm{I}=2\mathrm{I}$

Now, checking the given options, we look for a sum that equals $2\mathrm{I}$:
${\mathrm{I}}_1+{\mathrm{I}}_3=\mathrm{I}+\mathrm{I}=2\mathrm{I}$

Thus, the moment of inertia about the perpendicular axis is ${\mathrm{I}}_1+{\mathrm{I}}_3$.

Answer: ${\mathrm{I}}_1+{\mathrm{I}}_3$