Define moment of inertia. Explain the physical significance of it.

$ABC$ is a plane lamina of the shape of an equilateral triangle. $D, E$ are mid-points of $AB, AC$ and $G$ is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through $G$ and perpendicular to the plane $ABC$ is $I_0$. If part $ADE$ is removed, the moment of inertia of the remaining part about the same axis is $\frac{N{I}_0}{16}$ where $N$ is an integer. Value of $N$ is:

An massless equilateral triangle EFG of side 'a' (As shown in figure) has three particles of mass $\mathrm{m}$ situated at its vertices. The moment of inertia of the system about the line EX perpendicular to EG in the plane of EFG is $\frac{\mathrm{N}}{20}{\mathrm{ma}}^2$ where $\mathrm{N}$ is an integer. The value of $\mathrm{N}$ is ___________ .

Three identical spherical shells, each of mass $m$ and radius $r$ are placed as shown in the figure. Consider an axis $X{X}^{\mathit{\prime }}$ which is touching the two shells and passing through the diameter of the third shell. The moment of inertia of the system consisting of these three spherical shells about $X{X}^{\mathit{\prime }}$ axis is:

A thin uniform rectangular plate of mass $2 \mathrm{kg}$ is placed in $X-Y$ plane as shown in the figure. The moment of inertia about $x$-axis is $I_x=0.2 \mathrm{kg} {\mathrm{m}}^2$ and the moment of inertia about $y$ - axis is $I_y=0.3 \mathrm{kg} {\mathrm{m}}^2$. The radius of gyration of the plate about the axis passing through $O$ and perpendicular to the plane of the plate is

A thin ring has a mass $0.25 kg$ and a radius $0.5 m$. Its moment of inertia about an axis passing through its centre and perpendicular to its plane is ____.