The radius of gyration of a solid sphere of radius $\mathrm{r}$ about a certain axis is $\mathrm{r}$. Find the distance of this axis from the centre of the sphere.

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 ___________ .

A solid cube of wood of side $2a$ and mass $M$ is resting on a horizontal surface as shown below.

The cube is free to rotate about the fixed axis $AB$. A bullet of mass $m\left(<<M\right)$ and speed $v$ is shot horizontally at the face opposite to $ABCD$ at a height $h$ above the surface to impart the cube an angular speed ${\omega }_c$, so that the cube just topples over. Then, ${\omega }_c$ is (Note the moment of inertia of the cube about an axis perpendicular to the face and passing through the center of mass is $\frac{2M{a}^3}{3}$)

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

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: Moment of inertia of a circular disc of mass $M$ and radius $R$ about $X,Y$ axes (passing through its plane) and Z-axis which is perpendicular to its plane were found to be $I_{\mathrm{x}},I_{\mathrm{y}}$ and $I_{\mathrm{z}}$, respectively. The respective radii of gyration about all the three axes will be the same.

Reason R: A rigid body making rotational motion has fixed mass and shape. In the light of the above statements, choose the most appropriate answer from the options given below:

$ABC$ 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:

Four equal masses, $m$ each are placed at the corners of a square of length $\left(l\right)$ as shown in the figure. The moment of inertia of the system about an axis passing through $A$ and parallel to $DB$ would be :

A thin metal wire of length $L$ and uniform linear mass density $\rho$ is bent into a circular coil with $O$ as centre. The moment of inertia of a coil about the axis $XX\text{'}$ is

Which one of the following four graphs best depict the variation with $x$ of the moment of inertia $I$ of a uniform triangular lamina about an axis parallel to its base at a distance $x$ from it