The $\mathrm{M}.\mathrm{I}.$ of solid sphere about an axis passing through its centre is $2\mathrm{kg}-{\mathrm{m}}^2$. Calculate its $\mathrm{M}.\mathrm{I}.$ about a tangent passing through any point on its surface.

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 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 $\frac{2M{a}^3}{3}$)

Two identical spherical balls of mass $M$ and radius $R$ each are stuck on two ends of a rod of length $2R$ and mass $M$(see figure). The moment of inertia of the system about the axis passing perpendicularly through the centre of the rod 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:

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

A mass $m$ is supported by a massless string wound around a uniform hollow cylinder of mass m and radius R. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?