A solid metal cylinder $A$ and a hollow metal cylinder $B$ have same mass but their radii are in the ratio $2:1$. Then the ratio of their respective moments of inertia about their own axis 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:

The moment of inertia of a uniform horizontal solid cylinder of mass $M$ about an axis passing through its edge and perpendicular to the axis of the cylinder, when its length is $6$ times its radius $R$, is

A uniform solid cylinder with radius $R$ and length $L$ has moment of inertia $I_1$, about the axis of cylinder. A concentric solid cylinder of radius $R^{\text{'}}=\frac{R}{2}$  and length $L^{\text{'}}=\frac{L}{2}$ is carved out of the original cylinder. If $I_2$ is the moment of inertia of the carved out portion of the cylinder then $\frac{I_1}{I_2}=$  __________.

(Both $I_1$ and $I_2$ are about the axis of the cylinder)   

The solid cylinder of length $80 \mathrm{cm}$ and mass $M$ has a radius of $20 \mathrm{cm}.$ Calculate the density of the material used if the moment of inertia of the cylinder about an axis $CD$ parallel to $AB$ as shown in figure is $2.7 \mathrm{kg}{ \mathrm{m}}^2$.

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

Three solid spheres each of mass $m$ and diameter $d$ are stuck together such that the lines connecting the centres form an equilateral triangle of side of length $d$ . The ratio $\frac{I_0}{I_A}$ of moment of inertia $I_0$ of the system about an axis passing the centroid and about center of any of the spheres $I_A$ and perpendicular to the plane of the triangle is: