$A$ solid cylinder of length $L = 80 \, \text{cm}$ and mass $M$ has a radius $r = 20 \, \text{cm}$. Calculate the density of the material used if the moment of inertia of the cylinder about an axis $CD$ parallel to the central axis $AB$ (as shown in the figure) is $2.7 \, \text{kg m}^2$.

From a circular disc of radius $R$ and mass $9M$,a small disc of radius $\frac{R}{3}$ is removed. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through $O$ is:

The moment of inertia of a uniform circular disc of radius $R$ and mass $M$ about an axis passing through the edge of the disc and normal to the disc is

$A$ thin uniform square lamina of side $a$ is placed in the $xy$-plane with its sides parallel to $X$ and $Y$-axes and with its centre coinciding with the origin. Its moment of inertia about an axis passing through a point on the $Y$-axis at a distance $y = 2a$ and parallel to the $X$-axis is equal to its moment of inertia about an axis passing through a point on the $X$-axis at a distance $x = d$ and perpendicular to the $xy$-plane. Then the value of $d$ is:

Four spheres of diameter $2a$ and mass $M$ are placed with their centres on the four corners of a square of side $b$. The moment of inertia of the system about an axis passing through one of the sides of the square is ..........

$A$ solid sphere of mass $500\,g$ and radius $5\,cm$ is rotated about one of its diameters with an angular speed of $10\,rad\,s^{-1}$. If the moment of inertia of the sphere about its tangent is $x \times 10^{-2}$ times its angular momentum about the diameter,then the value of $x$ will be ..............