$A$ 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 $2.7 \, \text{kg m}^2$.

What is the moment of inertia of a square sheet of side $l$ and mass per unit area $\mu$ about an axis passing through the centre and perpendicular to its plane?

$A$ solid sphere has mass $M$ and radius $R$. Its moment of inertia about a parallel axis passing through a point at a distance $R/3$ from its centre is:

Identical rings are arranged in a hexagonal plane pattern such that each ring touches its neighbouring rings. Each ring has mass $M$ and radius $R$. The moment of inertia of the system of seven rings about an axis passing through the centre of the central ring and normal to the plane of all rings is: (in $MR^2$)

The moment of inertia of a rod of mass $40 \text{ kg}$ and length $3 \text{ m}$ about an axis $AB$ passing through its end and perpendicular to its length is equal to the moment of inertia of a solid sphere of mass $10 \text{ kg}$ and radius $R$ about an axis parallel to the $AB$ axis, with a separation of $3 \text{ m}$ between the axes, as shown in the figure. The value of $R$ is given as $\sqrt{\frac{\alpha}{2}}$. The value of $\alpha$ is . . . . . . .

The moment of inertia of a uniform thin rod of length $L$ and mass $M$ about an axis passing through a point at a distance of $\frac{L}{4}$ from one of its ends and perpendicular to the rod is: