Find the moment of inertia of a solid cylinder of mass $M$ and radius $R$ about a line parallel to the axis of the cylinder and on the surface of the cylinder.

Four identical uniform solid spheres,each of mass $M$ and radius $R$,are placed touching each other in a line as shown in the figure,with centers $A, B, C, D$. Let $I_A, I_B, I_C,$ and $I_D$ be the moment of inertia of the entire system of four spheres about an axis passing through the center of the respective sphere and perpendicular to the plane containing the centers. The difference $|I_A - I_B|$ is: (in $MR^2$)

$A$ small circular disc of radius $R/3$ is cut from a larger circular disc of radius $R$ and mass $9M$. 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 ring of mass $m = 3 \ gm$ and radius $r = 1 \ cm$ about an axis passing through its edge and parallel to its natural axis is:

$A$ thin square plate of side $4R$ has a mass $M$. Four circles of radius $R$ are cut from it as shown. Calculate the moment of inertia of the remaining part about the $z$-axis passing through the center of the square.

$A$ rigid body of mass $m$ rotates with angular velocity $\omega$ about an axis at a distance $d$ from its center of mass. The radius of gyration about an axis passing through the center of mass $G$ and parallel to the given axis is $K$. What is the rotational kinetic energy of the body?