For the given uniform square lamina $ABCD$,whose centre is $O$,

$A$ lamina is made by removing a small disc of diameter $2R$ from a bigger disc of uniform mass density and radius $2R$,as shown in the figure. The moment of inertia of this lamina about axes passing through $O$ and $P$ is $I_0$ and $I_P$,respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $\frac{I_P}{I_0}$ to the nearest integer is:

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

The moment of inertia of a solid cylinder about its own axis is equal to its moment of inertia about an axis passing through its center of gravity and perpendicular to its length. What is the relationship between its length $L$ and radius $R$?

$A$ 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 ..............

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 $MR^2$)