From a disc of radius $R$,a concentric circular portion of radius $r$ is cut out so as to leave an annular disc of mass $M$. The moment of inertia of this annular disc about the axis perpendicular to its plane and passing through its centre of gravity is

$A$ rod of length $L$ consists of two halves: one half is made of copper with mass $m_c$ and the other half is made of silver with mass $m_s$. What is the moment of inertia of the rod about an axis passing through its midpoint and perpendicular to the rod?

$A$ square lamina of side $b$ has the same mass as a disc of radius $R$. The moment of inertia of the two objects about an axis perpendicular to the plane and passing through the centre is equal. The ratio $\frac{b}{R}$ is

The moment of inertia of a rectangular plate of mass $M$,length $l$,and breadth $b$ about an axis perpendicular to its plane and passing through its center is:

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 $MR^2$)

Suppose there is a uniform circular disc of mass $M$ and radius $r$ shown in the figure. Two shaded circular regions, each of radius $r/4$, are cut out from the disc. The centers of these cut-out discs are at a distance of $3r/4$ from the center of the original disc. The moment of inertia of the remaining part about the axis $A$ (passing through the center of the disc and perpendicular to its plane) is given by $\frac{x}{256} Mr^2$. The value of $x$ is . . . . . . .