From a uniform circular disc of radius $R$ and mass $9M$,a small disc of radius $\frac{R}{3}$ is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through the centre of the original disc is

$I_{CM}$ is the moment of inertia of a circular disc about an axis $(CM)$ passing through its center and perpendicular to the plane of the disc. $I_{AB}$ is its moment of inertia about an axis $AB$ perpendicular to the plane and parallel to the axis $CM$ at a distance $\frac{2}{3}R$ from the center,where $R$ is the radius of the disc. The ratio of $I_{AB}$ and $I_{CM}$ is $x:9$. The value of $x$ is $........$

Seven identical coins are rigidly arranged on a flat table in the pattern shown below,so that each coin touches its neighbours. Each coin is a thin disc of mass $m$ and radius $r$. Note that the moment of inertia of an individual coin about an axis passing through its centre and perpendicular to the plane of the coin is $\frac{m r^2}{2}$. The moment of inertia of the system of seven coins about an axis that passes through the point $P$ (the centre of the coin positioned directly to the right of the central coin) and perpendicular to the plane of the coins is ..........$m r^2$.

$A$ rigid body of mass $m$ is rotating with angular velocity $\omega$ about an axis at a distance '$a$' from the center of mass $G$. The radius of gyration about an axis passing through $G$ is $K$. The rotational kinetic energy of this body about the new parallel axis will be:

Calculate the moment of inertia of an annular disc about an axis tangent to the outer circle and lying in the plane of the disc. The mass of the disc is $M$,the inner radius is $R_1$,and the outer radius is $R_2$.

Three rings each of mass $M$ and radius $R$ are arranged as shown in the figure. The moment of inertia of the system about the axis $YY'$ is: