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 $YY'$ will be

$A$ circular plate of mass $M$ and radius $R$ has its density varying as $\rho(r) = \rho_0 r$,where $\rho_0$ is a constant and $r$ is the distance from its center. The moment of inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is $I = aMR^2$. The value of the coefficient $a$ is:

Three solid spheres each of mass $M$ and radius $R$ are arranged as shown in the figure. The moment of inertia of the system about $YY'$ will be

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

The moment of inertia of a ring about an axis perpendicular to its plane and passing through its center is $4 \,kg \,m^2$. Its moment of inertia about the tangent in the plane is