Each spherical shell has a radius $R$ and mass $M$. They are connected by a light massless rod. Calculate the moment of inertia about the axis $xx'$.

$ABC$ is a plane lamina in the shape of an equilateral triangle. $D$ and $E$ are the midpoints of $AB$ and $AC$,respectively,and $G$ is the centroid of the lamina. The moment of inertia of the lamina about an axis passing through $G$ and perpendicular to the plane $ABC$ is $I_{0}$. If the part $ADE$ is removed,the moment of inertia of the remaining part about the same axis is $\frac{NI_{0}}{16}$,where $N$ is an integer. The value of $N$ 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:

$A$ thin wire of length $L$ has a uniform linear mass density $\rho$. It is bent into a circular loop with center $O$. Calculate the moment of inertia of the circular loop about the axis $XX'$ as shown in the figure.

$A$ solid sphere and a disc of same mass $M$ and radius $R$ are kept such that their curved surfaces are in contact and their centers lie along the same horizontal line. The moment of inertia of the two-body system about an axis passing through their point of contact and perpendicular to the plane of the disc is

The moment of inertia of a square frame about an axis passing through its center of mass and perpendicular to its plane is $20 \ kg \cdot m^2$. The moment of inertia about an axis touching its side and lying in the plane of the frame is ........ $kg \cdot m^2$.