$A$ solid hemisphere and a hemispherical shell are joined as shown. Both of them have a mass of $m/2$ individually. Find the moment of inertia about the axis $AB$.

$ABC$ is a triangular plate of uniform thickness. The sides are in the ratio shown in the figure. $I_{AB}, I_{BC}, I_{CA}$ are the moments of inertia of the plate about axes $AB, BC,$ and $CA$ respectively. Which one of the following relations is correct?

The moment of inertia of a rod of mass $M$ and length $L$ about an axis passing through its centre and perpendicular to it is $\frac{1}{12} ML^2$. The rod is bent in the middle so that the two halves make an angle of $60^o$. The moment of inertia of the bent rod about the same axis (passing through the original centre of the rod) would be:

Let $M$ and $L$ be the mass and length of a thin uniform rod,respectively. In the $1^{\text{st}}$ case,the axis of rotation passes through the center and is perpendicular to its length. In the $2^{\text{nd}}$ case,the axis of rotation passes through one end and is perpendicular to its length. The ratio of the radius of gyration in the first case to the second case is:

From a uniform wire,two circular loops are made: $(i)\ P$ of radius $r$ and $(ii)\ Q$ of radius $nr$. If the moment of inertia of $Q$ about an axis passing through its center and perpendicular to its plane is $8$ times that of $P$ about a similar axis,the value of $n$ is (diameter of the wire is very much smaller than $r$ or $nr$).

Four particles each of mass $M$ are placed at the corners of a square of side $L$. The radius of gyration of the system about an axis perpendicular to the square and passing through its centre is