Four small objects, each of mass $m$, are fixed at the corners of a rectangular wire-frame of negligible mass with sides $a$ and $b$ $(a > b)$. If the wire frame is now rotated about an axis passing along the side of length $b$, then the moment of inertia of the system for this axis of rotation is:

$A$ circular disc $X$ of radius $R$ is made from an iron plate of thickness $t$,and another disc $Y$ of radius $4R$ is made from an iron plate of thickness $\frac{t}{4}$. The ratio of the moment of inertia $\frac{I_Y}{I_X}$ is:

Let $M$ be the mass and $L$ be the length of a thin uniform rod. In the first case,the axis of rotation passes through the center and is perpendicular to the length of the rod. In the second case,the axis of rotation passes through one end and is perpendicular to the length of the rod. The ratio of the radius of gyration in the first case to the second case is

One circular ring and one circular disc,both having the same mass and radius. The ratio of their moments of inertia about the axis passing through their centers and perpendicular to their planes will be:

The moment of inertia of a uniform rod of mass $M$ and length $L$ about an axis passing through its center and perpendicular to it is $\frac{1}{12} ML^2$. The rod is bent at the center such that the two halves make an angle of $60^\circ$ with each other. Find the moment of inertia of the bent rod about the same axis passing through the center of the rod (the point of bending) and perpendicular to the plane containing the two halves.

Two circular loops $P$ and $Q$ are made from a uniform wire. The radii of $P$ and $Q$ are $R_1$ and $R_2$ respectively. The moments of inertia about their own axis are $I_{P}$ and $I_{Q}$ respectively. If $\frac{I_{P}}{I_{Q}}=\frac{1}{8}$,then $\frac{R_2}{R_1}$ is