Two circular loops $P$ and $Q$ of radii $r$ and $nr$ are made respectively from a uniform wire. The moment of inertia of loop $Q$ about its axis is four times that of loop $P$ about its axis. The value of $n$ is

The densities of two solid spheres $A$ and $B$ of the same radii $R$ vary with radial distance $r$ as $\rho_A(r) = k \left(\frac{r}{R}\right)$ and $\rho_B(r) = k \left(\frac{r}{R}\right)^5$,respectively,where $k$ is a constant. The moments of inertia of the individual spheres about axes passing through their centres are $I_A$ and $I_B$,respectively. If $\frac{I_B}{I_A} = \frac{n}{10}$,the value of $n$ is

Four masses are attached to a light circular frame of radius $a$ as shown in the figure. The radius of gyration of this system about an axis passing through the center $O$ of the circular frame and perpendicular to its plane is:

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:

We have a rectangular plate of uniform thickness. $E, F, G, H$ are the midpoints of $AB, BC, CD,$ and $AD$ respectively. About which axis will the moment of inertia be minimum?

The ratio of the moments of inertia of two uniform rings of radii $R$ and $nR$ about an axis passing through their centers and perpendicular to their planes is $1 : 8$. What is the value of $n$?