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 moment of inertia of a thin rod about an axis passing through its midpoint and perpendicular to the rod is $2400 \,g \,cm^2$. The length of the $400 \,g$ rod is nearly: (in $\,cm$)

Four hollow spheres,each with a mass of $1\, kg$ and a radius $R = 10\, cm$,are connected with massless rods to form a square with a side of length $L = 50\, cm$. In case-$1$,the masses rotate about an axis that bisects two sides of the square. In case-$2$,the masses rotate about an axis that passes through the diagonal of the square,as shown in the figure. Compute the ratio of the moments of inertia $I_1/I_2$ for the two cases.

$A$ solid sphere $A$ is rotating about an axis $PQ$. If the radius of the sphere is $5 \, cm$ and the distance of its center from the axis $PQ$ is $10 \, cm$,then its radius of gyration about $PQ$ will be $\sqrt{x} \, cm$. The value of $x$ is $................$.

$A$ uniform thin bar of mass $6 \,kg$ and length $2.4 \,m$ is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of the hexagon is ...... $\times 10^{-1} \,kg \cdot m^2$.

What is the moment of inertia of a solid sphere of radius $R$ and density $\rho$ about its diameter?