Two circular rings $A$ and $B$ of radii $nR$ and $R$ are made from the same wire. The moment of inertia of $A$ about an axis passing through the centre and perpendicular to the plane of $A$ is $64$ times that of the ring $B$. The value of $n$ is:

The moment of inertia of a semicircular ring about an axis passing through the center and perpendicular to the plane of the ring is $\frac{1}{x} MR^2$,where $R$ is the radius and $M$ is the mass of the semicircular ring. The value of $x$ will be $...........$

Two point masses of $0.3 \ kg$ and $0.7 \ kg$ are fixed at the ends of a rod of length $1.4 \ m$ and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum is located at a distance of

$A$ wire of mass $M$ and length $L$ is bent in the form of a circular ring. The moment of inertia of the ring about its axis is

$A$ ring of mass $m$ and radius $r$ is melted and recast into a sphere. The moment of inertia of the sphere will be:

Three particles of mass $m$ each are placed at the vertices of an equilateral triangle $ABC$ of side length $\ell$. What is the moment of inertia of the system about the axis $AX$ passing through vertex $A$ and perpendicular to side $AB$ in the plane of the triangle?