$A$ 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

$A$ long slender rod is welded to a thin circular disc of diameter $0.5 \ m$ at a point on its circumference. The rod is in the same plane as that of the disc and forms a tangent to the disc. The radius of gyration of the disc about the rod (in $m$) is

The moment of inertia of a semicircular plate of radius $R$ and mass $M$ about an axis $AA'$ in its plane passing through its centre is given by:

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

The moment of inertia of a cylinder of mass $M$,length $L$,and radius $R$ about an axis passing through its centre and perpendicular to the axis of the cylinder is $I = M \left(\frac{R^2}{4} + \frac{L^2}{12}\right)$. If such a cylinder is to be made for a given mass of material,the ratio $L/R$ for it to have the minimum possible $I$ is

Three point masses $m_1, m_2$,and $m_3$ are placed at the vertices of an equilateral triangle of side $a$. Find the moment of inertia of this system about the altitude passing through $m_1$.