The ratio of radii of gyration of a thin circular ring and a circular disc of same radius about a tangential axis in their own planes is $\sqrt{12}: \sqrt{K}$. The value of $K$ is

Four masses are fixed on a massless rod as shown in the figure. The moment of inertia about the axis $PQ$ is about ..... $kg-m^2$.

Three point masses,each of mass $m$,are placed at the corners of an equilateral triangle of side $\ell$. The moment of inertia of the system about an axis passing through one of the vertices and parallel to the side joining the other two vertices is:

$A$ solid sphere $A$ of radius $R$ and mass $M$ is attached at a point to a smaller solid sphere $B$ of radius $r < R$ and mass $m < M$. Assume that the line joining their centres lies along the horizontal. The moment of inertia of the system calculated about a vertical axis passing through the centre of $A$ is $I_A$ and that calculated about a vertical axis passing through the centre of $B$ is $I_B$. The difference $I_A - I_B$ is :

If $I_1$ is the moment of inertia of a thin rod of mass $M$ and length $l$ about an axis perpendicular to its length and passing through its centre of mass,and $I_2$ is the moment of inertia of the ring formed by bending the rod about an axis passing through its centre and perpendicular to its plane,then:

The radius of gyration of a uniform rod of length $l$ about an axis passing through one of its ends and perpendicular to its length is