$A$ ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to the plane of the ring. The ratio of the radius of the ring to that of the sphere is $\sqrt{\frac{2}{x}}$. The value of $x$ is $.......$

The moment of inertia of a uniform rod of length $2l$ and mass $m$ about an axis $xx'$ passing through its centre and inclined at an angle $\alpha$ is

Three particles,each of mass $m \; g$,are situated at the vertices of an equilateral triangle $ABC$ of side $l \; cm$ (as shown in the figure). The moment of inertia of the system about a line $AX$ perpendicular to $AB$ and in the plane of $ABC$ in $g \cdot cm^2$ units will be:

$A$ straight rod of length $L$ is made of a material having mass per unit length $m(x) = \lambda|x|$,where $x$ is measured from the center of the rod. The moment of inertia about an axis perpendicular to the rod and passing through one end of the rod is to be calculated. Given $L = 1 \ m$ and $\lambda = 16 \ kg/m^2$.

$A$ one-quarter sector is cut from a uniform circular disc of radius $R$. This sector has mass $M$. It is made to rotate about a line perpendicular to its plane and passing through the centre of the original disc. Its moment of inertia about the axis of rotation is

$A$ wheel of mass $10 \ kg$ has a moment of inertia of $160 \ kg \cdot m^2$ about its own axis. The radius of gyration is ........ $m.$