$A$ 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 $.......$

Of the two eggs which have identical sizes,shapes and weights,one is raw and the other is half-boiled. The ratio between the moment of inertia of the raw egg to the half-boiled egg about their central axis is:

Two solid spheres ($A$ and $B$) are made of metals of different densities $\rho_A$ and $\rho_B$ respectively. If their masses are equal, the ratio of their moments of inertia $(I_B / I_A)$ about their respective diameter is

Radius of gyration of a body depends on

Two loops $A$ and $B$ of radii $R_1$ and $R_2$ are made from a uniform wire. If the moment of inertia of $A$ is $I_A$ and that of $B$ is $I_B$,then find the ratio $R_2 / R_1$ given that $I_A / I_B = 27$.

$A$ 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 $\lambda = 16 \ kg/m^2$.