The radius of gyration of a cylindrical rod about an axis of rotation perpendicular to its length and passing through its center is $k$. Given that the length of the rod is $10 \sqrt{3} \ m$,find the value of $k$ in meters.

The moment of inertia of a solid sphere about its diameter is $I$. It is then recast into $27$ small spheres of the same diameter. The moment of inertia of each small sphere about its diameter is:

$A$ uniform square plate $S$ (side $c$) and a uniform rectangular plate $R$ (sides $b$,$a$) $(a > b)$ have identical areas and masses. Based on the figure,which of the following is correct?
$(a)\ I_{xR}/I_{xS} < 1$
$(b)\ I_{yR}/I_{yS} > 1$
$(c)\ I_{zR}/I_{zS} > 1$

From a circular ring of mass $M$ and radius $R$,an arc corresponding to a $90^{\circ}$ sector is removed. The moment of inertia of the remaining part of the ring about an axis passing through the centre of the ring and perpendicular to the plane of the ring is $K$ times $MR^{2}$. Then the value of $K$ is

$A$ uniform thin bar of mass $6 \,kg$ and length $2.4 \,m$ is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of the hexagon is ...... $\times 10^{-1} \,kg \cdot m^2$.

Four particles,each of mass $1 \ kg$,are placed at the four corners of a square of side $2 \ m$. The moment of inertia of the system about an axis perpendicular to its plane and passing through one of its vertices is . . . . . . $kg \ m^2$.