Three thin rods,each of mass $2M$ and length $L$,are placed along the $x, y,$ and $z$ axes,which are mutually perpendicular. One end of each rod is at the origin. The moment of inertia of the system about the $x$-axis is:

Radius of gyration of a thin uniform circular disc about the axis passing through its centre and perpendicular to its plane is $K_c$. Radius of gyration of the same disc about a diameter of the disc is $K_d$. The ratio $K_d : K_c$ is

The figure shows the variation of the moment of inertia of a uniform rod about an axis passing through its centre and inclined at an angle $\theta$ to the length of the rod. The moment of inertia of the rod about an axis passing through one of its ends and making an angle $\theta = \frac{\pi}{3}$ with the rod will be ........ $kg-m^2$.

Define moment of inertia,write its unit and dimensional formula.

Two circular loops $P$ and $Q$ of radii $r$ and $nr$ are made respectively from a uniform wire. The moment of inertia of loop $Q$ about its axis is $4$ times that of loop $P$ about its axis. The value of $n$ is:

$A$ wire of length $l$ and mass $m$ is bent in the form of a rectangle $ABCD$ with $(AB/BC) = 2$. The moment of inertia of this wire about the side $BC$ is