Three thin rods,each of length $L$ and mass $M$,are placed along the $x, y,$ and $z-$ axes such that one end of each rod is at the origin. The moment of inertia of this system about the $z-$ axis is:

The mass per unit area of a circular disc of radius $a$ depends on the distance $r$ from its centre as $\sigma(r) = A + Br$. The moment of inertia of the disc about the axis perpendicular to the plane and passing through its centre is

Weights of $1 \ kg$ and $4 \ kg$ are placed at the $20 \ cm$ and $70 \ cm$ marks of a light meter scale,respectively. What is the moment of inertia in $kg \ m^2$ about an axis perpendicular to the scale and passing through the $100 \ cm$ mark?

$A$ thin uniform rectangular plate of mass $2 \, kg$ is placed in the $xy$-plane as shown in the figure. The moment of inertia about the $x$-axis is $I_{x}=0.2 \, kg \cdot m^{2}$ and the moment of inertia about the $y$-axis is $I_{y}=0.3 \, kg \cdot m^{2}$. The radius of gyration of the plate about the axis passing through $O$ and perpendicular to the plane of the plate is (in $ \, cm$)

The moment of inertia of a thin semicircular disc (mass $= M$ and radius $= R$) about an axis passing through point $O$ and perpendicular to the plane of the disc is given by:

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: