For a uniform rectangular sheet shown in the figure,the ratio of moments of inertia about the axes perpendicular to the sheet and passing through $O$ (the centre of mass) and $O'$ (corner point) is

$A$ thin wire of length $l$ and uniform linear mass density $\rho$ is bent into a circular loop with centre $O$ and radius $r$ as shown in the figure. The moment of inertia of the loop about the axis $XX'$ is

The $X$ and $Z$ axes are mutually perpendicular in the plane of a disc,and the $Y$ axis is perpendicular to the plane of the disc. If the moments of inertia of the object about the $X$ and $Y$ axes are $30 \ kg \ m^2$ and $40 \ kg \ m^2$ respectively,then the moment of inertia about the $Z$ axis will be ....... $kg \ m^2$.

State the theorem of perpendicular axes and the theorem of parallel axes.

$A$ circular plate of mass $M$ and radius $R$ has its density varying as $\rho(r) = \rho_0 r$,where $\rho_0$ is a constant and $r$ is the distance from its center. The moment of inertia of the circular plate about an axis perpendicular to the plate and passing through its edge is $I = aMR^2$. The value of the coefficient $a$ is:

Four rods,each of mass $M$ and length $l$,are arranged to form a square as shown in the figure. What is the moment of inertia of this square about an axis passing through $O$ and perpendicular to the plane of the square?