Two discs,one of density $7.2 \, g/cm^3$ and the other of density $8.9 \, g/cm^3$,have the same mass and thickness. What is the ratio of their moments of inertia?

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

Four point masses,each of mass $m$,are fixed at the corners of a square of side $\ell$. The square is rotating with angular frequency $\omega$,about an axis passing through one of the corners of the square and parallel to its diagonal,as shown in the figure. The angular momentum of the square about this axis is $.......m\ell^2 \omega$.

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

For a rectangle $ABCD$ with $BC = 2AB$,about which axis will the moment of inertia be minimum?

If the radius of a solid sphere is $35\,cm,$ calculate the radius of gyration when the axis is along a tangent.