Find the moment of inertia of a uniform half-disc about an axis perpendicular to the plane and passing through its centre of mass. Mass of this disc is $M$ and radius is $R.$ 

Calculate the radius of gyration of a uniform circular disk of radius $r$ and thickness $t$ about a line perpendicular to the plane of this disk and tangent to the disk as shown in figure.

Calculate the net torque on the system about the point $O$ as shown in figure if $F_1=11 \mathrm{N}, F_2=9 \mathrm{N},\mathit{ }{F}_3=10 \mathrm{N}, a=10 \mathrm{cm}$and$b=20 \mathrm{cm}.$(All the forces along the tangent).

Assuming frictionless contacts, determine the magnitude of external horizontal force $P$ applied at the lower end for equilibrium of the rod as shown in figure. The rod is uniform and its mass is $m\mathit{.}$