State and prove parallel axis theorem.

For a thin flat circular disc, the radius of gyration about a diameter as axis is $k$. If the disc is cut along a diameter $AB$ as shown into two equal pieces, then find the radius of gyration of each piece about $AB$.

Show that $\overrightarrow{a}.\left(\overrightarrow{b}\times \overrightarrow{c}\right)$ is equal in magnitude to the volume of the parallelepiped formed on the three vectors $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$.

A rope of negligible mass is wound round a hollow cylinder of mass $3 \mathrm{kg}$ and radius $40 \mathrm{cm}$. What is the angular acceleration of the cylinder if the rope is pulled with a force of $30 \mathrm{N}$? What is the linear acceleration of the rope? Assume that there is no slipping.

A coin is kept at a distance of $10 \mathrm{cm}$ from the centre of a circular turntable. If the coefficient of static friction between the table and the coin is $0.8$, find the frequency of rotation of the disc at which the coin just begin to slip.