A grindstone is in the form of a flat circular disc of diameter $0.5 \mathrm{m}$ and mass $16 \mathrm{kg}$. What constant torque should it be applied so that it may attain an angular velocity of $120$ rpm in $8$ seconds?

A boy stands at the centre of a turntable with his two arms stretched outward. The turntable is rotating with a frequency of $40$ rpm. How much is the frequency of revolution of the boy if he folds his arms back and thereby reduces his moment of inertia by $\frac{2}{5}$ times the initial value? Ignore friction

A horizontal disc is revolving about a vertical axis at $100$ rpm rate. A small piece of wax falls from a height and gets stuck to the disc at a distance of $9 \mathrm{cm}$ from the axis. If the mass of the piece be $10 \mathrm{g}$ and the moment of inertia of the disc $7.3\times {10}^3 \mathrm{g}-{\mathrm{cm}}^2$, what will be the changed angular velocity of the disc?

A uniform plank $AB$, $5 \mathrm{m}$ long and weighing $60 \mathrm{kg}$ rests on two supports, one $1.5 \mathrm{m}$ from $A$ and the other $2 \mathrm{m}$ from $B$. A man weighing $70 \mathrm{kg}$ is moving about the plank. How near could he approach A and B respectively without upsetting the plank?

A beam $AB$ of weight $20 \mathrm{kg}$ rests on two supports at distances $4 \mathrm{cm}$ and $16 \mathrm{cm}$ from the end $A$. Two loads of masses $10 \mathrm{kg}$ and $4 \mathrm{kg}$ are hung from the end $A$ and $B$ respectively. Find the reactions at the two supports. The length of the beam is $20 \mathrm{cm}$.

Find the centre of gravity of a piece of a uniform wire $98 \mathrm{cm}$ long bent into $L$-shape with the upright part $56 \mathrm{cm}$ long.