A pulley of radius $2 \mathrm{m}$ is rotated about its axis by a force $F=\left(20t-5t^2\right)$ Newton (where, $t$ is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is $10 {\mathrm{kgm}}^2,$ the number of rotations made by the pulley before its direction of motion is reversed, is

A mass $m$ supported by a massless string wound around a uniform hollow cylinder of mass $m$ and radius $R$. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?

A roller is made by joining together two cones at their vertices $0.$ It is kept on two rails $\mathrm{AB}$ and $\mathrm{CD}$ which are placed asymmetrically (see figure), with its axis perpendicular to $\mathrm{CD}$ and its centre $\mathrm{O}$ at the centre of line joining $\mathrm{AB}$ and $\mathrm{CD}$ (see figure). It is given a light push so that it starts rolling with its centre $\mathrm{O}$ moving parallel to $\mathrm{CD}$ in the direction shown. As it moves, the roller will tend to 

A slender uniform rod of mass $M$ and length $l$ is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. the angular acceleration of the rod when it makes an angle $\theta$ with the vertical is

Seven identical circular planar disks, each of mass $M$ and radius $R$ are welded symmetrically as shown. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point $P$ is 

From a uniform circular disc of radius $R$ and mass $9 \mathrm{M}$, a small disc of radius $\frac{R}{3}$ is removed as shown in the figure. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through centre of disc is: