When a ceiling fan is switched off,its angular velocity falls to half while it makes $36$ rotations. How many more rotations will it make before coming to rest?

Four masses of $2 \ kg$ each are attached to an axle by spokes of length $1/4 \ m$ as shown in the figure. $A$ force $F = 24 \ N$ is applied at the end of a spoke of length $1/2 \ m$ at an angle of $30^{\circ}$ to produce an angular acceleration $\alpha$. The value of $\alpha$ is ........ $rad/s^2$.

$A$ wheel with a moment of inertia of $5 \times 10^{-3} \ kg \cdot m^2$ is rotating at a rate of $20 \ rev/s$. The torque required to bring it to rest in $10 \ s$ is:

$A$ solid disc of radius $20 \, \text{cm}$ and mass $10 \, \text{kg}$ is rotating with an angular velocity of $600 \, \text{rpm}$ about an axis normal to its circular plane and passing through its centre of mass. The retarding torque required to bring the disc to rest in $10 \, \text{s}$ is $x \pi \times 10^{-1} \, \text{Nm}$. Find the value of $x$.

$A$ disc of radius $0.4 \,m$ and mass $1 \,kg$ rotates about an axis passing through its center and perpendicular to its plane. The angular acceleration of the disc is $10 \,rad/s^2$. The tangential force applied to the rim of the disc is (in $\,N$)

$A$ solid cylinder of mass $50\, kg$ and radius $0.5\, m$ is free to rotate about a horizontal axis. $A$ massless string is wound around the cylinder with one end attached to it and the other hanging freely. The tension in the string required to produce an angular acceleration of $2\, \text{revolutions } s^{-2}$ is ...... $N$.