$A$ 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 is $10 \,rad \,s^{-2}$. The tangential force applied to the rim of the disc is (in $\,N$)

$A$ rope of negligible mass is wound round a hollow cylinder of mass $3 \, kg$ and radius $40 \, cm$. If the rope is pulled with a force of $30 \, N$,then the angular acceleration produced in the cylinder is ........ $rad \, s^{-2}$.

An air compressor is powered by a $200\,rad\,s^{-1}$ electric motor using a $V-$belt drive. The motor pulley is $8\,cm$ in radius,and the tension in the $V-$belt is $135\,N$ on one side and $45\,N$ on the other. The power of the motor will be ...... $kW$.

An electric motor is started from rest to produce a torque. It acquires an angular acceleration given by $\alpha = 3t - t^2$. The angular acceleration becomes zero after $2 \ s$. Calculate the angular velocity after $6 \ s$.

$A$ pulley of radius $2 \ m$ is rotated about its axis by a force $F = (20t - 5t^2) \ N$ (where $t$ is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is $10 \ kg \ m^2$,the number of rotations made by the pulley before its direction of motion is reversed is approximately equal to: (in $.5$)

$A$ wheel having a moment of inertia of $2 \, kg \cdot m^2$ about its vertical axis rotates at the rate of $60 \, rpm$ about the axis. The torque required to stop the wheel's rotation in one minute would be