$A$ constant torque is applied to a wheel undergoing uniform circular motion,causing its angular momentum to change from $A_0$ to $4A_0$ in $4 \ s$. The magnitude of this torque is .......

The moment of inertia of a body about a given axis is $1.2 \ kg \cdot m^2$. Initially,the body is at rest. To produce a rotational kinetic energy of $1500 \ J$,an angular acceleration of $25 \ rad/s^2$ must be applied to the body for a duration of ........ $(s)$.

$A$ thin solid disk of $1 \ kg$ is rotating along its diameter axis at the speed of $1800 \ rpm$. By applying an external torque of $25 \pi \ Nm$ for $40 \ s$,the speed increases to $2100 \ rpm$. The diameter of the disk is . . . . . . $m$.

$A$ wheel of mass $20 \,kg$ and radius $30 \,cm$ is rotating at an angular speed of $80 \,rev/min$ when the motor is turned off. Neglecting the friction at the axis, calculate the force that must be applied tangentially to the wheel to bring it to rest in $5 \,rev$. (in $\pi \,N$)

If the external torque acting on a system is $ \tau = 0 $, then:

$A$ 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 ........ $rad \, s^{-2}$.