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

$A$ same torque is applied to a disc and a ring of equal mass and radii. Then:

$A$ constant torque of $31.4 \, Nm$ is exerted on a pivoted wheel. If the angular acceleration of the wheel is $4\pi \, rad/s^2$,then the moment of inertia will be ....... $kg \cdot m^2$.

$A$ pulley has the shape of a uniform solid disc of mass $2 \ kg$ and radius $0.5 \ m$. $A$ string is wrapped over its rim and is pulled by a force of $2.5 \ N$. The pulley is free to rotate about its axis. Initially,the pulley is at rest. Find the angular velocity of the pulley just after $10 \ s$. (in $rad/s$)

$A$ disc of mass $10 \ kg$ and radius $0.1 \ m$ is rotating at $120 \ rpm$. $A$ retarding torque brings it to rest in $10 \ s$. If the same torque is due to a force applied tangentially on the rim of the disc, then the magnitude of the force is: (in $\pi \ N$)

Torques of equal magnitude are applied to a hollow cylinder and a solid sphere,both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry,and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?