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

$A$ 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 $.5$)

$A$ thin uniform rod is pivoted at $O$. It rotates in a horizontal plane with a constant angular velocity $\omega$. At $t = 0$,a small insect starts moving from $O$ towards the other end with a constant speed $v$ relative to the rod. It reaches the other end at $t = T$ and stops. The angular velocity $\omega$ of the system remains constant. Which of the following graphs best represents the magnitude of the torque $|\tau|$ at $O$ as a function of time $t$?

If $I$,$\alpha$,and $\tau$ are the moment of inertia,angular acceleration,and torque of an object,respectively,and the object is rotating about an axis with an angular velocity $\omega$,then:

$A$ rod $PQ$ of mass $M$ and length $L$ is hinged at end $P$. The rod is kept horizontal by a massless string tied to point $Q$ as shown in the figure. When the string is cut,the initial angular acceleration of the rod is

An automobile engine develops $200 \ hp$ at $6000 \ rpm$. The corresponding torque is ......... $N \cdot m$.