$A$ 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 ...... $N$.

$A$ rope is wound around a solid cylinder of mass $1 \,kg$ and radius $0.4 \,m$. What is the angular acceleration of the cylinder, if the rope is pulled with a force of $25 \,N$? (The cylinder is rotating about its own axis.)

$A$ uniform rod of mass $M$ and length $L$ is hinged at one end as shown in the figure. Initially,the rod is kept horizontal by a massless string. When the string is cut,the angular acceleration of the rod is:

$A$ crew of scientists has built a new space station. The space station is shaped like a wheel of radius $R,$ with essentially all its mass $M$ at the rim. When the crew arrives,the station will be set rotating at a rate that causes an object at the rim to have radial acceleration $g,$ thereby simulating Earth's surface gravity. This is accomplished by two small rockets,each with thrust $T$ newtons,mounted on the station's rim. How long a time $t$ does one need to fire the rockets to achieve the desired condition?

$A$ uniform rod $AB$ of length $l$ and mass $m$ is free to rotate about point $A.$ The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about $A$ is $ml^2/3$,the initial angular acceleration of the rod will be:

$A$ disc of mass $M \ kg$ and radius $R \ m$ is rotating at an angular speed of $\omega \ rad/s$ when the motor is switched off. Neglecting the friction at the axle,the force that must be applied tangentially to the wheel to bring it to rest in time $t$ is ...............