$A$ uniform rod of length $l$ and mass $m$ is pivoted at point $A$. The rod is released from a horizontal position. If the moment of inertia of the rod about point $A$ is $ml^2/3$,then its initial angular acceleration 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?

$A$ non-uniform rod $OA$ of linear mass density $\lambda = \lambda_0 x$ (where $\lambda_0$ is a constant) is suspended from the ceiling with a hinge joint at $O$ and a light string as shown in the figure. Find the angular acceleration of the rod just after the string is cut.

Obtain the relation between angular momentum of a particle and torque acting on it.

$A$ pulley of radius $1.5\,m$ is rotated about its axis by a force $F = (12t - 3t^2)\,N$ applied tangentially (where $t$ is measured in seconds). If the moment of inertia of the pulley about its axis of rotation is $4.5\,kg\cdot m^2$,the number of rotations made by the pulley before its direction of motion is reversed will be $\frac{K}{\pi}$. The value of $K$ is $.....$

The angular momentum of a wheel changes from $2L$ to $5L$ in $3$ seconds. The magnitude of the torque acting on the wheel is: