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

$A$ rigid body rotates about a fixed axis with variable angular velocity equal to $(\alpha - \beta t)$ at time $t,$ where $\alpha$ and $\beta$ are constants. The angle through which it rotates before it comes to rest is

$A$ wheel of radius $0.2 \ m$ rotates freely about its center when a string that is wrapped over its rim is pulled by a force of $10 \ N$ as shown in the figure. The established torque produces an angular acceleration of $2 \ rad/s^2$. The moment of inertia of the wheel is . . . . . . $kg \ m^2$. (Acceleration due to gravity $= 10 \ m/s^2$)

Four masses of $2 \ kg$ each are attached to an axle by spokes of length $1/4 \ m$ as shown in the figure. $A$ force $F = 24 \ N$ is applied at the end of a spoke of length $1/2 \ m$ at an angle of $30^{\circ}$ to produce an angular acceleration $\alpha$. The value of $\alpha$ is ........ $rad/s^2$.

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

$A$ massless rod of length $2l$ has equal point masses $m$ attached to its two ends as shown in the figure. The rod is rotating about an axis passing through its centre and making an angle $\alpha$ with the axis. The magnitude of the rate of change of angular momentum of the rod,i.e.,$\left|\frac{dL}{dt}\right|$,is equal to: