$A$ disc has mass $M$ and radius $R$. How much tangential force should be applied to the rim of the disc so as to rotate the disc with angular velocity $\omega$ in 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:

$A$ 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 $rad/s$)

$A$ 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 $.....$

$A$ torque of $30 \, N \cdot m$ is applied for $10 \, s$ on a wheel of mass $5 \, kg$ and moment of inertia $2 \, kg \cdot m^2$. The angular displacement of the wheel in $10 \, s$ will be ....... radians.

$A$ wheel having a moment of inertia $2 \; kg \cdot m^2$ about its vertical axis rotates at the rate of $60 \; rpm$ about this axis. The torque required to stop the wheel's rotation in one minute is: