If a disc of mass $M$ and radius $R$ rotates with an angular acceleration $a$, the torque acting on the disc is

When a ceiling fan is switched off,its angular velocity falls to half while it makes $36$ rotations. How many more rotations will it make before coming to rest?

$A$ disc of radius $0.4 \,m$ and mass $1 \,kg$ rotates about an axis passing through its center and perpendicular to its plane. The angular acceleration of the disc is $10 \,rad/s^2$. The tangential force applied to the rim of the disc is (in $\,N$)

$A$ rope of negligible mass is wound round a hollow cylinder of mass $3\; kg$ and radius $40\; cm$. What is the angular acceleration of the cylinder if the rope is pulled with a force of $30\; N$? What is the linear acceleration of the rope? Assume that there is no slipping.

$A$ thin solid disk of $1 \ kg$ is rotating along its diameter axis at the speed of $1800 \ rpm$. By applying an external torque of $25 \pi \ Nm$ for $40 \ s$,the speed increases to $2100 \ rpm$. The diameter of the disk is . . . . . . $m$.

$A$ 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 $.....$