Write the formula of work done by torque in a rotational rigid body about a fixed axis.

$A$ flywheel of moment of inertia $I$ is rotating at $n$ revolutions per second. The work needed to double the frequency would be

$A$ disc is rotating with angular velocity $\vec{\omega}$. $A$ force $\vec{F}$ acts at a point whose position vector with respect to the axis of rotation is $\vec{r}$. The power associated with the torque due to the force is given by ..........

To maintain a rotor at a uniform angular speed of $200 \; rad \; s^{-1}$,an engine needs to transmit a torque of $180 \; N \; m$. What is the power required by the engine (in $; kW$)? (Note: uniform angular velocity in the absence of friction implies zero torque. In practice,applied torque is needed to counter frictional torque). Assume that the engine is $100 \%$ efficient.

$A$ uniform solid cylinder of mass $M$ and radius $R$ can freely rotate around its axis $O$. There is an elastic string of relaxed length $L$ and stiffness $K$ attached to the cylinder and a static wall. Initially,the string is relaxed. As the cylinder starts rotating,the string will wind around the cylinder. The surface of the cylinder is very rough,so the string does not slip on the cylinder's surface. The minimum initial angular speed of the cylinder,${\omega _0}$,so that it can rotate through an angle $2\pi$ is (Assume Hooke's law to be valid.)

$A$ disc of mass $1\,kg$ and radius $R$ is free to rotate about a horizontal axis passing through its centre and perpendicular to the plane of the disc. $A$ body of the same mass as that of the disc is fixed at the highest point of the disc. Now the system is released. When the body comes to the lowest position,its angular speed will be $4 \sqrt{\frac{x}{3 R}} \text{ rad s}^{-1}$ where $x=$ (Given $g = 10 \text{ m s}^{-2}$)