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.

$A$ thin uniform rod $AB$ of mass $m$ and length $l$ is hinged at one end $A$ to the ground level. Initially,the rod stands vertically and is allowed to fall freely to the ground in the vertical plane. The angular velocity of the rod when its end $B$ strikes the ground is ($g$ = acceleration due to gravity).

$A$ 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.)

$A$ tangential force $F$ is applied on a disc of radius $R$,due to which it deflects through an angle $\theta$ from its initial position. The work done by this force would be

$A$ rod of length $1 \ m$ is held vertically and allowed to fall such that its lower end remains fixed on the ground. The velocity of the other end of the rod when it hits the ground is ......... $m/s$ $(g = 9.8 \ m/s^2)$.

$A$ cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of the wheel is $10 \ kg$ and the radius is $10 \ cm$. It can rotate freely without any friction. Initially,the wheel is at rest. If a steady pull of $20 \ N$ is applied on the cord,the angular velocity (in $rad/s$) of the wheel,after the cord is unwound by $1 \ m$,would be: