The angular velocity of a body is $\vec{\omega} = 2\hat{i} + 3\hat{j} + 4\hat{k}$ and a torque $\vec{\tau} = \hat{i} + 2\hat{j} + 3\hat{k}$ acts on it. The rotational power will be .......... $W$.

$A$ 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 rod of length $2L$ is placed with one end in contact with a horizontal surface. The other end is released from an angle $\alpha$ with the horizontal,such that the end in contact does not slip. What will be its angular velocity when it becomes horizontal?

The $M.I.$ of a body about a given axis is $1.2 \ kg \cdot m^2$ and 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 time duration of $....... \ s$.

$A$ tangential force $F$ is applied on the rim of a ring of radius $R$. If the ring rotates by an angle $\theta$,what is the work done by the force?

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.