$A$ sphere of mass $m$ and moment of inertia $I$ about its center of mass is rolling without slipping down an inclined plane from rest. Which of the following represents its total kinetic energy?

$A$ disc with a moment of inertia of $\frac{9.8}{\pi^2} \ kg \ m^2$ is rotating at $600 \ rpm$. If the rotational frequency changes from $600 \ rpm$ to $300 \ rpm$,the work done is .......... $J$.

$A$ circular disc of radius $R$ meter and mass $M$ kg is rotating around the axis perpendicular to the disc. An external torque is applied to the disc such that $\theta(t) = 5t^2 - 8t$,where $\theta(t)$ is the angular position of the rotating disc as a function of time $t$. How much power is delivered by the applied torque,when $t = 2$ s (in $MR^2$)?

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.

$A$ constant power is supplied to a rotating disc. The angular velocity $(\omega)$ of the disc varies with the number of rotations $(n)$ made by the disc as:

$A$ wheel is rotating with an angular speed of $20\ rad/s$. It is stopped to rest by applying a constant torque in $4\ s$. If the moment of inertia of the wheel about its axis is $0.20\ kg\cdot m^2$,then the work done by the torque in two seconds will be .......... $J$.