A spherical solid ball of $10\,kg$ mass and radius $3\,cm$ is rotating about an axis passing through its centre with an angular velocity of $50\,radian/s$ the kinetic energy of rotation is ....... $J.$

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?

(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 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

The moment of inertia of a body about a given axis is $2.4\ kg-m^2$.  To produce a rotational kinetic energy of $750\ J$, an angular acceleration of $5\ rad/s^2$ must be applied about that axis for.......... $\sec$

A solid cylinder $P$ rolls without slipping from rest down an inclined plane attaining a speed $v_p$ at the bottom. Another smooth solid cylinder $Q$ of same mass and dimensions slides without friction from rest down the inclined plane attaining a speed $v_q$ at the bottom. The ratio of the speeds $\frac{v_q}{v_p}$ is

For the pivoted slender rod of length $l$ as shown in figure, the angular velocity as the bar reaches the vertical position after being released in the horizontal position is