A solid sphere is rolling down an inclined plane. Then the ratio of its translational kinetic energy to its rotational kinetic energy is

A solid sphere of mass $500\ gm$ and radius $10\ cm$ rolls without slipping with the velocity $20\ cm/s$.  The total kinetic energy of the sphere will be  ........ $J$

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 rod of length $l$ is hinged from one end. It is brought to a horizontal position and released. The angular velocity of the rod when it is in the vertical position is

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

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