A ring of radius $0.5\, m$ and mass $10 \,kg$ is rotating about its diameter with an angular velocity of $20 \,rad/s.$  Its kinetic energy is .......... $J$

A cord is wound round the circumference of wheel of radius $r$. The axis of the wheel is horizontal and moment of inertia about it is $I$. A weight $mg$ is attached to the end of the cord and falls from rest. After falling through a distance $h$, the angular velocity of the wheel will be

A circular disc of mass $2 \,kg$ and radius $10 \,cm$ rolls without slipping with a speed $2 \,m / s$. The total kinetic energy of disc is .......... $J$

Two discs of moments of inertia $I_1$ and $I_2$ about their respective axes (normal to  the disc and passing through the centre), and rotating with angular speed $\omega _1$ and $\omega _2$ are  brought into contact face to face with their axes of rotation coincident. What is the  loss in kinetic energy of the system in the process ?

Point masses $m_1$ and $m_2$ are placed at the opposite ends of a rigid rod of length $L$, and negligible  mass. The rod is to be set rotating about an axis perpendicular to it. The position of point $P$ on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity $\omega_0$ is minimum, is given by

A thin hollow cylinder open at both ends:

$(i)$ Slides without rotating

$(ii)$ Rolls without slipping, with the same speed