A thin uniform circular disc of mass ' $M$ ' and radius ' $R$ ' is rotating with angular velocity ' $\omega$ ' in a horizontal plane about an axis passing through its centre and perpendicular to its plane. Another disc of same radius but of mass $\left(\frac{\mathrm{M}}{3}\right)$ is placed gently on the first disc co-axially. The new angular velocity will be

A solid cylinder of mass ' $M$ ' and radius ' $R$ ' rolls down an inclined plane of height ' $h$ '. When it reaches the foot of the plane, its rotational kinetic energy is ( $\mathrm{g}=$ acceleration due to gravity)

A disc and a ring both have same mass and radius. The ratio of moment of inertia of the disc about its diameter to that of a ring about a tangent in its plane is

A rotating body has angular momentum ' $L$ '. If its frequency is doubled and kinetic energy is halved, its angular momentum will be