$A$ uniform disc of radius $R$ is placed on a smooth horizontal surface. $A$ particle of mass $m$ is moving with velocity $v$ on the surface and hits the disc at the periphery and sticks to it. The angular velocity of the system after the collision is:

$A$ boy runs and sits on a rotating platform. Which of the following quantities will be conserved?

$A$ disc of mass $M$ and radius $R$ is rolling with angular speed $\omega$ on a horizontal plane as shown. The magnitude of angular momentum of the disc about the origin $O$ is

$A$ uniform disc of mass $100 \ kg$ and radius $2 \ m$ is rotating at $1 \ rad/s$ about a perpendicular axis passing through its centre. $A$ boy of mass $60 \ kg$ standing at the centre of the disc suddenly jumps to a point which is $1 \ m$ from the centre of the disc. The final angular velocity of the boy (in $rad/s$) is

When a dancer pulls their arms inward while spinning on ice,they spin faster. This is due to:

$A$ small object is attached to a light string which passes through a hollow tube. The tube is held by one hand and the string by the other. The object is set into rotation in a circle of radius $r_1$. The string is then pulled down,shortening the radius of the circle to $r_2$. The ratio of the new kinetic energy to the original kinetic energy is