$A$ hot solid sphere is rotating about a diameter at an angular velocity $\omega_0$. If it cools so that its radius reduces to $\frac{1}{\eta}$ of its original value,its angular velocity becomes .............

$A$ smooth tube of certain mass closed at both ends is rotated in a gravity-free space and released. The two balls shown in the figure move towards the ends of the tube and stay there. Which statement is incorrect about this whole system?

$A$ child with mass $m$ is standing at the edge of a disc with moment of inertia $I$,radius $R$,and initial angular velocity $\omega$. See the figure given below. The child jumps off the edge of the disc with tangential velocity $v$ with respect to the ground. The new angular velocity of the disc is

$A$ disc of mass $1 \, kg$ and radius $0.1 \, m$ is rotating with angular velocity $20 \, rad/s$. What is the angular velocity (in $rad/s$) if a mass of $0.5 \, kg$ is placed on the circumference of the disc?

$A$ 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

Two identical beads are placed at vertex $A$ of an equilateral triangle $ABC$ made of a uniform wire. The triangle is rotated about the axis $AO$. The beads are then released from rest and move along $AB$ and $AC$ respectively (see figure). Neglecting friction,which of the following quantities will be conserved during the downward motion of the beads?