$A$ 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 the same radius but of mass $\frac{M}{3}$ is placed gently on the first disc co-axially. The new angular velocity will be:

Consider a disc of mass $5 \,kg$ and radius $2 \,m$, rotating with an angular velocity of $10 \,rad/s$ about an axis perpendicular to the plane of rotation and passing through its center. An identical disc is placed gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate together without slipping is . . . . . . $J$.

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?

$A$ person is sitting on a rotating stool with their arms outstretched. Suddenly, they fold their arms inward.

$(a)$ $A$ child stands at the centre of a turntable with his two arms outstretched. The turntable is set rotating with an angular speed of $40\; rev/min$. How much is the angular speed of the child if he folds his hands back and thereby reduces his moment of inertia to $2/5$ times the initial value? Assume that the turntable rotates without friction.
$(b)$ Show that the child's new kinetic energy of rotation is more than the initial kinetic energy of rotation. How do you account for this increase in kinetic energy?

$A$ man standing on a turn-table is rotating at a certain angular frequency with his arms outstretched. He suddenly folds his arms. If his moment of inertia with folded arms is $75 \%$ of that with outstretched arms,then his rotational kinetic energy will