An object of mass $m$ is tied to a thin string,which is passed through a hollow tube. The tube is held in one hand and the string in the other. The object is rotated in a circle of radius $R$ with a velocity $v$. If the string is pulled downwards to decrease the radius,which of the following quantities remains conserved?

$(a)$ $A$ 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 is standing on the edge of a circular platform. The platform is rotating with a constant angular velocity about an axis passing through its center and perpendicular to its plane. If the man walks along the radius towards the axis, the angular velocity will ........ .

$Assertion$ : There are very small sporadic changes in the speed of rotation of the earth.
$Reason$ : Shifting of large air masses in the earth's atmosphere produces a change in the moment of inertia of the earth, causing its speed of rotation to change.

$A$ dancer on a smooth floor is spinning about a vertical axis at an angular velocity of $20 \ rad/s$ with her arms folded. When she stretches her arms out,the angular speed decreases to $10 \ rad/s$. If the initial moment of inertia of the dancer is $I$,what will be the new moment of inertia?

If the radius of a rotating disc is suddenly halved while its mass remains the same,its angular velocity will become: