In a playground,there is a merry-go-round of mass $120\ kg$ and radius $4\ m$. The radius of gyration is $3\ m$. $A$ child of mass $30\ kg$ runs at a speed of $5\ m/s$ tangent to the rim of the merry-go-round when it is at rest and then jumps on it. Neglecting friction,find the angular velocity of the merry-go-round and child in $rad/s$.

An equilateral triangle $ABC$ is made of a uniform wire. Two identical beads are initially at $A$. The triangle is set to rotate about a vertical axis $AO$. The beads are then released simultaneously from rest to slide down along $AB$ and $AC$. Neglecting friction,which quantities are conserved as the beads slide down?

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

$A$ wheel is rotating freely with an angular speed $\omega$ on a shaft. The moment of inertia of the wheel is $I$ and the moment of inertia of the shaft is negligible. Another wheel of moment of inertia $3I$ initially at rest is suddenly coupled to the same shaft. The resultant fractional loss in the kinetic energy of the system is:

$A$ circular stage is free to rotate about a vertical axis passing through its centre. $A$ tortoise is sitting at the corner of the stage. The stage is provided with an initial angular velocity $\omega_0$. If the tortoise starts moving along one chord at a constant speed with respect to the stage,then how does the angular velocity of the stage $\omega(t)$ vary with time $t$?

$A$ horizontal disc rotating freely about a vertical axis through its centre makes $90$ revolutions per minute. $A$ small piece of wax of mass $m$ falls vertically on the disc and sticks to it at a distance $r$ from the axis. If the number of revolutions per minute reduces to $60$,then the moment of inertia of the disc is .........