In the above problem,the angular velocity of the system after the particle sticks to it will be ....... $rad/s$. (in $.3$)

$A$ thin uniform circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with an angular velocity $\omega$. Another disc of same dimension but of mass $M/4$ is placed gently on the first disc coaxially. The angular velocity of the system now is

If the earth were to suddenly contract to $1/n^{th}$ of its present radius without any change in its mass,the duration (in $hrs.$) of the new day will be nearly

$A$ particle is moving in a uniform circular motion. At which point in the plane of the circle will the angular momentum of the particle be conserved?

$A$ small bead of mass $m$ moving with velocity $v$ gets threaded on a stationary semicircular ring of mass $m$ and radius $R$ kept on a horizontal table. The ring can freely rotate about its centre. The bead comes to rest relative to the ring. What will be the final angular velocity of the system?

An equilateral triangle $ABC$ formed from a uniform wire has two small identical beads initially located at $A$. The triangle is set rotating about the vertical axis $AO$. Then the beads are released from rest simultaneously and allowed to slide down,one along $AB$ and the other along $AC$ as shown. Neglecting frictional effects,the quantities that are conserved as the beads slide down,are