$A$ thin uniform rod of mass $M$ and length $L$ is rotating about a perpendicular axis passing through its centre with a constant angular velocity $\omega$. Two objects,each of mass $\frac{M}{3}$,are attached gently to the two ends of the rod. The rod will now rotate with an angular velocity of

The total angular momentum of all particles of a system is conserved if:

$A$ thin circular ring of mass $M$ and radius $r$ is rotating about its axis with a constant angular velocity $\omega$. Two objects,each of mass $m$,are attached gently to the opposite ends of a diameter of the ring. The ring will now rotate with an angular velocity:

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

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?

$A$ uniform rod of mass $M$ and length $L$ is suspended from a rigid support. $A$ small bullet of mass $m$ hits the rod with velocity $v$ and gets embedded into the rod. The angular velocity of the system just after impact is