$A$ uniform rod of length $2L$ has one end on a horizontal floor. It is inclined at an angle $\alpha$ to the horizontal floor. It falls without slipping,rotating about the point of contact. Its angular velocity when it hits the horizontal floor will be:

Two cylindrical hollow drums of radii $R$ and $2R$ and of a common height $h$ are rotating with angular velocities $\omega$ (anti-clockwise) and $\omega$ (clockwise) respectively. Their axes,which are fixed,are parallel and in a horizontal plane separated by $3R$. They are now brought into contact.
$(a)$ Show the frictional forces just after contact.
$(b)$ Identify forces and torques external to the system just after contact.
$(c)$ What would be the ratio of final angular velocities when friction ceases?

$A$ particle performs uniform circular motion with an angular momentum $L$. If the angular frequency of the particle is doubled and its kinetic energy is halved,what will be its new angular momentum?

One ice skater of mass $m$ moves with speed $2v$ to the right,while another of the same mass $m$ moves with speed $v$ toward the left,as shown in figure $I$. Their paths are separated by a distance $b$. At $t = 0$,when they are both at $x = 0$,they grasp a pole of length $b$ and negligible mass. For $t > 0$,consider the system as a rigid body of two masses $m$ separated by distance $b$,as shown in figure $II$. Which of the following is the correct formula for the motion after $t = 0$ of the skater initially at $y = b/2$?

Choose the correct statement.

State whether the following statements are true or false:
$(1)$ In the law of conservation of momentum, the center of mass velocity remains constant.
$(2)$ "If the resultant of internal forces on a system is zero, then the total linear momentum of the system remains constant." This is the statement of the law of conservation of linear momentum.
$(3)$ The position of the center of mass of a rigid body must always be inside the rigid body.
$(4)$ All particles of a rigid body undergoing rotational motion have the same linear velocity.