Choose the correct statement.

$A$ ring and a disc have the same mass and the same radius. The ratio of the moment of inertia of a ring about a tangent in its plane to that of the disc about its diameter is: (in $: 1$)

$A$ uniform rod of mass $M$ is hinged at its upper end. $A$ particle of mass $m$ moving horizontally strikes the rod at its mid-point elastically. If the particle comes to rest after the collision,find the value of $M/m$.

Fill in the blanks:
$(1)$ In rotational motion,the role played by ............ is analogous to the role played by mass in linear motion.
$(2)$ For a rigid body in rotational motion,if a particle at a distance of $10 \ cm$ from the fixed axis of rotation has an angular velocity of $10 \ rad/s$,then the linear velocity of a particle at a distance of $5 \ cm$ from the axis of rotation is ............
$(3)$ The $SI$ unit $J \cdot s^{-2}$ is the unit of the physical quantity ............
$(4)$ The condition for a body to roll without slipping down an inclined plane with friction is ............

$A$ particle is moving in a uniform circular motion with angular momentum $L$. If its angular frequency is doubled and its kinetic energy is halved,then the new angular momentum of the particle will be .......

Fill in the blanks:
$(1)$ If the velocity of the center of mass of a body is $v_{cm} = 0$ and the angular velocity is $\omega = 0$,the body is said to be in ............. equilibrium.
$(2)$ Angular momentum is generated in a body when a ............. acts on it.
$(3)$ If a barrel is filled half with water,its center of gravity will move ............. .
$(4)$ The point at which the entire mass of a body is assumed to be concentrated is called the ............. .