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 ............. .

$A$ ring of mass $M$ and radius $R$ sliding with a velocity $v_0$ suddenly enters a rough surface where the coefficient of friction is $\mu$,as shown in the figure. Choose the correct statement$(s)$.

$A$ disc of radius $R$ is rolling purely on a flat horizontal surface with a constant angular velocity $\omega$. The angle between the velocity and acceleration vectors of point $P$ (which is at the same horizontal level as the center $C$) is

$A$ thin uniform straight rod of mass $2 \, kg$ and length $1 \, m$ is free to rotate about its upper end when at rest. It receives an impulsive blow of $10 \, Ns$ at its lowest point,normal to its length as shown in the figure. The kinetic energy of the rod just after impact is ........ $J$.

Two masses of $0.3 \, kg$ and $0.7 \, kg$ are attached to the ends of a rod of length $1.4 \, m$ and negligible mass. The rod is rotated about an axis perpendicular to its length with a constant angular speed. The point on the rod through which the axis should pass so that the work required to rotate the rod is minimum is:

$A$ thin circular coin of mass $5 \text{ g}$ and radius $4/3 \text{ cm}$ is initially in a horizontal $xy$-plane. The coin is tossed vertically up ($+z$ direction) by applying an impulse of $J = \sqrt{\frac{\pi}{2}} \times 10^{-2} \text{ N-s}$ at a distance $r = 2/3 \text{ cm}$ from its center. The coin spins about its diameter and moves along the $+z$ direction. By the time the coin reaches back to its initial position,it completes $n$ rotations. The value of $n$ is. . . . . [Given: The acceleration due to gravity $g = 10 \text{ m/s}^2$]