$A$ thin uniform bar of length $L$ and mass $8m$ lies on a smooth horizontal table. Two point masses $m$ and $2m$ move in the same horizontal plane from opposite sides of the bar with speeds $2v$ and $v$ respectively. The masses stick to the bar after collision at distances $L/3$ and $L/6$ respectively from the center of the bar. If the bar starts rotating about its center of mass as a result of the collision,the angular speed of the bar will be

$A$ uniform cube of mass $m$ and side $a$ is placed on a frictionless horizontal surface. $A$ vertical force $F$ is applied to the edge as shown in the figure. Match the following (most appropriate choice):

$(a)$ $\frac{mg}{4} < F < \frac{mg}{2}$	$(i)$ Cube will move up
$(b)$ $F > \frac{mg}{2}$	$(ii)$ Cube will not exhibit motion
$(c)$ $F > mg$	$(iii)$ Cube will begin to rotate about $A$
$(d)$ $F = \frac{mg}{4}$	$(iv)$ Normal reaction effectively at $a/3$ from $A$,no motion

$A$ metal rod of length $L$ and mass $m$ is pivoted at one end. $A$ thin disk of mass $M$ and radius $R$ $(R < L)$ is attached at its center to the free end of the rod. Consider two ways the disc is attached: (case $A$) The disc is not free to rotate about its center and (case $B$) the disc is free to rotate about its center. The rod-disc system performs $SHM$ in a vertical plane after being released from the same displaced position. Which of the following statement$(s)$ is (are) true?

Statement-$1$: $A$ body is rotating about an axis with angular velocity $\omega$ and moment of inertia $I$. Its angular momentum $L$ remains constant,but its rotational kinetic energy $K$ decreases,provided no external torque is applied.
Statement-$2$: $L = I\omega$ and $K = \frac{L^2}{2I} = \frac{1}{2} I\omega^2$.

$A$ ladder is leaned against a smooth wall and it is allowed to slip on a frictionless floor. Which figure represents the track of its centre of mass?

$A$ rod $AB$ is free to rotate in a vertical plane about a horizontal axis through $A$ as shown in the figure. It is slightly disturbed from rest in its position of unstable equilibrium and when it is next vertical,the end $B$ collides with a fixed peg and rebounds. If the rod comes to instantaneous rest when $AB$ is horizontal (as shown in the figure),then: