$A$ uniform rod of mass $M$ and length $L$ is placed on a smooth horizontal surface. $A$ horizontal force $F$ is applied at one end of the rod at an angle $\theta$ with the rod. The acceleration of the centre of mass of the rod is:

$A$ piece of wood of mass $0.03\, kg$ is dropped from the top of a $100\, m$ height building. At the same time,a bullet of mass $0.02\, kg$ is fired vertically upward,with a velocity $100\, ms^{-1}$,from the ground. The bullet gets embedded in the wood. Then the maximum height to which the combined system reaches above the top of the building before falling below is ........ $m$. $(g = 10\, ms^{-2})$

Two blocks of masses $2 \text{ kg}$ and $1 \text{ kg}$ respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in $2 \text{ s}$ is . . . . . . $\text{m}$. (Take $g = 10 \text{ m/s}^2$)

$A$ balloon of mass $M$ with a light rope and a monkey of mass $m$ are at rest in mid-air. If the monkey climbs up the rope and reaches the top of the rope,the distance by which the balloon descends will be (Total length of the rope is $L$):

$A$ man of mass $M$ stands at one end of a plank of length $\ell$,which lies at rest on a frictionless surface. The man walks to the other end of the plank. If the mass of the plank is $3M$,what is the distance moved by the man relative to the ground?

Two men $A$ and $B$ are standing on a plank. $B$ is at the middle of the plank and $A$ is at the left end of the plank. The bottom surface of the plank is smooth. The system is initially at rest and the masses are as shown in the figure. $A$ and $B$ start moving such that the position of $B$ remains fixed with respect to the ground and $A$ meets $B$. Then the point where $A$ meets $B$ is located at