$A$ frictionless cart $A$ of mass $100 \ kg$ carries two other frictionless carts $B$ and $C$ having masses $8 \ kg$ and $4 \ kg$ respectively,connected by a string passing over a pulley as shown in the figure. What horizontal force $F$ must be applied on the cart $A$ so that the smaller carts do not move relative to it (in $N$)? (Take $g = 10 \ m/s^2$)

Find the velocity of block $B$ at the instant shown in the figure (in $m/s$).

The velocity of end $A$ of a rigid rod placed between two smooth perpendicular walls moves with velocity $u$ along the vertical direction. Find the velocity of end $B$ of the rod,given that the rod always remains in contact with the walls.

The machine as shown has $2$ rods of length $1\, m$ connected by a pivot at the top. The end of one rod is connected to the floor by a stationary pivot and the end of the other rod has a roller that rolls along the floor in a slot. As the roller moves back and forth,a $2\, kg$ weight moves up and down. If the roller is moving towards the right at a constant speed,the weight moves up with a:

If the blocks $A$ and $B$ are moving towards each other with accelerations $a$ and $b$ respectively,find the net acceleration of block $C$.

If block $A$ has a velocity of $0.6\,m/s$ to the right,determine the velocity of block $B$.