For the given fig. find the speed of block $A$ when $\theta  = {60^o}$

If acceleration of $A$ is $2 \,\,m/s^2$ to left and acceleration of $B$ is $1\,\,m/s^2$ to left, then acceleration of $C$ is

Figure shows a boy on a horizontal platform $A$ on a smooth horizontal surface, holding a rope attached to a box $B$ . Boy pulls the rope with a constant force of $50\ N$ . (boy does not slip over the platform). The combined mass of platform $A$ and boy is $250\ kg$ and that of box $B$ is $500\ kg$ . The velocity of $A$ relative to the box $B$ , $5\ s$ after the boy on $A$ begins to pull the rope, will be ............ $m/s$

A ladder rests against a frictionless vertical wall, with its upper end $6\,m$ above the ground and the lower end $4\,m$ away from the wall. The weight of the ladder is $500 \,N$ and its C. G. at $1/3^{rd}$ distance from the lower end. Wall's reaction will be, (in Newton)

In the figure shown the velocity of different blocks is shown. The velocity of $C$ is ......... $m/s$

A uniform metal chain of mass $m$ and length ' $L$ ' passes over a massless and frictionless pulley. It is released from rest with a part of its length ' $l$ ' is hanging on one side and rest of its length ' $L -l$ ' is hanging on the other side of the pulley. At a certain point of time, when $l=\frac{L}{x}$, the acceleration of the chain is $\frac{g}{2}$. The value of $x$ is ........