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 ........

Two particles $A$ and $B$ are connected by rigid rod $A B$. The rod slides along perpendicular rails as shown here. The velocity of $A$ to the left is $10\; m / s$. What is the velocity of $B$(in $m/s$) when angle $\alpha=60^{\circ}$?

In the arrangement shown in fig. the ends $P$ and $Q$ of an unstretchable string move downwards with uniform speed $U$. Pulleys $A$ and $B$ are fixed. Mass $M$ moves upwards with a speed.

A balloon of mass $m$ is descending down with an acceleration $\frac{g}{2}$.  How much mass should be removed from it so that it starts moving up with same acceleration?

Find velocity of block ' $B$ ' at the instant shown in figure $........\,m/s$

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