$A$ block of mass $M$ on a horizontal smooth surface is pulled by a load of mass $M/2$ by means of a rope $AB$ and string $BC$ as shown in the figure. The length and mass of the rope $AB$ are $L$ and $M/2$ respectively. As the block is pulled from $AB = L$ to $AB = 0$,its acceleration changes from:

$A$ mass $1\, kg$ is suspended by a thread. It is
$(i)$ lifted up with an acceleration $4.9\, m/s^2$
$(ii)$ lowered with an acceleration $4.9\, m/s^2$.
The ratio of the tensions is

Two blocks of $2 \,kg$ and $1 \,kg$ are in contact on a frictionless table. If a force of $3 \,N$ is applied on the $2 \,kg$ block, then the force of contact between the two blocks will be: (in $\,N$)

The minimum acceleration with which a fireman can slide down a rope of breaking strength two-third of his weight is

Two masses of $4\, kg$ and $5 \,kg$ are connected by a string passing through a frictionless pulley and are kept on a frictionless table as shown in the figure. The acceleration of the $5\, kg$ mass is ........... $m/s^2$.

Two spheres $P$ and $Q$ of equal masses $m$ are attached to a string of total length $2\, m$ as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular velocity $\omega$. The distance $OP = 1\, m$ and $PQ = 1\, m$. What is the value of the ratio $\left( \frac{\text{Tension in the string between } P \text{ and } Q}{\text{Tension in the string between } O \text{ and } P} \right)?$