$A$ rod of length $L$ and mass $M$ is acted on by two unequal forces $F_1$ and $F_2$ $(F_2 < F_1)$ as shown in the figure. The tension in the rod at a distance $y$ from the end $A$ is given by:

$A$ mass of $2 \, kg$ is suspended with a thread $AB$ as shown in the figure. $A$ thread $CD$ of the same type is attached to the other end of the $2 \, kg$ mass. The lower thread is pulled gradually,harder and harder in the downward direction,so as to apply force on $AB$. Which of the threads will break and why?

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 $\left( \frac{\text{Tension in the string between } P \text{ and } Q}{\text{Tension in the string between } O \text{ and } P} \right)?$

An elevator of mass $500 \,kg$ is ascending upwards with a constant acceleration $a=2 \,m/s^2$. What is the work done by the tension in the elevator cable during its climb by $12 \,m$ (in $\,kJ$)? (Take $g=10 \,m/s^2$)

$A$ small block of mass $m$ is placed on a wedge of mass $M$ as shown,which is initially at rest. All the surfaces are frictionless. The spring attached to the other end of the wedge has a force constant $k$. If $a'$ is the acceleration of $m$ relative to the wedge as it starts coming down and $A$ is the acceleration acquired by the wedge as the block starts coming down,then the maximum retardation of $M$ is:

$A$ rope of mass $5 \,kg$ is moving vertically in a vertical position with an upward force of $100 \,N$ acting at the upper end and a downward force of $70 \,N$ acting at the lower end. The tension at the midpoint of the rope is ......... $N$.