Explain with the illustration of cranes the applications of the elastic behaviour of materials.

(A) In all engineering designs,the elastic behaviour of materials plays an important role. Let us consider the illustration of cranes for this.
Cranes used for lifting and moving heavy loads from one place to another have a thick metal rope to which the load is attached,and thus the rope (cable) is under stress.
Suppose we want to design a crane with a lifting capacity of $10$ tonnes or metric tons ($1$ metric ton $= 1000 \text{ kg}$). How thick should the steel rope be?
To ensure the load does not deform the rope permanently,the extension should not exceed the elastic limit.
This means the stress produced in the rope must be less than the yield strength $(S_y)$. Suppose the minimum cross-sectional area of the mild steel rope is $A$ and the yield strength of mild steel $(S_y)$ is $300 \times 10^6 \text{ N m}^{-2}$.
$\therefore A \geq \frac{W}{S_y}$
$= \frac{Mg}{S_y}$
$= \frac{10^4 \text{ kg} \times 10 \text{ m s}^{-2}}{300 \times 10^6 \text{ N m}^{-2}}$
$= 3.3 \times 10^{-4} \text{ m}^2$
$\therefore A \geq 3.3 \times 10^{-4} \text{ m}^2$
If $g = 3.1 \pi \text{ m s}^{-2}$ and $A = \pi r^2$,then the radius $r$ can be calculated to determine the thickness of the rope.