In a tensile test on a metal bar of diameter $0.015 \ m$ and length $0.2 \ m$, the relation between the load and elongation within the proportional limit is found to be $F = 97.2 \times 10^6 (\Delta L)$, where $F$ is the load in $N$ and $\Delta L$ is the elongation in $m$. The Young's modulus of the material in $GPa$ is:

$A$ steel wire and a copper wire are joined end to end having equal cross-sections. The elongation of the two wires is found to be equal under tension. What is the ratio of the length of the steel wire to the length of the copper wire? (Young's modulus of steel $= 2.0 \times 10^{11} \ N \ m^{-2}$ and Young's modulus of copper $= 1.1 \times 10^{11} \ N \ m^{-2}$)

$A$ uniform rod of length $L$ is rotated in a horizontal plane about a vertical axis through one of its ends. The angular speed of rotation is $\omega$. Find the increase in length of the rod, if $\rho$ and $Y$ are the density and Young's modulus of the rod respectively.

Two different wires made with the same material have their radii in the ratio $1:2$. Their lengths are also in the ratio $1:2$. If the extensions produced are equal when subjected to different loads,find the ratio of the loads applied.

There are two wires of the same material and same length. The diameter of the second wire is two times the diameter of the first wire. What is the ratio of the extensions produced in the wires by applying the same load?

Consider the situation shown in the figure. The force $F$ is equal to $m_2g/2$. If the area of cross-section of the string is $A$ and its Young's modulus is $Y$,find the strain developed in it. The string is light and there is no friction anywhere.