Question

Two bar $A$ and $B$ of circular cross section and of same volume made of the same material are subjected to tension. If the diameter of $A$ is half that of $B$ and if the force applied to both the rods is the same and it is within the elastic limit. What will be the ratio of extension of $A$ to that of $B$?

Answer 1

Given,$d_A=\frac{d_B}{2}$

Young modulus, $Y=\frac{\frac{F}{\pi r^2}}{\frac{△l}{l}}$

$\Rightarrow \frac{△l}{l}=\frac{\frac{F}{\pi r^2}}{Y}$

$\therefore \frac{{\left(\frac{△l}{l}\right)}_A}{{\left(\frac{△l}{l}\right)}_B}=\frac{r_B^2}{r_A^2}$

$\Rightarrow \frac{d_A^2}{d_B^2}=\frac{1}{4}$

$\therefore {\left(\frac{△l}{l}\right)}_A:{\left(\frac{△l}{l}\right)}_B=1:4$