Question

When a certain weight is suspended from a long uniform wire(massless), its length increases by $1\text{cm}$. If the same weight is suspended from another wire(massless) of the same material and length but having a diameter half of the first one, the increase in length will be (in $\text{cm}$)

Answer 1

Since, force, material and length are same thus, extension is dependent only on area and inversely proportional to it.
Young's modulus : $y=\frac{F/A}{\Delta L/L}$
$\Rightarrow \Delta L_1=1\text{cm},A_1=\pi r_1^2,A_2=\pi r_2^2$
$\Rightarrow \frac{F/A_1}{\Delta L_1/L}=\frac{F/A_2}{\Delta L_2/L}$
$\Rightarrow \frac{\Delta L_2}{\Delta L_1}=\frac{A_1}{A_2}=\frac{\pi r_1^2}{\pi r_2^2}$
$\Rightarrow \Delta L_2=\frac{r_1^2}{r_2^2}\times \Delta L_1$
$\Rightarrow \Delta L_2=\frac{r_1^2}{{\left(\frac{r}{2}\right)}^2}\times \Delta L_1$
$\Rightarrow \Delta L_2=4\times 1$
$\Rightarrow \Delta L_2=4\text{cm}$