Question

Two separate wires A and B are stretched by 2 mm and 4 mm respectively, when they are subjected to a force of 2 N. Assume that both the wires are made up of same material and the radius of wire B is 4 times that of the radius of wire A. The length of the wires A and B are in the ratio of a : b. Then a/b can be expressed as 1/x where x is __________.

Hint:

$L \propto r^2 \Delta L$ when force and material are constant.

Answer 1

Correct Answer: 32

Step 1: Young's Modulus $Y = \frac{F/A}{\Delta L / L} = \frac{FL}{A \Delta L}$.
Step 2: Since same material, $Y_A = Y_B \implies \frac{F L_A}{\pi r_A^2 \Delta L_A} = \frac{F L_B}{\pi r_B^2 \Delta L_B}$.
Step 3: $\frac{L_A}{L_B} = \frac{r_A^2 \Delta L_A}{r_B^2 \Delta L_B} = (\frac{r_A}{4r_A})^2 \times (\frac{2}{4}) = \frac{1}{16} \times \frac{1}{2} = \frac{1}{32}$... Let's re-calculate. Wait, $\frac{L_A}{L_B} = \frac{r_A^2 \Delta L_A}{r_B^2 \Delta L_B} = \frac{r^2 \cdot 2}{(4r)^2 \cdot 4} = \frac{2}{16 \cdot 4} = \frac{2}{64} = \frac{1}{32}$. If $x=8$ is the target, check if $r_B=2r_A$. For $r_B=4r_A$, $x=32$.