Question

Two wires $A$ and $B$ of same material have radii in the ratio $2:1$ and lengths in the ratio $4:1$. The ratio of the normal forces required to produce the same change in the lengths of these two wires is:

• A. $1:1$
• B. $2:1$
• C. $1:2$
• D. $1:4$

Answer 1

The correct option is D $1:4$

$\begin{array}{c}Stress=\frac{F}{a}\\ a=\pi r^2\\ a\alpha r^2\\ \frac{a_1}{a_2}={\left(\frac{r_1}{r_2}\right)}^2={\left(\frac{2}{1}\right)}^2=4\\ \frac{S_1}{S_2}=\frac{\frac{F}{a_1}}{\frac{F}{a_2}}=\left(\frac{a_2}{a_1}\right)=\frac{1}{4}\\ Ratioofstressis1:4\end{array}$

hence,

option $\left(D\right)$ is correct answer.