Question

The Young's module for brass and steel are, respectively, $1.0\times {10}^{10}N/m^2$ and $2\times {10}^{10}N/m^2$. A brass wire and a steel wire, both of the same length, are extended by $1$ mm under the same force, the radii of the brass and steel wires being $R_B$ and $R_S$ respectively. Then.

• A. $R_S=\sqrt{2}{R}_B$
• B. $R_S=\frac{R_B}{\sqrt{2}}$
• C. $R_S=4R_B$
• D. $R_S=\frac{R_B}{4}$

Answer 1

The correct option is B $R_S=\frac{R_B}{\sqrt{2}}$

Here,

$\Delta L=1,L,F$ are same for two wires,

$Y_S=2Y_B$

$\frac{Y_S}{Y_B}=\frac{F_S{L}_S}{A_S\Delta L_S}\times \frac{A_B\Delta L_B}{F_B{L}_B}$

$2=\frac{A_B}{A_S}=\frac{\pi R_B^2}{\pi R_S^2}$

$R_B=\sqrt{2}{R}_S$