Question

If the ratio of lengths, radii and Young's moduli of steel and brass wires in the figure are $a,b$ and $c$ respectively, then the corresponding ratio of increase in their lengths is :

• A. $\frac{3c}{2a{b}^2}$
• B. $\frac{2a^{2}c}{b}$
• C. $\frac{3a}{2b^{2}c}$
• D. $\frac{2ac}{b^2}$

Answer 1

Answer: (C)

According to questions,
$\frac{{\ell }_s}{{\ell }_b}=a,\frac{r_s}{r_b}=b,\frac{Y_s}{Y_b}=\frac{\Delta {\ell }_s}{\Delta {\ell }_b}=?$
As, $Y=\frac{F\ell }{A\Delta \ell }\Rightarrow \Delta \ell =\frac{F\ell }{AY}$
$\Delta {\ell }_s=\frac{3mg{\ell }_s}{\pi r_s^2.Y_s}\left[\because F_s=\left(M+2M\right)g\right]$
$\Delta {\ell }_b=\frac{2Mg{\ell }_b}{\pi r_b^2.Y_b}\left[\because F_b=2Mg\right]$
$\therefore \frac{\Delta {\ell }_s}{\Delta {\ell }_b}=\frac{\frac{3Mg{\ell }_s}{\pi r_s^2.Y_s}}{\frac{2Mg{\ell }_b}{\pi r_b^2.Y_b}}=\frac{3a}{2b^{2}C}$