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. $\frac{3c}{2ab^{2}}$
• 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, $Y = \frac{F\ell }{A\Delta\ell} \Rightarrow \Delta\ell = \frac{F\ell}{AY}$
$\Delta \ell_{s} = \frac{3mg\ell_{s}}{\pi r_{s}^{2}. Y_{s}}\left[\because F_{s}=\left(M + 2M\right)g \right]$
$\Delta \ell_{b} = \frac{2Mg\ell _{b}}{\pi r_{b}^{2}. Y_{b}}\left[\because F_{b}=2Mg \right]$
$\therefore \frac{\Delta \ell _{s}}{\Delta \ell _{b}} = \frac{\frac{3Mg\ell_{s}}{\pi r_{s}^{2}. Y_{s}}}{\frac{2Mg\ell _{b}}{\pi r_{b}^{2}. Y_{b}}} = \frac{3a}{2b^{2}C}$