Question

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

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

Answer 1

The correct option is B $\frac{3a}{2b^{2}c}$

Extension in a rod due to force $F$ is
$\Delta l=\frac{FL}{AY}$

Ratio of extension is steel and brass can be considered as
$\frac{\Delta l_s}{\Delta l_B}=\frac{F_s}{F_B}\times \frac{L_s}{L_B}\times \frac{A_B}{A_S}\times \frac{Y_B}{Y_S}$

From figure,
tension is steel wire is $3mg$
tension is brass wire is $2mg$
$\frac{\Delta l_s}{\Delta l_B}==\frac{3mg}{2mg}\times a\times \frac{1}{b^2}\times \frac{1}{c}=\frac{3a}{2b^{2}c}$
Hence, the correct answer is option (b).