Question

In figure the upper wire is made of steel and the lower of copper. The wires have equal cross section. Find the ratio of the longitudinal strains developed in the two wires.

Answer 1

Given that both wires are of equal length and equal cross-sectional area,
the block applies equal tension on both of them.

∴  \[L_{\text{ steel} } = L_{\text{ Cu }} \]
\[ A_{\text{ steel}} = A_{\text{ Cu }} \]
\[ F_{\text{ Cu}} = F_{\text{ Steel }} \]

\[\frac{\text{ Strain of Cu }}{\text{ Strain of steel }} = \frac{\frac{∆ L_{\text{ Steel }}}{L_{\text{ Steel }}}}{\frac{∆ L_{\text{cu}}}{L_{\text{cu}}}} = \frac{F_{\text{Steel}} L_{\text{Steel}} A_{\text{cu}} Y_{\text{cu}}}{A_{\text{Steel}} Y_{\text{ Steel }} F_{\text{ cu } } L_{\text{ cu}}}\]

\[ \left( \text{ Using }\frac{∆ L}{L} = \frac{F}{AY} \right)\]

\[ \Rightarrow \frac{\text{ Strain of Cu }}{\text{ Strain of steel }} = \frac{Y_{\text{ cu }}}{Y_{\text{ Steel }}} = \frac{1 . 3 \times {10}^{11}}{2 \times {10}^{11}}\]
\[ \Rightarrow \frac{\text{ Strain of steel }}{\text{Strain of Cu }} = \frac{20}{13} = 1 . 54\]

Hence, the required ratio of the longitudinal strains is 20 : 13.