Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
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A Steel Wire and a Copper Wire of Equal Length and Equal Cross-sectional Area Are Joined End to End and the Combination is Subjected to a Tension. Find the Ratio of the Strains Developed. - Physics

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Short Note

A steel wire and a copper wire of equal length and equal cross-sectional area are joined end to end and the combination is subjected to a tension. Find the ratio of the strains developed. Y of steel = 2 × 1011N m−2. Y of copper = 1.3 × 10 11 N m−2

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Young's modulus of steel = 2 × 1011 N m−2
Young's modulus of copper = 1.3 × 10 11 N m−2
Both wires are of equal length and equal cross-sectional area. Also, equal tension is applied on them.
 As per the question :

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

Here: Lsteel and LCu denote the lengths of steel and copper wires, respectively.
           Asteel and ACu denote the cross-sectional areas of steel and copper wires, respectively.
           Fsteel and FCu  denote the tension of steel and cooper wires, respectively.

\[\frac{\text{Strain of Cu}}{\text{Strain of steel}} = \frac{\frac{∆ L_{\text{Steel}}}{L_{\text{Steel}}}}{\frac{∆ L_{\text{cu}}}{L_{\text{cu}}}} = \frac{F_{\text{Steel}} L_{\text{Steel}} A_{\text{cu}} Y_{\text{cu}}}{A_{\text{Steel}} Y_{\text{Steel}} F_{\text{cu}} L_{\text{cu}}}\]
                                 \[ \left( \text{Using } \frac{∆ L}{L} = \frac{F}{AY} \right)\]

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

\[ \Rightarrow \frac{\text{ Strain of Cu} }{ \text{ Strain of steel }} = \frac{13}{20}\]

\[ \Rightarrow \frac{\text{ Strain of steel }}{\text{Strain of Cu }} = \frac{20}{13}\]

Hence, the required ratio is 20 : 13.

Concept: Elastic Moduli - Determination of Young’s Modulus of the Material of a Wire
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HC Verma Class 11, Class 12 Concepts of Physics Vol. 1
Chapter 14 Some Mechanical Properties of Matter
Q 4.2 | Page 300
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