Question

Two wires P and Q of same cross-sectional area $A=10m{m}^2$ and the same length $l_0=1$ m but made of different materials are joined together and their ends are rigidly clamp between two walls with negligible tension. Their respective Young's modulus and the coefficients of linear expansion are as $Y_1,{\alpha }_1$ and $Y_2,{\alpha }_2$ respectively.
Given $Y_1=2\times {10}^{9}N/m^2,{\alpha }_1=3\times {10}^{-4}{/}^{\circ }C,Y_2={10}^{9}N/m^2,{\alpha }_2=6\times {10}^{-4}{/}^{\circ }C$,

Displacement of joint O is

• A. 1 mm
• B. 0.5 mm
• C. 2 mm
• D. Zero

Answer 1

The correct option is D

Zero

Let displacement of joint O is towards right by x. Then
Wire $P:{\sigma }_P=\frac{F}{A}$

$=Y_1\frac{\Delta l_1}{l_0}=\frac{Y_1[l_0{\alpha }_1\Delta T+X]}{l_0}$

Wire $Q:{\sigma }_Q=\frac{F}{A}$

$=Y_2\frac{\Delta l_2}{l_0}=\frac{Y_2[l_0{\alpha }_2\Delta T-X]}{l_0}$

Comparing gives x = 0.