Question

A steel wire of cross-sectional area $0.5m{m}^2$ is held between two fixed supports. If the wire is just taut at ${20}^{\circ }C$, determine the tension when the temperature falls to $0^{\circ }C$. Coefficient of linear expansion of steel is $1.2\times {10}^{-5}{}^{\circ }{C}^{-1}$ and its Young's modulus is $2.0\times {10}^{11}N{m}^{-2}$

Answer 1

Given $A=0.5m{m}^2=0.5\times {10}^{-6}{m}^2$

$T_1={20}^{\circ },T_2=0^{\circ }C$

${\alpha }_s=1.2\times {10}^{-5}/{}^{\circ }C$

$\gamma =2\times {10}^{11}n/m^2$

Decrease in length due to compression

$L\alpha \Delta \theta$ . . .(1)

$\gamma =\frac{\text{stress}}{strain}=\frac{F}{A}\times \frac{L}{\Delta L}$

$\Rightarrow \Delta L=\frac{FL}{AY}$

Tension is developed due to (1) and (2). Equating (1) and (2).

$L\alpha \Delta \theta =\frac{FL}{AY}$

$\Rightarrow F=\alpha \Delta \theta AY$

$=1.2\times {10}^{-5}\times (20-0)\times 0.5\times {10}^{-6}\times 2\times {10}^{11}$

$=1.2\times 20=24N$