Question

A steel wire of cross sectional area $0.5m{m}^2$ is held between two fixed support. If tension in the wire is negligible and it is just taut at ${20}^{o}C$, determine tension when temperature falls $0^{o}C$. Young's modulus of elasticity is $21\times {10}^{11}dyn/{cm}^{-2}$ and coefficient of linear expansion is $12\times {10}^{-6}$ per degree centigrade. Assume that the distance between the supports remains the same.

Answer 1

Given : $Y=21\times {10}^{11}dyn/c{m}^2$ $\alpha =12\times {10}^{-6}{/}^{o}C$

Change in temperature $\Delta T={20}^{o}C$

Cross-section area of the wire $A=0.5m{m}^2=0.005c{m}^2$

Using $\Delta L=L\alpha \Delta T$

$⟹$ Strain $=\frac{\Delta L}{L}=\alpha \Delta T=12\times {10}^{-6}\times 20=2.4\times {10}^{-4}$

From Hooke's law, $Stress=Y\times Strain$

$⟹$ Tension $T=Y\times Strain\times A$

$\therefore$ $T=21\times {10}^{11}\times 2.4\times {10}^{-4}\times 0.005=252\times {10}^{4}dyn$

$⟹$ $T=25.2N$ $(1N={10}^{5}dyn)$