Question

A steel wire of cross-sectional area 0.5 mm² is held between two fixed supports. If the wire is just taut at 20°C, determine the tension when the temperature falls to 0°C. Coefficient of linear expansion of steel is 1.2 × 10^–5 °C^–1 and its Young's modulus is 2.0 × 10^–11 Nm^–2.

Answer 1

Given:
Cross-sectional area of the steel wire, A = 0.5 mm² = 0.5 × 10^–6 m²
The wire is taut at a temperature, T₁ = 20 °C,
After this, the temperature is reduced to T₂ = 0 °C
​So, the change in temperature, Δθ = T₁$-$T₂ = 20 °C
Coefficient of linear expansion of steel, α = 1.2 ×10^–5 °C^​-1
Young's modulus, γ = 2 ×10¹¹ Nm​^$-2$
Let L be the initial length of the steel wire and L' be the length of the steel wire when temperature is reduced to 0°C.
Decrease in length due to compression, ΔL = L'$-$L= LαΔθ ...(1)
Let the tension applied be F.

^{$$\begin{gathered} \gamma =\frac{\mathrm{stress}}{\mathrm{strain}}=\raisebox{1ex}{$\left(\frac{F}{A}\right)$}\!\left/ \!\raisebox{-1ex}{$\left({\displaystyle \frac{\Delta L}{L}}\right)$}\right. \\ \Rightarrow \gamma =\frac{F}{A}\times \frac{L}{\Delta L} \\ \Rightarrow \Delta L=\frac{FL}{AY}...\left(2\right) \end{gathered}$$}
Change in length due to tension produced is given by (1) and (2).
So, on equating (1) and (2), we get:
$$\begin{gathered} L\alpha \Delta \theta =\frac{FL}{AY} \\ \Rightarrow F=\alpha \Delta \theta AY \\ =1.2\times {10}^{-5}\times \left(20-0\right)\times 0.5\times {10}^{-6}\times 2\times {10}^{11} \\ =1.2\times 20 \\ \Rightarrow F=24\mathrm{N} \end{gathered}$$
Therefore, the tension produced when the temperature falls to 0°C is 24 N.