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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.

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Solution

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

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