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 = T_1-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.

`γ = "stress"/"strain" =("F"/"A")/((triangle"L")/"L")` 

`=> γ = "F"/"A" xx "L"/(triangle"L")`

`=> triangle"L" = "FL"/("A"gamma)`     ..(2)

Change in length due to tension produced is given by (1) and (2).
So, on equating (1) and (2), we get:

`"L"αΔθ ="FL"/("A"gamma)`

⇒ F = αΔθAγ

= 1.2 × 10^-5 × (20-0) × 0.5 ×10^-6 × 2 ×10¹¹

= 1.2 × 20

⇒ F = 24 N

Therefore, the tension produced when the temperature falls to 0°C