A steel wire of cross-sectional area 0.5 mm^{2}^{ }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}.

#### Solution

Given:

Cross-sectional area of the steel wire, A = 0.5 mm^{2} = 0.5 × 10^{–6} m^{2}

The wire is taut at a temperature, T_{1}_{ }= 20 °C,

After this, the temperature is reduced to T_{2} = 0 °C

So, the change in temperature, Δθ = T_{1-}T_{2}_{ }= 20 °C

Coefficient of linear expansion of steel, α = 1.2 ×10^{–5} °C^{-1}

Young's modulus, γ = 2 ×10^{11} 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 ×10^{11}

= 1.2 × 20

⇒ F = 24 N

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