Question

A steel rod is rigidly clamped at its two ends. The rod is under zero tension at 20°C. If the temperature rises to 100°C, what force will the rod exert on one of the clamps? Area of cross-section of the rod is 2.00 mm². Coefficient of linear expansion of steel is 12.0 × 10^–6 °C^–1 and Young's modulus of steel is 2.00 × 10¹¹ Nm^–2.

Answer 1

Given:
Temperature of the rod at zero tension, T₁ = 20 °C
Final temperature, T₂ = 100 °C
​Change in temperature, $\Delta \theta$ = 80 °C
Cross-sectional area of the rod, A = 2 mm² = 2 × 10^$-$6 m²
Coefficient of linear expansion of steel, α = 12 ×10^–6 °C^​$-$1
Young's modulus of steel, Y = 2 × 10¹¹ Nm^$-2$
Let L be the length of the steel rod at 20 °C and L' be the length of steel rod at 100 °C.
Change of length of the rod, $∆L$ = L'$-$ L
If F be the force exerted by the rod on one of the clamps due to rise in temperature, then

$$\begin{gathered} Y=\frac{\mathrm{stress}}{\mathrm{strain}}=\frac{F/A}{{\displaystyle \raisebox{1ex}{$\Delta L$}\!\left/ \!\raisebox{-1ex}{$L$}\right.}} \\ \Rightarrow F=\frac{Y\times \Delta L}{L}\times A \\ \Delta L=L\alpha \Delta \theta \\ \Rightarrow F=\frac{YL\alpha \Delta \theta A}{L} \\ \Rightarrow F=YA\alpha \Delta \theta \\ =2\times {10}^{11}\times 2\times {10}^{-6}\times 12\times {10}^{-6}\times 80 \\ =48\times 80\times {10}^{-1} \\ \mathrm{So},F=384\mathrm{N} \end{gathered}$$
Therefore, the rod will exert a force of 384 N on one of the clamps.