Question

A steel rod is rigidly clamped at its two ends. The rod is under zero tension at ${20}^{\circ }C$. If the temperature rises to ${100}^{\circ }C$, what force will the rod exert on one of the clamps? Area of cross section of the rod = $2.0m{m}^2$. Coefficient of linear expansion of steel $=12.0\times {10}^{-6}{}^{\circ }{C}^{-1}$ and Young's modulus of steel = $2.00\times {10}^{11}N{m}^{-2}$.

Answer 1

${\theta }_1={20}^{\circ }C,{\theta }_2={100}^{\circ }C$

$A=2m{m}^2=2\times {10}^{-6}{m}^2$

${\alpha }_{steel}=12\times {10}^{-6}{/}^{\circ }C$

$Y_{steel}=2\times {10}^{11}N/m^2$

Force exerted on the clamps = ?

$\frac{\frac{F}{A}}{\text{strain}}=Y$

$F=\frac{Y\times \Delta L}{L}\times A$

$\frac{YL\alpha \Delta \theta A}{L}=YA\alpha \Delta \theta$

$=2\times {10}^{11}\times 2\times {10}^{-6}\times 12\times {10}^{-6}\times 80$

$=48\times 80\times {10}^{-1}=384N$