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A steel rail of length $5\,m$ and area of cross-section $40\,cm^2$ is prevented from expanding along its length while the temperature rises by $10\,^oC$. If coefficient of linear expansion and Young's modulus of steel are $1.2\times10^{-5}\, K^{-1}$ and $2\times10^{11}\, Nm^{-2}$ respectively, the force developed in the rail is approximately
$2\times10^7\, N$
$1\times10^5\, N$
$2\times10^9\, N$
$3\times10^{-5}\, N$
Solution
$Young's\,modulus$
$ = \frac{{Thermal\,stress}}{{Strain}} = \frac{{F/A}}{{\Delta L/L}}$
$Y = \frac{F}{{A.\alpha .\Delta \theta }}$
$\left( {\frac{{\Delta L}}{L} = \alpha \,\Delta \,\theta } \right)$
Force developed in the rail $F = YA\,\alpha \,\Delta t$$ = 2 \times {10^{11}} \times 40 \times {10^{ – 4}} \times 1.2 \times {10^{ – 5}} \times 10$
$ = 9.6 \times {10^4} = 1 \times {10^5}N$
