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(B) The thermal expansion of the rod if it were free to expand is given by $\Delta L = \alpha L T$.To prevent this expansion,a compressive force $F$ m

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$\frac{Y A \alpha T}{(1+\alpha T)}$

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(B) The thermal expansion of the rod if it were free to expand is given by $\Delta L = \alpha L T$.To prevent this expansion,a compressive force $F$ must be applied.The stress produced is $\sigma = \frac{F}{A}$.The strain produced is $\epsilon = \frac{\Delta L}{L_{new}}$,where $L_{new} = L(1 + \alpha T)$.Using Young's modulus $Y = \frac{	ext{Stress}}{	ext{Strain}} = \frac{F/A}{\Delta L / L(1 + \alpha T)}$.Rearranging for $F$,we get $F = \frac{Y A \Delta L}{L(1 + \alpha T)}$.Substituting $\Delta L = \alpha L T$,we get $F = \frac{Y A (\alpha L T)}{L(1 + \alpha T)}$.Therefore,$F = \frac{Y A \alpha T}{1 + \alpha T}$.

$A$ metal rod of length $L$ and cross-sectional area $A$ is heated through $T^{\circ} C$. What is the force required to prevent the expansion of the rod lengthwise? $[Y=$ Young's modulus of the material of rod,$\alpha=$ coefficient of linear expansion $]$

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