A rod of length L and radius r is held betwee | vedclass

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(C) When a rod is heated,it tends to expand by an amount $\Delta L = L \alpha \Delta T$,where $\alpha$ is the coefficient of linear expansion and $\De

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$r^2$

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(C) When a rod is heated,it tends to expand by an amount $\Delta L = L \alpha \Delta T$,where $\alpha$ is the coefficient of linear expansion and $\Delta T$ is the change in temperature.Since the rod is held between two rigid walls,it cannot expand. This constraint induces a compressive stress and a corresponding force $F$ in the rod.The thermal stress is given by $\sigma = Y \frac{\Delta L}{L}$,where $Y$ is Young's modulus.Substituting $\Delta L = L \alpha \Delta T$,we get $\sigma = Y \alpha \Delta T$.The force $F$ is given by $F = \sigma A$,where $A = \pi r^2$ is the cross-sectional area.Therefore,$F = (Y \alpha \Delta T) (\pi r^2)$.Since $Y$,$\alpha$,$\Delta T$,and $\pi$ are constants for a given material and temperature change,the force $F$ is proportional to $r^2$.

$A$ rod of length $L$ and radius $r$ is held between two rigid walls so that it is not allowed to expand. If its temperature is increased,then the force developed in it is proportional to .........

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