A compressive force, $F$ is applied at the two ends of a long thin steel rod. It is heated, simultaneously, such that its temperature increases by $\Delta T$. The net change in its length is zero. Let $l$ be the length of the rod, $A$ its area of cross- section, $Y$ its Young's modulus, and $\alpha $ its coefficient of linear expansion. Then, $F$ is equal to
JEE MAIN 2017, Medium
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Due to thermal exp., change in length $\left( {\Delta l} \right)$
$ = l\alpha \Delta T$ $...(i)$
$Young's\,modulus (Y)$
$ = \frac{{Normal\,stress}}{{Longitudinal\,strain}}$
$Y = \frac{{F/A}}{{\Delta l/l}} \Rightarrow \frac{{\Delta l}}{l} = \frac{F}{{AY}}$
$\Delta l = \frac{{Fl}}{{AY}}$
$From\,e{q^n}(i),\,\frac{{Fl}}{{AY}} = l\,\alpha \,\Delta T$
$F = AY\,\alpha \,\Delta T$
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