A compressive force, F is applied at two ends of a long thin steel rod. It is heated, simultaneously

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8.Mechanical Properties of Solids

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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

A

${l^2}\,Y\alpha \Delta T$

B

$lA\,Y\alpha \Delta T$

C

$A\,Y\alpha \Delta T$

D

$\frac{{AY}}{{\alpha \Delta T}}$

(JEE MAIN-2017)

Solution

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}}$

$Y = \frac{{F/A}}{{\Delta l/l}} \Rightarrow \frac{{\Delta l}}{l} = \frac{F}{{AY}}$

$\Delta l = \frac{{Fl}}{{AY}}$

$From\,e{q^n}(i),\,\frac{{Fl}}{{AY}} = l\,\alpha \,\Delta T$

$F = AY\,\alpha \,\Delta T$

Standard 11

Physics

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