Question 436: Thermodynamics & KTG

Question

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 $\ell $ 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 :

Accepted Answer

Because of thermal expansion,  change in length
($\Delta $$\ell $) = $\ell $ $\alpha $ $\Delta $T . . . . .(1)
Because of compressive force, the compansion is $\Delta $$\ell $ '  ,
$	herefore\,\,\,$ Young's Modulus (y) = ${{{F \over A}} \over {{{\Delta \ell} \over \ell }}}$
$ \Rightarrow $$\,\,\,$ F = YA ${{\Delta \ell '} \over \ell }$
As net change in length is 0 . So,
$\Delta \ell '$ = $\Delta \ell $ = $\ell \alpha \,\Delta T$
$	herefore\,\,\,$ F = YA $ 	imes $ ${{\ell \alpha \,\Delta T} \over \ell }$ = AY$\alpha $$\Delta $T

Answer

$\ell $
2
Y$\alpha $ $\Delta $T

Answer

$\ell $A Y$\alpha $ $\Delta $T

Answer

A Y$\alpha $ $\Delta $T

Answer

${{AY} \over {\alpha \,\Delta T}}$

Thermodynamics & KTG

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