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A steel rod of length $\ell$, cross sectional area $A$, young's modulus of elasticity $Y$, and thermal coefficient of linear expansion $'a'$ is heated so that its temperature increases by $t\,^oC$. Work that can be done by rod on heating will be
$(YA\, \alpha\, t)\times(\ell\, \alpha\, t)$
$\frac{1}{2}\left( {YA\,\alpha \,t} \right) \times \left( {\ell \,\alpha \,t} \right)$
$\frac{1}{2}\left( {YA\,\alpha \,t} \right) \times \left( {1/2} \right)\left( {\ell \,\alpha \,t} \right)$
$2(YA\, \alpha\, t)\,(\ell\, \alpha\, t)$
Solution
Work done $=\frac{1}{2} \mathrm{Y}(\text { strain })^{2}(\mathrm{A} \ell)$
$=\frac{1}{2} \mathrm{Y}\left(\frac{\Delta \ell}{\ell} \mathrm{A}\right)(\mathrm{A} \ell)=\frac{1}{2} \mathrm{Y}\left(\frac{\Delta \ell}{\ell} \mathrm{A}\right)\left(\frac{\Delta \ell}{\ell} \times \ell\right)$
$=\frac{1}{2} \mathrm{Y}(\alpha \text { t } \mathrm{A})(\alpha \text { t } \ell)[\therefore \Delta \ell=\ell \alpha \mathrm{f}]$
$=\frac{1}{2}(\mathrm{YA} \propto \mathrm{t})(\ell \propto \mathrm{t})$
