A light uniform rod of Young's modulus Y, cross sectional area A, coefficient of linear expansion α and length ℓ0 is rigidly connected to support at one end and the other end of the rod is connected to the spring of constant K as shown in the figure. The temperature of the rod is increased by Δθ with supports remaining fixed. Initially, the spring is in natural length position. Spring force on the rod acts uniformly over the cross section during elongation of the rod. Find the net elongation of the rod. (Assume thermal strain to be small)
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\begin{align} &\text{Rod (length }\ell_0,\; \text{area }A,\; Y\text{), spring (}K\text{); walls fixed.}\\ &\text{Let }x=\text{net change in rod length. Then spring change } \delta=-x.\\ &\text{Spring force on rod: }F=K\delta=-Kx\quad(\text{compressive}).\\[4pt] &\text{Rod free thermal expansion: }(\Delta\ell)_{\rm th}=\alpha \ell_0 \Delta\theta.\\ &\text{Elastic change under force }F: (\Delta\ell)_{\rm el}=\frac{F\ell_0}{AY}=\frac{(-Kx)\ell_0}{AY}.\\ &\text{Total change: } x=(\Delta\ell)_{\rm th}+(\Delta\ell)_{\rm el} =\alpha \ell_0 \Delta\theta-\frac{K\ell_0}{AY}x.\\[4pt] &\Rightarrow x\!\left(1+\frac{K\ell_0}{AY}\right)=\alpha \ell_0 \Delta\theta \;\;\Longrightarrow\;\; \boxed{x=\alpha \ell_0 \Delta\theta\;\frac{AY}{AY+K\ell_0}}. \end{align} ```