### Solved Problem on Thermal Expansion

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An iron rod, average linear coefficient of expansion $$12 \times 10^{-6}\;^{\text{o}}\text{C}^{-1}$$, has a length of 15 m at 20 ° C, if the bar is heated to 150 ° C, determine
ä) The expansion of the bar; b) The final length of the bar.

Problem data
• coefficient of linear expansion of iron:     $$\alpha =12 \times 10^{-6}\;^{\text{o}}\text{C}^{-1}$$
• ;
• initial length of the bar:     $$L_{0}=15\;\text{m}$$
• ;
• initial bar temperature:     $$t_{\text{i}}=20^{\text{o}}\text{C}$$
• ;
• final bar temperature:     $$t_{\text{f}}=150^{\text{o}}\text{C}$$
• .
Problem sketch

figure 1

Solution

(a) The expansion of the bar
$\bbox[#99CCFF,10px] {\Delta L=\alpha L_{0}\Delta t}$
$\Delta L=\alpha L_{0}(\;t_{\text{f}}-t_{\text{i}}\;)\\ \Delta L=12 \times 10^{-6} \times 15 \times(\;150-20\;)\\ \Delta L=180 \times 10^{-6} \times 130\\ \Delta L=23400 \times 10^{-6}$
$\bbox[#FFCCCC,10px] {\Delta L=0.02\;\text{m}}$

b) The final length of the bar
$\bbox[#99CCFF,10px] {\Delta L=L-L_{0}}$
$L=L_{0}+\Delta L\\ L=15+0.02$
$\bbox[#FFCCCC,10px] {L=15.02\;\text{m}}$