Solved Problem on Thermal Expansion

An iron rod, average linear coefficient of expansion 12 × 10⁻⁶ °C⁻¹, has a length of 15 m at 20 °C, if the bar is heated to 150 °C find:
ä) The expansion of the bar;
b) The final length of the bar.

Problem data:

Coefficient of linear expansion of iron: α = 12 × 10⁻⁶ °C⁻¹;
Initial length of the bar: L₀ = 15 m;
Initial temperature of the bar: t_i = 20 °C;
Final temperature of the bar: t_f = 150 °C.

Problem diagram:

Figure 1

Solution

a) The expansion of the bar

\[ \bbox[#99CCFF,10px] {\Delta L=\alpha L_{0}\Delta t} \]

\[ \begin{gather} \Delta L=\alpha L_{0}(t_{f}-t_{i})\\ \Delta L=12 \times 10^{-6} \times 15 \times(150-20)\\ \Delta L=180 \times 10^{-6} \times 130\\ \Delta L=23400 \times 10^{-6} \end{gather} \]

\[ \bbox[#FFCCCC,10px] {\Delta L=0.02\;\text{m}} \]

b) The final length of the bar

\[ \bbox[#99CCFF,10px] {\Delta L=L-L_{0}} \]

\[ \begin{gather} L=L_{0}+\Delta L\\ L=15+0.02 \end{gather} \]

\[ \bbox[#FFCCCC,10px] {L=15.02\;\text{m}} \]

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