Solved Problem on Thermal Expansion

A square aluminum plate has an area of 2 m² at 50 °C, if the plate is cooled to 0 °C its area decreases from 0.0044 m². Find the coefficients of surface and linear expansion of aluminum.

Problem data:

Initial area of the plate: A₀ = 2 m²;
Surface area change: Δ A = −0,0044 m²;
Initial plate temperature: t_i = 50 °C;
Final plate temperature: t_f = 0 °C.

Problem diagram:

Figure 1

As the plate decreases, upon cooling, the signal of the surface area change is negative.

Solution

Writing the equation of area thermal expansion, we have

\[ \bbox[#99CCFF,10px] {\Delta A=\beta A_{0}\Delta t} \]

\[ \begin{gather} \Delta A=\beta A_{0}(\;t_{\text{f}}-t_{\text{i}}\;)\\ -0.0044=\beta \times 2 \times (\;0-50\;)\\ -0,0044=-100\beta \\ \beta =\frac{-0.0044}{-100}\\ \beta =\frac{4.4 \times 10^{-3}}{1 \times 10^{2}}\\ \beta =4.4 \times 10^{-3} \times 10^{-2}\\ \beta =4.4 \times 10^{-5}\\ \beta=0.000044 \end{gather} \]

\[ \bbox[#FFCCCC,10px] {\beta =4.4 \times 10^{-5} °{\text{C}}^{-1}} \]

The coefficient of linear thermal expansion of aluminum

\[ \bbox[#99CCFF,10px] {\beta =2\alpha } \]

\[ \begin{gather} \alpha=\frac{\beta }{2}\\ \alpha=\frac{4.4 \times 10^{-5}}{2} \end{gather} \]

\[ \bbox[#FFCCCC,10px] {\alpha =2.2 \times 10^{-5} °{\text{C}}^{-1}} \]

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