Solved Problem on Thermal Expansion

### Solved Problem on Thermal Expansion

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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. Determine the coefficients of surface and linear expansion of aluminum

Problem data
• initial area of the plate:     $$A_{0}=2\;\text{m}^{2}$$;
• surface area change:     $$\Delta A=-0.0044\;\text{m}^{2}$$;
• initial plate temperature:     $$t_{\text{i}}=50\;^{\text{o}}\text{C}$$;
• final plate temperature:     $$t_{\text{f}}=0\;^{\text{o}}\text{C}$$.
Problem sketch

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
$\bbox[#99CCFF,10px] {\Delta A=\beta A_{0}\Delta t}$
$\Delta A=\beta A_{0}(\;t_{\text{f}}-t_{\text{i}}\;)\\ -0,0044=\beta \times 2 \times (\;0-50\;)\\-0.0044=-100\beta \\ \beta =\frac{-0.0044}{-100}\\ \beta=0.000044$
$\bbox[#FFCCCC,10px] {\beta =4.4 \times 10^{-5}\;^{\text{o}}\text{C}^{-1}}$
The coefficient of linear thermal expansion of aluminum
$\bbox[#99CCFF,10px] {\beta =2\alpha }$
$\alpha=\frac{\beta }{2}\\ \alpha=\frac{4.4 \times 10^{-5}}{2}$
$\bbox[#FFCCCC,10px] {\alpha =2.2 \times 10^{-5}\;^{\text{o}}\text{C}^{-1}}$