Solved Problem In Thermal Expansion

Solved Problem in 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. 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:     ti = 50 °C;
• final plate temperature:     tf = 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
$\bbox[#99CCFF,10px] {\Delta A=\beta A_{0}\Delta t}$
$\Delta A=\beta A_{0}(\;t_{\text{f}}-t_{\text{i}}\;)\\ -0.0044=\beta \times 2 \times (\;0-50\;)\\ -0,0044=-100\beta \\ \beta =\frac{-0.0044}{-100}\\ \beta =\frac{4.4 \times 10^{-3}}{1 \times 10^{2}}\\ \beta =4.4 \times 10^{-3} \times 10^{-2}\\ \beta =4.4 \times 10^{-5}\\ \beta=0.000044$
$\bbox[#FFCCCC,10px] {\beta =4.4 \times 10^{-5} °{\text{C}}^{-1}}$
The coefficient of linear thermal expansion of aluminum
$\bbox[#99CCFF,10px] {\beta =2\alpha }$
$\alpha=\frac{\beta }{2}\\ \alpha=\frac{4.4 \times 10^{-5}}{2}$
$\bbox[#FFCCCC,10px] {\alpha =2.2 \times 10^{-5} °{\text{C}}^{-1}}$