Solved Problem on Thermal Expansion

   Solved Problem on Thermal Expansion

Bandeira do Brasil  Portugu√™s    Flag of the United Kingdom  English
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
Problem sketch

Aluminum plate with 2 square meters of area at 50 degrees Celsius and under a decrease in temperature to 0 degrees Celsius its area decreases from 0.0044 square meters.
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
\[ \bbox[#99CCFF,10px] {\beta =2\alpha } \]
\[ \alpha=\frac{\beta }{2}\\ \alpha=\frac{4.4 \times 10^{-5}}{2} \]
\[ \bbox[#FFCCCC,10px] {\alpha =2.2 \times 10^{-5}\;^{\text{o}}\text{C}^{-1}} \]


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