Question

A brass sheet whose coefficient of superficial expansion is 3.1 x 10-4 oC-1

has its length 40cm

at 10oC. If the surface area at 110oC is 320.1cm2

find the width of the sheet.

Answer 1

The coefficient of superficial expansion can be used to find the coefficient of linear expansion and then find how much did the length change when the T increased:

$\alpha=\beta/2=1.55\times10^{-4}{°C}^{-1} \\ L_1=40\,cm; L'_1=L_1(1+\alpha(T-10°C) ) \\ L'_1=(40\,cm)(1+ (1.55\times10^{-4}\,{°C}^{-1})(100°C) ) \\ L'_1=(40\,cm)(1.0155)=40.62\,cm$

We proceed to use $L'_1=40.62\,cm$ to find the width of the sheet at 110 °C or L^'₂, and then we can find L₂:

$\\ A_2=L'_1\cdot L'_2=320.1\,{cm}^{2} \\ \implies L'_2=A_2/L'_1=\frac{320.1\,{cm}^{2}}{(40.62\,cm)}=7.88\,cm \\ L'_2=L_2(1+ (1.55\times10^{-4}\,{°C}^{-1})(100°C) ) \\ \\ L'_2=L_2(1+ 1.55\times10^{-2}) \\ \implies L_2=L'_2/1.0155=(7.88\,cm)/1.0155=7.76\,cm$

In conclusion, the width of the sheet (at 10 °C) is L₂ = 7.76 cm.

Reference:

• Sears, F. W., & Zemansky, M. W. (1973). University physics.