Question

A uniform, rectangular laminar has sides L, and L. It is made of a material with constant coefficient of linear expansion α. If we define the coefficient of area expansion as γ,,M/ (AT), how are γ and α related? Explain the approximation that you have made.

Answer 1

γ ≈ 2 α

Step-by-step explanation:

The thermal expansion of solids in one dimension is given by

ΔL = α L ΔT

If we fear a square side L * L we must find the square area to see its expansion

A = L * L

Square the body warms up the area increases a quantity DA and each side increases a quantity DL

A + DA = (L + ΔL) (L + ΔL)

Let's replace the linear expansion equation

A + Da = (L + α L ΔT) (L + α L ΔT)

A + DA = L² + 2 α L Δt + (α L ΔT)²

A + Da = L2 (1 + 2 α ΔT + α² ΔT²

)

A = L2

ΔA / A + DA / A = (1 + 2 α DT + α² DT²)

1+ DA / A = (1 + 2 α DT + α² ΔT²)

ΔA / A = (2 α ΔT + α² ΔT²)

In general alpha is small (10⁻⁶ C⁻¹ ) and the temperature changes are not very large (100ºC), so we can neglect the quadratic term

ΔA = γ A Δt

γ ≈ 2 α