11 Example 10.1

Example

Question:

Show that the coefficient of area expansion, \(\frac{\Delta A}{A} / \Delta T\), of a rectangular sheet of the solid is twice its linear expansivity, \(\alpha_l\).

Solution:

Consider a rectangular sheet of solid material with length \(a\) and breadth \(b\).
When the temperature increases by \(\Delta T\):
- \(a\) increases by \(\Delta a = \alpha_l a\Delta T\)
- \(b\) increases by \(\Delta b = \alpha_l b \Delta T\)
From the geometry, \[ \Delta A = \Delta A_1 + \Delta A_2 + \Delta A_3 \] where \(\Delta A_1 = a\Delta b\), \(\Delta A_2 = b\Delta a\), \(\Delta A_3 = (\Delta a)(\Delta b)\).
Substituting, \[ \Delta A = a\Delta b + b\Delta a + (\Delta a)(\Delta b) \] \[ = \alpha_l a b \Delta T + \alpha_l a b \Delta T + (\alpha_l a\Delta T)(\alpha_l b\Delta T) \] \[ = ab[2\alpha_l \Delta T + (\alpha_l \Delta T)^2] \] For small \(\alpha_l \Delta T\), \((\alpha_l \Delta T)^2\) is negligible compared to \(2\), so: \[ \frac{\Delta A}{A} \approx 2\alpha_l \Delta T \] Therefore, \[ \frac{\Delta A}{A} \frac{1}{\Delta T} \approx 2\alpha_l \] The coefficient of area expansion is twice the linear expansivity.