13.2 Thermal expansion of solids and liquids  (Page 5/10)

Another example of thermal stress is found in the mouth. Dental fillings can expand differently from tooth enamel. It can give pain when eating ice cream or having a hot drink. Cracks might occur in the filling. Metal fillings (gold, silver, etc.) are being replaced by composite fillings (porcelain), which have smaller coefficients of expansion, and are closer to those of teeth.

Two blocks, A and B, are made of the same material. Block A has dimensions $l\times w\times h=L\times 2L\times L$ and Block B has dimensions $2L\times 2L\times 2L$ . If the temperature changes, what is (a) the change in the volume of the two blocks, (b) the change in the cross-sectional area $l\times w$ , and (c) the change in the height $h$ of the two blocks?

(a) The change in volume is proportional to the original volume. Block A has a volume of $L\times 2L\times L={2L}^3\text{.}$ ^. Block B has a volume of $2L\times 2L\times 2L={8L}^3,$ which is 4 times that of Block A. Thus the change in volume of Block B should be 4 times the change in volume of Block A.

(b) The change in area is proportional to the area. The cross-sectional area of Block A is $L\times 2L={2L}^2,$ while that of Block B is $2L\times 2L={4L}^2\text{.}$ Because cross-sectional area of Block B is twice that of Block A, the change in the cross-sectional area of Block B is twice that of Block A.

(c) The change in height is proportional to the original height. Because the original height of Block B is twice that of A, the change in the height of Block B is twice that of Block A.

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Section summary

• Thermal expansion is the increase, or decrease, of the size (length, area, or volume) of a body due to a change in temperature.
• Thermal expansion is large for gases, and relatively small, but not negligible, for liquids and solids.
• Linear thermal expansion is
  $\text{Δ}L=\mathrm{\alpha L}\text{Δ}T,$
  where $\text{Δ}L$ is the change in length $L$ , $\text{Δ}T$ is the change in temperature, and $\alpha$ is the coefficient of linear expansion, which varies slightly with temperature.
• The change in area due to thermal expansion is
  $\text{Δ}A=2\mathrm{\alpha A}\text{Δ}T,$
  where $\text{Δ}A$ is the change in area.
• The change in volume due to thermal expansion is
  $\text{Δ}V=\mathrm{\beta V}\text{Δ}T,$
  where $\beta$ is the coefficient of volume expansion and $\beta \approx \mathrm{3\alpha }$ . Thermal stress is created when thermal expansion is constrained.

Conceptual questions