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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 × w × h = L × 2 L × L size 12{l times w times h=L times 2L times L} {} and Block B has dimensions 2 L × 2 L × 2 L size 12{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 × w size 12{l times w} {} , and (c) the change in the height h size 12{h} {} of the two blocks?

There are two rectangular blocks. Block A has its dimensions labeled length equals L, width equals two times L, height equals L. Block B has its dimensions labeled length, width, and height all equal to two times L.

(a) The change in volume is proportional to the original volume. Block A has a volume of L × 2 L × L = 2 L 3 . size 12{L´2L´L=2L rSup { size 8{3} } "." } {} . Block B has a volume of 2 L × 2 L × 2 L = 8 L 3 , size 12{2L´2L´2L=8L rSup { size 8{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 × 2 L = 2 L 2 , size 12{L´2L=2L rSup { size 8{2} } ,} {} while that of Block B is 2 L × 2 L = 4 L 2 . size 12{2L´2L=4L rSup { size 8{2} } "." } {} 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
    Δ L = αL Δ T , size 12{ΔL=αLΔT} {}
    where Δ L size 12{ΔL} {} is the change in length L size 12{L} {} , Δ T size 12{ΔT} {} is the change in temperature, and α size 12{α} {} is the coefficient of linear expansion, which varies slightly with temperature.
  • The change in area due to thermal expansion is
    Δ A = 2 αA Δ T , size 12{ΔA=2αAΔT} {}
    where Δ A size 12{ΔA} {} is the change in area.
  • The change in volume due to thermal expansion is
    Δ V = βV Δ T , size 12{ΔV=βVΔT} {}
    where β size 12{β} {} is the coefficient of volume expansion and β size 12{β approx 3α} {} . Thermal stress is created when thermal expansion is constrained.

Conceptual questions

Thermal stresses caused by uneven cooling can easily break glass cookware. Explain why Pyrex®, a glass with a small coefficient of linear expansion, is less susceptible.

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Water expands significantly when it freezes: a volume increase of about 9% occurs. As a result of this expansion and because of the formation and growth of crystals as water freezes, anywhere from 10% to 30% of biological cells are burst when animal or plant material is frozen. Discuss the implications of this cell damage for the prospect of preserving human bodies by freezing so that they can be thawed at some future date when it is hoped that all diseases are curable.

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One method of getting a tight fit, say of a metal peg in a hole in a metal block, is to manufacture the peg slightly larger than the hole. The peg is then inserted when at a different temperature than the block. Should the block be hotter or colder than the peg during insertion? Explain your answer.

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Questions & Answers

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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