Thermal Expansion linear vs area expansion

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SUMMARY

The discussion focuses on the relationship between linear expansion and area expansion, specifically addressing why the area coefficient of expansion is twice the linear coefficient of expansion. Participants analyze a square object with side length L, applying the linear expansion formula to derive the change in area. The conclusion confirms that the increase in area is indeed 2(αLΔT), where α represents the linear coefficient of expansion and ΔT is the change in temperature.

PREREQUISITES
  • Understanding of linear expansion and its mathematical representation
  • Familiarity with the concept of coefficients of thermal expansion
  • Basic knowledge of geometry, particularly regarding area calculations
  • Ability to apply algebraic manipulation to physical formulas
NEXT STEPS
  • Study the derivation of area expansion from linear expansion in detail
  • Explore the implications of thermal expansion in different materials
  • Learn about the coefficients of thermal expansion for various substances
  • Investigate real-world applications of thermal expansion in engineering
USEFUL FOR

Students in physics or engineering, material scientists, and professionals involved in design and manufacturing processes that require an understanding of thermal expansion principles.

Searay330
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this is a picture of my notes for thermal expansion for linear vs area.
my question is why does the area coefficient of expansion for the area = 2(liner coefficient of expansion).
any insight would be appreciated.
 
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As an example to ponder, consider an object with a flat, square surface that is L units of length on each side. Apply the linear expansion concept to each dimension and work out an expression for the change in area. Then consider that the coefficient of linear expansion ##\alpha## is typically on the order of a few parts per million per degree C. Is there an obvious simplification?
 
im not sure the change in area would be equal to the new L2 and that is the only dimension that changes
 
Searay330 said:
im not sure the change in area would be equal to the new L2 and that is the only dimension that changes
No, the change in area would not equal L2.
If the original side of the square is L, the original area is L2.
If the new side length is ##L+\alpha L\Delta T##, what is new area? How much has the area increased by?
 
its increased by 2(αLΔT) one for each side
 
Searay330 said:
its increased by 2(αLΔT) one for each side
Right. Does that answer your original question?
 
yes
 

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