Thermal dilation formula discrepancy?

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Discussion Overview

The discussion revolves around the linear thermal dilation formula ΔL=Li*a*ΔT, specifically questioning its applicability when temperature changes are reversed. Participants explore the implications of using the formula in both directions and whether it can lead to nonsensical results, such as negative lengths.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions the validity of the thermal dilation formula when applied in reverse, noting that it leads to an illogical conclusion of negative length after repeated temperature changes.
  • Another participant points out that the formula is an approximation valid over small temperature ranges, suggesting that it may not be accurate for larger changes.
  • A third participant introduces a mathematical perspective, indicating that the algebraic manipulation of the formula may not hold true for larger values, referencing the approximation of 1/(1+x).
  • One participant inquires about a more complete formula for thermal dilation, recalling a mention of a "real" thermal dilation formula from a theorem.
  • Another participant provides a link to a Wikipedia page that discusses conditions under which the linear expansion formula is valid and suggests that integration may be necessary if those conditions do not hold.
  • A later reply acknowledges the need for calculus to fully understand the more complex aspects of thermal dilation.

Areas of Agreement / Disagreement

Participants express differing views on the applicability and limitations of the linear thermal dilation formula, with no consensus reached on a definitive solution or alternative formula.

Contextual Notes

Participants note that the linear thermal dilation formula is an approximation and may not be valid for larger temperature changes. The discussion also highlights the potential need for calculus to derive a more accurate model.

Who May Find This Useful

This discussion may be useful for students and practitioners interested in thermal expansion, mathematical modeling in physics, and those seeking to understand the limitations of linear approximations in thermal dilation.

lookez
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Considering the linear thermal dilation formula ΔL=Li*a*ΔT (length change equals initial length times thermal dilation coefficient times temperature change), I was wondering why does it not work backwards? am I using it wrong or is there something missing?

For instance if we assume a=5*10^-5 , ΔT=100 and Li=20 then ΔL will be = 20*0.00005*100 = 0.1 which gives us a new length of 20.1, now if we do ΔT=(-100) we get ΔL = 20.1*0.00005*(-100) = -0.1005 instead of -0.1!

The way I see it this implies that if you repeatedly raise and lower the temperature of an object it will get smaller and smaller until the length reaches zero or negative. And obviously that's impossible. What's going on? I do realize this formula seems to be only used for thermal expansion, when ΔT > 0, but isn't it supposed to work backwards too?
 
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hello lookez, :welcome:

Good deduction! But that's why they make a reservation that this expression is valid over a small temperature range anyway. It's not high precision stuff, just an approximation -- good one.

And even though your expansion coefficient is pretty big, it still gives an accuracy of five microns on 20 cm for a hefty temperature change. Not bad at all.
 
This has nothing to do with thermal expansion per se. It's algebra - just that 1/(1+x) is not 1-x (but for small x it's close).
 
Well is there another more complete formula for this? I remember my teacher saying something about the "real" thermal dilation formula that comes from a theorem, but I can't find anything in my notes.
 
lookez said:
Well is there another more complete formula for this?
See the note at the end of the "Linear Expansion" section of this Wikipedia page:
https://en.wikipedia.org/wiki/Thermal_expansion#Linear_expansion
If either of these conditions does not hold, the equation must be integrated.
Do you know any integral calculus? The way a physics textbook would proceed is to make the changes infinitesimal, and then integrate over matching ranges in L and T: $$ \int_{L_1}^{L_2} \frac {dL} L = \alpha_L \int_{T_1}^{T_2} dT$$ (assuming that ##\alpha_L## is constant)
 
I see, my teacher did mention that we would need Calculus to understand, I'm not there yet. Thank you all for the answers!
 
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