Thermal dilation formula discrepancy?

[lookez]

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?

 

[BvU]

hello lookez,

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.

 

[Vanadium 50]

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).

 

[lookez]

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.

 

[jtbell]

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)

 

[lookez]

I see, my teacher did mention that we would need Calculus to understand, I'm not there yet. Thank you all for the answers!