Change in density due to thermal expansion

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1. Feb 7, 2016

empty_set

1. The problem statement, all variables and given/known data
I want to find the mathematical proof to show that the density of an object changes with thermal expansion. My professor showed this in class and it was horribly wrong because he let a few things out. The book I'm reading showed it in 4 steps and left out a lot of the crucial parts. I've been working on this for a few days.

How do I go from, Δρ=m/ΔV, to Δρ/ρ= -3(ΔL/L) ?
I think these are the relevant equations:
V=(1+βΔT)Vi
L=(1+αΔT)Li
V=m/ρ

3. The attempt at a solution
This is how I got it but I don't think it's correct: m.imgur.com/a/ZOfCp

2. Feb 8, 2016

CrazyNinja

My friend, dont struggle so much for it.

You know that ρ=m/V.
Let ρ°=m/V°.
Then V°= V( 1+γΔT) where γ=3α.
Put that in to the equation above, divide by ρ, subtract 1. You have your answer.

3. Feb 8, 2016

Staff: Mentor

It isn't clear what you did. I suggest doing the analysis on a cube. $$V_0=L_0^3$$
$$V=L^3=[L_0(1+\alpha \Delta T)]^3$$
and neglect non-linear terms in $\Delta T$.

4. Feb 9, 2016

empty_set

For both of those methods I got Δρ/ρ = 3(ΔL/L). Would the negative be implied or is my algebra wrong?

5. Feb 9, 2016