Change in density due to thermal expansion

[empty_set]

Homework Statement

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)V_i
L=(1+αΔT)L_i
V=m/ρ

The Attempt at a Solution

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

 

[CrazyNinja]

My friend, don't 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 into the equation above, divide by ρ, subtract 1. You have your answer.

 

[Chestermiller]

Homework Statement

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)V_i
L=(1+αΔT)L_i
V=m/ρ

The Attempt at a Solution

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

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 $\Delta T$.

 

[empty_set]

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

 

[Chestermiller]

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

There should be a minus sign in this equation.

 

[CrazyNinja]

Your algebra is wrong. My method WILL give a minus sign inherently. Check again. You probably got confused with the ρ and ρ°.