# Homework Help: Thermal Physics question regarding the thermal expansion coefficients

1. Jul 8, 2012

### Stolbik

1. The problem statement, all variables and given/known data

I am working on this ahead of my fall class and don't actually want the answer...
just pointers to help me understand something.. Thanks guys! :)
I am really rusty with my general physics and calculus knowledge =(

The original question asks me to prove that, for a solid, the linear thermal expansion coefficients (in x, y, z directions) add together to give the thermal expansion coefficient as such:

B=ax+ay+az

where B is the thermal expansion coefficient
B=(deltaV/V)/deltaT
deltaV= change in volume
V=volume
deltaT=change in temperature in Kelvin

and a is the linear thermal expansion coefficient
a=(deltaL/L)/deltaT
deltaL= change in length
deltaL=length

So here are my questions:

Shouldn't the equation be B=ax*ay*az instead? for a solid like a cube you have to multiply the lengths to get the volume... Why isn't it the same here?

Also is there a way to get the volume from the lengths of an object with calculus? I don't remember =( Just remember you can get the area under a curve from doing the integral or the volume of an object made by a curve somehow too...

2. Relevant equations

B=ax+ay+az

where B is the thermal expansion coefficient
B=(deltaV/V)/deltaT
deltaV= change in volume
V=volume
deltaT=change in temperature in Kelvin

and a is the linear thermal expansion coefficient
a=(deltaL/L)/deltaT
deltaL= change in length
deltaL=length

3. The attempt at a solution

uh.. well my attempt so far has been to understand the question. I tried to define the Volume as Lx+Ly+Lz but then got stuck. Please don't give me the answer though!! I got 3 months to work this out :)

2. Jul 8, 2012

### Andrew Mason

You are trying to find the relationship between the change in volume and change in temperature.

You are given the relationship between change in length in the x, y and z directions. From that you can work out what an infinitesimal change of dx, dy and dz does to the change in volume (ie for an infinitesimal change in temperature dT):

V + dV = (x + dx)(y+dy)(z+dz) = (xy + xdy + dxy + dxdy)(z+dz)

Work that out and ignore the second and third order infinitesimal terms.

AM