## December 13, 2010

### What can you do with this?

I just wrote this up over at the home.drewsday blog, but the real reason for taking the photos was to use this in class next quarter. We start with thermodynamics; one of the first topics is thermal expansion and contraction.

I regularly read Dan Meyer's blog. He was (is?) a math teacher, but I'm consistently inspired by his ideas. One of his regular features is something he calls "What can you do with this?" [WCYDWT]. The idea is that he finds an example of something in the world which illustrates a math concept and brings it into the classroom.

When I saw the contraction of the vinyl siding on my house, I knew I could bring it into the class next quarter.  The question is: What can you do with it?

I want to present this to class, so I have to think of the questions that would be appropriate.  Usually, when I start a problem in class I make a list of everything I know and everything I don't know (or want to know).

What I know

• temperature outside today was 12° F. ( T)
• nominal length of the siding was 12 feet. ( L)
• change of length (on one side) was 1/8 inch. ( ΔL )
What I don't know (would like to know)

• temperature when the siding was painted ( T)
• coefficient of linear expansion for the vinyl siding (α)
Relevant equation

• ΔL = αL0ΔT
The problem is that I have two unknowns (ΔT and α).

I have no idea what the temperature was when the house was painted. I didn't even know the house existed when it was painted.  I suppose I could come up with a reasonable estimate, but realistically, there is a pretty wide range of temperatures to work with. Conservatively, I would guess that the painting could have been done when the siding was anywhere between 60° F and 90° F. That is a pretty wide range, so I'm not sure I want to try to estimate that and use it to find α.  So let's make the initial temperature something we want to find.

This means that I need to figure out what the coefficient of linear expansion is for the siding. In class we discuss how the thermal expansion process is approximately linear over a certain range. I have no idea if the expansion of vinyl siding is linear or not.  I'm going to cross my fingers and assume it's linear-ish.

With a bit of googling, I learned that vinyl siding is made of unplasticized polyvinyl chloride (uPVC). I also found the coefficient of linear expansion is listed as 50 ×10-6 °C-1 for PVC. I couldn't find uPVC thermal coefficient, but every listed coefficient I could find was close to this value.

Using that thermal expansion coefficient, and solving for the initial temperature, I found that the initial temperature was 23.6° C, which is 74.5° F.  (Right in the middle of my estimated range, hmmm...)

I think that the value for the coefficient is reasonable.  I plan to work this example in class, then extend it by asking the class to calculate what the maximum expansion/contraction range would be for a 12' length of siding in Illinois weather.  I'll be looking for other ways to extend this.

Side notes:

1. The units specified online for the coefficient of thermal expansion are often given as [L]/[L][T] (e.g. m/mK). Of course, the length units cancel and all that remains is inverse temperature units, which is how our textbook lists the units.
2. Coefficients of linear expansion for common materials varied quite a bit (sorry, not scientifically quantified) from one table to another. Surprisingly, the PVC coefficient seemed to always be within 2% of 50 ×10-6 °C-1.