# Thermal expansion of metals question.

1. Jun 16, 2011

### thepatient

1. The problem statement, all variables and given/known data
You use a steel measuring tape and measure the length of an aluminum bar. You measure a length of 20.700 m when the temperature is 21.2 degrees C. What is the measured length when the temperature is 29.4 degrees C?

2. Relevant equations
L = L0(1+$\alpha$$\Delta$T

Coefficient of linear expansion:
Aluminum: 2.4x10^-5 C^-1
Steel: 1.2x10^-5 C^-1

3. The attempt at a solution
So I've never taken a course that introduced heat and thermodynamics, but I read through my Physics book and became interested in this problem in particular. I wasn't sure if my logic was correct.

First, I calculated the new length of the aluminum bar.
L = 20.700m*(1+2.4x10^-5 C^-1 *(29.4-21.2)C ) = 20.704m

Now, the steel measuring tape:
L = 20.700m*(1+1.2x10^-5C^-1*(29.4-21.2)C) = 20.702m

Now, since the measuring tape also increases its size, the true measurement of 20.702m at 29.4 degrees will actually read 20.700m, so I assumed that you can take the change in measurement and find out the change of measurement per meter.

Change in measurement per meter = 0.002/20.700m. For every 20.700m marked on the measuring tape, there is an increase of 0.002m in the measurement. So I assumed I can use this ratio of change in length divided by marked length to find the new change in length of the aluminum bar, using the steel measuring tape.

20.704m*0.002m/20.700m = 2.0003865x10^-3m, meaning that from the original measurement of 20.700m, the reading increases by this amount so:

On the steel tape:
L = 20.700m + 2.0003865x10^-3m = 20.70200039m. But since this is a measurement correct to the thousandth digit, then L = 20.702m

Is this correct? Does what I did made sense? Is there a more straightforward way of approaching this problem? Thanks.