Thermal expansion of metals question.

[thepatient]

Homework Statement

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?

Homework Equations

L = L₀(1+$\alpha$$\Delta$T

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

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.

 

[anathema]

Hello, great question! Your logic is correct and your approach is a valid way to solve this problem. However, there is a more straightforward way to approach this problem using the equation you listed in your attempt at a solution.

L = L0(1+\alpha\DeltaT)

We know the initial length (L0) is 20.700m, the coefficient of linear expansion of aluminum is 2.4x10^-5 C^-1, and the change in temperature is 29.4-21.2 = 8.2 C. Therefore, we can plug these values into the equation to find the new length (L) at 29.4 degrees C.

L = 20.700m(1+2.4x10^-5 C^-1 * 8.2C) = 20.70556m

This is very close to the value you calculated in your attempt at a solution, so your method is also correct. However, this approach may be a bit easier and more straightforward for those who may not be as familiar with thermodynamics equations.

Great job thinking through this problem and using your knowledge from the textbook to come up with a solution! Keep up the good work.