How Can I Calculate the Coefficient of Thermal Expansion for a Material?

[satchmo05]

Homework Statement

At 20°C, a rod is exactly 21.03 cm long on a steel ruler. Both the rod and the ruler are placed in an oven at 270°C, where the rod now measures 21.26 cm on the same ruler. What is the coefficient of thermal expansion for the material of which the rod is made? The linear expansion coefficient of steel is 11 x 10-6 /C°.

Homework Equations

change in length = rod length * coefficient of linear expansion * change in temperature (C)

The Attempt at a Solution

I can find out the thermal expansion of the rod, but how does the coefficient of steel go into that equation?! I cannot figure it out. Here is my attempt:

0.0023m = 0.2103 * coefficient of linear expansion * 250
Therefore, the coefficient of linear expansion is about 4.4e-5, but that answer is wrong. Please help! Thanks in advance to all help!

 

[mgb_phys]

The steel ruler also expands so the change in length of the rod isn't simply 0.23cm

 

[satchmo05]

so what would the equation look like then? thanks for the help! much appreciated!

 

[mgb_phys]

You have the correct equation
change in length = rod length * coefficient of linear expansion * change in temperature (C)

The only extra complexity is that you have to calculate the increase in length of the steel ruler (using the same equation) and then use that to work out the real increase in length of the rod. Then use the same equation to work out the expansivity of the rod.

You could also use the difference in exapansivity between rod and steel but you are more likely to make a mistake that way

 

[satchmo05]

so do i add them together? i don't quite understand what you mean...

to calculate the change in length of the steel, i just calculate:

0.2103m * 11e-6 * 250 degrees celsius = change in length of the steel

+

0.0023m / (0.2103m * 250 degrees celsius) = coefficient of linear expansion of the rod

= what my answer should be?!? Please let me know ASAP! Thanks again for the help!

 

[LowlyPion]

so do i add them together? i don't quite understand what you mean...

to calculate the change in length of the steel, i just calculate:

0.2103m * 11e-6 * 250 degrees celsius = change in length of the steel

+

0.0023m / (0.2103m * 250 degrees celsius) = coefficient of linear expansion of the rod

= what my answer should be?!? Please let me know ASAP! Thanks again for the help!

You start out with both being cold. You have the measurement on the ruler. When both are heated the ruler gradations expand by the coefficient of thermal expansion, which distorts the measurement at the higher temperature. So ...

Take the measurement you get from the hot ruler and recognize that in "cold" terms this measurement represents what that measure has expanded to. In your case, then your original Rod measures .2103. And on the hot ruler they say it measures .2126. So that means .2126 cold + its thermal expansion should be what the heated rod measures in terms of a "cold" measure.

Length of rod = .2126*(1 + 250*11*10^-6)

Knowing the amount then that the rod has changed in length from the original should allow you to now calculate the thermal expansion coefficient.