How does the change in diameter relate to the change in length?

[demonslayer42]

Homework Statement

Find the change in diameter of the metal rod.

So I did a lab in class. We heated up a metal rod by putting it in a "jacket" and heating it with steam and it expanded. We were asked to calculate the liner coefficient. Well I did that just fine. I determine that the rod was copper. But how do I fined the change in diameter?

Length of Rod: 600mm

Temperature initial : 22.5 C
Temperature Final : 99.4 C
Change in Temperature :76.9 C

X initial: 5.849mm
X final 6.622mm
Change in length = .773mm

Change in Length = coefficient*length*change in Temperature

\DeltaL = \alpha L \DeltaT

.773 = coefficient*600*76.9

So anyways my theoretical calculations coefficient = 16.75*10^-6 / C

Copper's coefficient is 17*10^-6/ C

I took the percent difference 1.47% off of actual. Great.

Homework Equations

The Attempt at a Solution

I have no idea how to figure this out without the radius, circumference, or volume. Is it even possible?

 

[p21bass]

At some point - preferably before heating - you should have measured the diameter of the rod. If not, then you can only give a theoretical Δd, using variables. Diameter is a linear dimension. Thus, it will change according to the same rule as the length.

 

[demonslayer42]

That's exactly what I was thinking, but neither my professor nor did the lab manual specify that. Oh actually, I just read the question again and it says : "Estimate" the change in diameter of the rod. Maybe it means I'm not suppose to actually calculate it but make a theoretical guess? Can I assume that the diameter increased equal to the length?

 

[p21bass]

If, by "equal to length" you mean using the same equation, yes. Just change your "L"s to "d"s.

 

[Astronuc]

That's exactly what I was thinking, but neither my professor nor did the lab manual specify that. Oh actually, I just read the question again and it says : "Estimate" the change in diameter of the rod. Maybe it means I'm not suppose to actually calculate it but make a theoretical guess? Can I assume that the diameter increased equal to the length?

This is a fundamental concept in materials. The linear coefficient of thermal expansion applies to a direction (linear = in a line). If one measures the LCTE in one direction, then one can apply to the normal directions - assuming the material behaves isotropically. Some materials, e.g., hcp metals, are anisotropic.

The diameter is simply another linear dimension. p21bass describes it well.

If one didn't measure the diameter or volume, one could measure the mass, and using the density, determine the volume. From the volume and length, one can determine the area with reasonable certainty, and from area, one can determine diameter.