Thermal Dynamics - Ring Heated Up

[Jordan D]

Homework Statement

This is more of a question that doesn't require formulas as much as common sense.
The question goes,"If you have a ring that is heated up, would the hole in the middle get smaller or larger."

Homework Equations

N/A

The Attempt at a Solution

I know that when objects are heated the object undergoes expansion. What I also know is expansion in our universe isn't just in one direction. My physics professor said that the hole would get larger. But if the ring expands, it will expand in all directions, including towards the hole. Which means the hole would get smaller. I do not understand this question, it seems to be counter-intuitive.

 

[TSny]

Welcome to PF!

Suppose you had square tiles as shown below with one tile missing (the "hole"). If you heat all the tiles, what happens to the size of the hole? Assume the tiles are allowed to expand.

 

[Jordan D]

Thank you for the welcoming!

The corners would expand squeezing the middle(top,left,right, bottom) tiles to where the only direction the inner blocks can go is towards the hole and away from the hole. So because of this I would still say the hole gets smaller from the initial area that it had. I see that this thinking is wrong, but I just don't understand why.

 

[TSny]

Imagine the tiles are resting on a frictionless floor. So they are free to slide on the floor. Consider one of the corner tiles as it expands. If it pushes against one of the other tiles, the other tile will push back on the corner tile and the corner tile will slide a little to prevent stress from building up. Similarly, the other tiles can shift positions. So, the inner tiles do not feel any stress and the hole is not "squeezed" to a smaller size.

You could imagine disassembling the tiles before heating them. Heat each tile individually and then reassemble the tile pattern.

Or, imagine taking a circular ring. Before heating it, cut the ring and bend the ring into a line. Heat the line and then bend it back into a circle as indicated below.

Also, if you know the equation for thermal expansion, you might compare how much the circumference of the ring expands to how much the width of the material of the ring expands.

 

[Jordan D]

The ring example confuses me more. If I had a ring and turned in into a line, then added heat, the line would be bigger as your picture shows. But if I were to turn it into a ring again, I would still expect the hole be smaller than the original hole.

For instance, say the line initially had a one centimeter width, and after heating it, the line has a two centimeter width. When I bend the line back into a ring, the center of the end and beginning would come together. So that way only thing that changed was the width. In which the hole in the middle still becomes smaller, in addition to the expansion outside the ring.

 

[Nidum]

Thermal expansion is proportional to initial dimensions of object .

So your strip expands very little in width and a lot in length .

 

[Jordan D]

I see... If it expands more on the length, the hole would definitely expand. The formula only accounts for length, why not width too or area? The formula appears to not seem to care about width. I do apologize for not including the formula, I felt as if it couldn't help in this problem due to not having numbers associated with it.
If it isn't too much trouble for you, can you help me understand why width doesn't matter? Is it because an object expands more in length than in width?

ΔL/L = αΔT

 

[Nidum]

L = Lo ( 1 + αΔT)

 

[TSny]

The formula ΔL/L = αΔT shows that the fractional (or percent) change in length is the same for any length. Thus, if the circumference were to double (not likely) then the width would also double. It is exactly like a photographic enlargement. In the figure below, I took the ring on the left and just resized it by a certain percentage. The width and circumference increased by the same percent. The diameter of the hole also increased by the same percent.

 

[Jordan D]

So we don't include width because it expands just as much as length, and because the formula gives a percent change the expansion will also change by the same percent? That would definitely explain why the hole gets larger.

Thank you for your time, I appreciate it!

 

[TSny]

OK.

I'll leave you with one more way to think about it. Imagine you had a solid circular disk as shown on the left below. It is all made of one material, but I shaded part of the disk so that you can think of it as a ring and inner disk. On the right you can imagine that you cut out the inner disk to produce the ring.

If you heat the solid disk on the left, the material will expand. The inner disk on the left will expand just like the separated inner disk on the right. And the ring portion on the left will expand just like the separated ring on the right.

 

[Jordan D]

Didn't think of it like that, it certainly is easier to imagine the disk expanding and just taking out the middle. I think I was just too focused on width and not length.

Thanks again!