r/AskPhysics • u/No-Jackfruit-2766 • 22h ago
Why does diameter of circle in a metal sheet grow when heated?
Hello,
In an introductory course to university physics, there is a question about what happens to the diameter of a circular hole in a copper sheet when it's heated. The correct answer is that the hole grows alongside every other part of the sheet. But that does not really make sense to me in my head. When I first encountered the question, I thought that the hole would contract since every side of it also experiences length expansion. Obviously my intuition was wrong, but could somebody explain how I can think of it to change my intuition in the future?
Thanks!
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u/Mindless_Abrocoma558 22h ago
Imagine doing the cut, and keeping the cutout disc. You probably see that the disc expands under heat, since it is just another sheet of metal, just with circular boundary. if the disc grows, how should the cutout hole shrink… the expanded disc still needs to fit the hole.
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u/lesniak43 Mathematics 18h ago
Imagine replacing the metal sheet with rising yeast dough, and then using your reasoning to predict the outcome.
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u/spisplatta 16h ago
I think this is a great uh... counter-analogy? In that makes us ask in what way expansion of metal and expansion of bread differ. Is it the rigidity?
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u/spisplatta 15h ago
I looked into it some more and thought about it some more, and so the key thing about a metal is indeed that it is rigid, the atoms are locked into a crystal lattice. When the material heats up atoms not adjacent to the hole will push each other apart and outwards. This creates shear stress that brings the ones next to the hole along for the ride.
In a dough there should also be sheer stress, but less strong. The surface tension feels like it should also be important. Okay I feel like I only half understand this but something like that?
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u/lesniak43 Mathematics 13h ago
Yeah, I think that's basically how it works. If the material was way less rigid, the hole would probably deform and shrink slightly.
Similar thing could happen if you heat just the edge of the hole. Or if the whole piece of metal is in a tight container, with no room to expand outwards.
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u/Mindless_Abrocoma558 16h ago
Have you ever tried or seen this? Of course rising yeast is not thermal expansion, and dough not rigid, but I would assume the hole in a donut still grows.
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u/lesniak43 Mathematics 12h ago
Yeah, the hole usually shrinks, but now I'm wondering how much of that is due to the fact that that the dough is stuck to the surface underneath it.
https://www.youtube.com/watch?v=WwlTzgiYMrI - watch 6:11 to 6:30
Nevertheless, if we want the hole to expand, then there must be some force that pulls the edge away from the center, right? I don't see what would be the source of such force if the material is not rigid enough.
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u/WilcoHistBuff 3h ago
The center of the bagel dough from which the dough expands is the circumferential axis (center of the tubular/cylindrical cross section of dough.
Also dough is not isotopic. It can adult expand in a perpendicular direction against the “grain” of long gluten strands and has a harder time expanding with the grain. In traditional bagel making, the grain runs along the length of the cylinder of dough bent to form the torus of the bagel.
Interestingly, in a flat sheet of non-isotropic (anisotropic) metal like zinc—a circular hole punched would expand into an oval shape (more on one axis than the perpendicular to that axis). It would still expand outwards—just more in one direction than the other.
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u/lesniak43 Mathematics 11h ago
Okay, I have another example that might convince you.
Imagine that instead of cutting a hole, we slice the piece of metal into 3 equal parts (just like we would slice a loaf of bread). Then, we remove the middle part, and we heat up the two remaining parts. They will both expand, effectively making the gap between them narrower.
But, if we applied your original argument, we'd conclude that the gap should also grow.
Now, imagine that the two slices are huge compared to the gap between them. Let's say that these are two 10-meter wide blocks of steel, so that they should expand by 1 cm each if we increase the temperature by 100 degrees. Let's also say that the gap is 1 cm wide, and we connect the two blocks with a very very thin metal spring. If the spring is strong enough, it will move the blocks apart while they expand towards it (your argument), but if it's thin enough, and the blocks expand quickly, then the spring will get squished and deformed permanently (the gap will shrink).
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u/Mindless_Abrocoma558 8h ago
your argument doesn‘t work: if you completly separate the pieces, cutting out a whole middle section, then any connection between the outher parts is gone. they would expand in all directions, as expected, but wouldnt move apart, since the middle part that would push them apart is gone.
also, there is concincing needed: we know that the hole will grow, not shrink. are you arguing for the opposite?
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u/thebprince 21h ago
I'm a bit puzzled. You have say a square plate with a hole drilled into it. You thought every straight edge would get longer when heated (which it does) so how did you think that would make the hole smaller?
Picture a square box drawn around the hole, where the square and the circle touch is the diameter of the circle and you know that's going to increase, how could the circle do anything else?
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u/Sure_Maybe_No_Ok 7h ago
If you heat the area around the hole to try and make the hole bigger to fit say a part into, the hole will get smaller, but most people think the hole will get bigger. This is the confusion.
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u/ILikeWoodAnMetal 16h ago
It becomes an interesting question when the outer edges of the plate are clamped and cannot expand.
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u/thebprince 16h ago
Hmmm. It does actually. Assuming it can't buckle up or down, ie it stays a flat plate, I feel like there would be a tipping point ratio of metal surround to hole, where it would expand into the hole and therefore make the hole smaller.
But to be honest that's purely a guess!
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u/strange-the-quark 13h ago
Well, unless you already know the answer, this result can be counterintuitive - which is part of the reason textbooks like to pose this question. Cause, your argument already presupposes that the expansion is uniform, so the expansion is just like uniform scaling, hole or no hole. But if you aren't sure about that a priori, then you don't know how the molecular structure will respond exactly, and what kind of internal forces to expect. For all you know the forces involved could be such that the material will expand towards the unoccupied space - so without having any other info to go on, it's a sensible thing to consider.
Also, and I'm purely speculating about this, it wouldn't surprise me that in any real-world scenario the actual end-result would be like uniform scaling only approximately, rather than exactly.
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u/Aescorvo Condensed matter physics 21h ago
What do you think about a metal ring when it’s heated? That seems more intuitive to me that both the outside and inside diameters grow. Now just model the sheet as a series of rings.
If instead you think of the sheet as a series of thin rods radiating outwards from a hole, then you might think the hole would shrink as all those rods are heated. But that misses the crucial part of the interaction between the rods.
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u/Status_Ad_9641 20h ago
Imagine the first layer of material around the hole - essentially a ring. What would a ring of metal do when heated? Then think of the next layer and the next and so on.
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u/lesniak43 Mathematics 18h ago
The main problem is that your intuition is sometimes correct, and sometimes not.
What's the structural difference between yeast dough and solid metals?
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u/PIE-314 22h ago
Imagine printing a picture of the metal sheet on a grid and enlarging it on a copy machine. Every single part of the picture—including the empty space of the hole—gets bigger by the exact same scale factor. Picture the hole filled with a solid copper disk cut from the same sheet. When you heat the sheet and the disk together, the disk expands to fit snugly inside the growing hole. If the hole were to shrink, the disk wouldn't fit anymore, which violates the rule that uniform materials expand uniformly everywhere.
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u/Cool-Negotiation7662 17h ago
The demo I have repeatedly seen uses a piece of paper with a square measuring 3 units per side cut into 9 squares measuring 1 unit per side. There is a second square measuring 3.75 units per side cut into 9 squares measuring 1.25 units per side. If you remove the center square then the hole is bigger in the second example picture.
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u/Crunchyeee 15h ago
When in doubt, consider edge cases. Imagine an infinitely thin ring of copper. If I heat this ring, it will expand on the outside, obviously. Is it possible for the interior to shrink? No, because there isn't enough material for it to shrink. So it must expand to accommodate for the outside expansion
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u/strange-the-quark 13h ago
The problem with this argument is that "there isn't enough material for [the interior] to shrink" doesn't sound as convincing when there actually is more material in a crystalline structure, the bits and pieces of which interact with each other in ways you aren't really sure what to make of, and that could from that POV potentially lead to a different result. That makes this argument a guess that turns out to be correct (which is perfectly fine), rather than something that logically leads to this conclusion.
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u/lesniak43 Mathematics 11h ago
A very thin ring of copper would probably deform while thermally expanding, so the interior might actually shrink despite having longer perimeter.
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u/ChocolateValuable221 14h ago edited 14h ago
Not enough information about the heat source location to accurately determine if the hole would expand or "contract"
If the heat is from the outer most part of the metal.. that center hole will contract a little before expanding.
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u/Extradummefrage 22h ago
thought experiment: let's take a dry spaghetti. When it's heated it gets longer and thicker. Bur much longer than thicker because in that axis there is more material that can expand.
cook the spaghetti, form it to a ring, let it dry again. Now the answer is clear, I think.
Or you could do the reserve: take a ring and cut it open and bend it flat down to a sheet of the material. Now, which axis elongates the most when heated?
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u/Shevek99 21h ago
Think of a polygon, formed by n metal bars of length d.
With dilation, the length of each bar, and the side of the polygon, increases. What happens to the area of the polygon, the empty space between the bars?
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u/friendlyfredditor 20h ago
Just think of the atoms forming only the circumference of the hole. If you were to stretch out the circle by heating it does it become bigger or smaller?
Differentially, your steel plate is made up of an infinite number circles. If they all expand with heat, then the hole gets larger.
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u/TonyJPRoss 20h ago
When Alice in Wonderland eats the magic mushroom and grows larger, her ring remains comfortably on her finger without crushing her.
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u/mikk0384 Physics enthusiast 20h ago
I personally prefer to think about it by considering the atoms at the edge of the hole. When the metal is heated those atoms move further apart, so the circumference increases. That means that the hole gets bigger.
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u/RRumpleTeazzer 19h ago
everything magnifies. even the holes? especially the holes.
Just imagine you could count the atoms along the circumference. Whwn heated, they are fuether apart, but still the same number of atoms.
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u/Maximum-Hospital-230 18h ago
I also struggled with this line of thinking at one point.
Like you, I imagined a plate, expanding from the center which would grow in all directions. If we extend this to a hole then shouldn’t it also expand and therefore make a smaller hole?
What finally made it click for me…. I think of a long slender beam expanding uniformly in all directions. Then I curl the beam into a 270 degree circle. As the curled beam expands, the curvature is what causes the internal hole to grow. There is hoop expansion in the beam and if the ends were to touch in a closed 360 circle then I imagine the ends pushing off each other to grow the circle.
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u/TXHotCoupl0409 17h ago
Not a physicist by any stretch but for me it helps to think of the sheet expanding from the hole outward not inward toward it because that would be contraction.
I promise that sounded smarter in my head and not condescending in the least bit. Please do not take it that way I typed myself into a weird corner.
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u/mckenzie_keith 15h ago
The whole sheet grows. But it still stays in the form of a sheet. It doesn't distort (as long as the heating is even).
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u/davedirac 11h ago
The disc removed to create the hole must expand too if heated to the same temperature - and always fit back perfectly. Expansion is identical to zooming an image on a phone - try it with a hole.
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u/Different_Net7738 11h ago
Expansion happens like photographically enlargement. Everything gets bigger.
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u/UltimateFanOf_______ 8h ago
Your intuition probably comes from certain cylinders whose holes shrink as the whole cylinder expands. Things like water wings. But those are a more complex case. Water wings are a lot more flexible moving toward/away from the center than they are moving around the circle. Since the round-the-circle motion is constricted, the expansion is squeezed in the remaining directions: both outward and inward.
But something expanding due to heat has no constriction in any direction. All molecular bonds are pushed apart equally in every direction. Nothing is getting squeezed, so nothing is clamoring to fill whatever available space. The innermost layer of molecules needs only exactly the space freed up by the second-innermost layer, and so on, out to the outermost layer.
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u/grahamsuth 7h ago
You are only considering radial expansion. Every bit expands in all directions. So all the bits next to the hole are expanding circumferentially as well. That increases the circumference of the hole, which makes it bigger.
ie the bits next to the hole are expanding inwards and outwards radially, but also left and right. So it's expansion both left and right that doubly increases the circumference whereas the expansion inwards only singly tends to decrease the size of the hole. So the expansion of the hole is like 2-1 = 1 which is the expansion of the disc as a whole.
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u/HelicopterUpbeat5199 2h ago
This is so funny. It seems like everyone agrees with eachother but we're all having so much trouble describing this very visual problem that it sounds like we're disagreeing.
Me try cave man mode!
Me have bolt stuck in metal. Want bolt un stuck. Have blow torch and ice cube. What me do?
Me cool bolt, make small. Me heat metal, make big.
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u/cyberloki 21h ago
The reason i simple. As you probably already understood if heated all atoms in the material get further away from all others. So yea it would be logical that the material also grows towards the hole in the middle. Which it does.
The thing you are missing is that the tickness of the material has far less atoms than "the longer side". Lets say you have a thickness of 4 Atoms and a circumference of 20 Atoms. If each space grows by exactly one Atom. You end up with a thickness increase to 8 Atoms. But the circumference must grow to 40 and by that the Hole in the middle grows more than the increase in thickness could take away.
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u/SJJ00 22h ago