Thread: Expansion of metal rings puzzle

1. Expansion of metal rings puzzle

I saw a logic puzzle and I am having doubt about the explanation that was given as the correct answer.

Say you have a coin with a hole in the center, or a washer that goes on a screw or a bolt. When it is heating up, the metal is expanding, like all metal object do. The question is: Does the diameter of the hole get bigger or smaller as the metal expands?

The answer that was given was that the diameter of the hole gets larger, with two reasons why that is the case:
1. Imagine instead of a hole, you simply draw a circle on the metal disk. When the disk expands, the circle of course has to get larger.
2. Propose the disk would be virtually two-dimensional and consist of only a single layer of atoms. The atoms that form the edge of the hole are arranged in a ring, and as the space between atoms increases, this ring has to expand.

I wouldn't bet my life on it (though I probably would 5€), but I feel that this seems wrong. And both explanations that are given for why the diameter has to expand seem to be based on false premises.
1. If you don't have a hole but simply a circle drawn on the surface, then the material inside the circle expands and pushes the atoms outside the circle away from the center. But if you have a hole instead, then atoms on the edge of the hole would have free room to expand into the hole.
2. Even before looking at the answer I was thinking about this: Metal disks are not two dimensional object. You can not simply abstract away the third dimension and expect it to still behave the same way.

One very important trait of metal is that they bend. They undergo plastic deformation. The atoms are not existing in a rigid crystal lattice. As the Disk expands under heat, the atoms on the outer edge of edge would see a greater increase of distance between them than the atoms on the inner edge of the hole, which I think necessarily will cause some kind of deformation.

It could still be the case that the hole gets actually smaller or it might be the same, but I think these explanations are completely wrong.

2. Re: Expansion of metal rings puzzle

I can confirm that the hole does expand, as unintuitive as it may seem. Ive actually done this. My 6th grade science teacher, of all things, had us do this to demonstrate that heat expands metal.

3. Re: Expansion of metal rings puzzle

To be fair, if you only heat the metal near the hole, and not the whole disk, then the hole will have to get smaller (as the disk deforms because only part of it is expanding, and that part is constrained by the part that is not).

Another way to envisage this is the same model used to show how the universe can expand with all parts moving away from all other parts.
1. Take an uninflated balloon.
2. Draw on the balloon a disk with a hole in the middle.
3. Inflate the balloon.

As the balloon expands every part of its surface expands, and thus the disk with a hole also expands - but as it does so the hole gets bigger.

4. Re: Expansion of metal rings puzzle

Originally Posted by Yora
I saw a logic puzzle and I am having doubt about the explanation that was given as the correct answer.

Say you have a coin with a hole in the center, or a washer that goes on a screw or a bolt. When it is heating up, the metal is expanding, like all metal object do. The question is: Does the diameter of the hole get bigger or smaller as the metal expands?

The answer that was given was that the diameter of the hole gets larger, with two reasons why that is the case:
1. Imagine instead of a hole, you simply draw a circle on the metal disk. When the disk expands, the circle of course has to get larger.
That definitely seems a bit like it's slightly begging the question, in a way that could get away with murder (it's not, but it could be).

If instead you instantly released a cloud of chemicals in a disk shape they would expand (diffuse), whereas if you released the chemicals in a doughnut it would fill the hole.
And yet the same 'logic' would superficially apply.

Alternatively instead, cut the ring into segments and remove the other segments (insist this is surely more logical), one of them clearly expands inside and outside (does the other person expect it to magically move), so clearly when I put the other segments back (here you need to somehow stop them noticing them overlapping).

Of course if you then accept that this overlap means there forces pushing the segment outward, till it's new circumference fits, I think it fairly easily follows that the segment is moved and is moved out faster than it expands (that is the answer is right, but it shows that there could be anything hiding in initial justification)

___
I think if you follow through with the drawn circle, you'd find some contradiction occuring fairly quickly. If you look at three cases it might be better:
1) A annulus with a void inside
2) A small disk with an annulus exactly touching
3) A larger disk with the ring drawn on.
A) Now it is 'clearly' the case that in situation 3 the ring has to either expand at least a little, or something weird has to happen
B) If that's the case it's 'clearly' also the case in case 2
C) And in that case it's also 'clearly' the case in case 1

Somewhere in B&C we need to show that the gap doesn't close entirely, (which we could do using a third annulus, and some well set up induction). 'clearly' is still doing a lot of work here, but it's reduced.

5. Re: Expansion of metal rings puzzle

Think about it this way: if you're heating the entire metal ring evenly, then the whole thing has to expand. If the whole thing expands, then perforce the ring has to get longer--and if it gets longer, then its internal diameter has to increase to accommodate that. It doesn't actually matter how thick the ring is, because every part of it has to get longer as it expands, including the inner edge. For the inner hole to get smaller you'd have to have a situation where the ring was expanding sideways but was not getting any longer, and barring some sort of weird SF material, thermal expansion is entirely symmetric.

6. Re: Expansion of metal rings puzzle

It's... An end-run around the thought exercise entirely, but I just opened up Paint, drew two concentric circles, and resized them larger, looks like the inner circle is larger than when it first began. Granted, that runs afoul of point 2 in the original post.

Maybe it might be better to think of two separate bars of metal, which are both shaped like a rectangular prism. For the sake of demonstration, let's say they're 4 units long by 1 unit wide by 1 unit tall. Now we'll heat one of them. These numbers are made up for demonstration as well, but let's say heating made it expand 25%. That one is now 5 units wide by 1.25 units long by 1.25 units tall. Since one side is longer than the others, that side is more affected by any universal change in volume. Now, take both bars, and do some magical science voodoo and twist them into a ring. The longer bar would result in a larger overall ring, and with this comes a larger inner ring. To bring this back to the beginning, the logic puzzle just assumes those bars were rings to begin with, and whether you're using bars or rings shouldn't change how expansion happens.

Besides, most logic puzzles are just designed to kick you in the groin regardless.

I, uh, was also thinking of an exercise where you flip the question around, "does cooling a metal ring cause the inner ring to contract or expand," with the concept that heating would result in the opposite of the conclusion, but then my head started hurting.

7. Re: Expansion of metal rings puzzle

Originally Posted by jayem
If instead you instantly released a cloud of chemicals in a disk shape they would expand (diffuse), whereas if you released the chemicals in a doughnut it would fill the hole.
And yet the same 'logic' would superficially apply.

Alternatively instead, cut the ring into segments and remove the other segments (insist this is surely more logical), one of them clearly expands inside and outside (does the other person expect it to magically move), so clearly when I put the other segments back (here you need to somehow stop them noticing them overlapping).
And the difference with both of these cases is that they are multiple unconnected objects, not one object made of connected atoms - and that is the reason why the hole gets bigger, do it with anything not connected and the hole will shrink. (Subject to the reolution of the size of the objects - it they are too big to move into the hole then they won't.)

8. Re: Expansion of metal rings puzzle

Originally Posted by Yora
1. If you don't have a hole but simply a circle drawn on the surface, then the material inside the circle expands and pushes the atoms outside the circle away from the center. But if you have a hole instead, then atoms on the edge of the hole would have free room to expand into the hole.
2. Even before looking at the answer I was thinking about this: Metal disks are not two dimensional object. You can not simply abstract away the third dimension and expect it to still behave the same way.
1) The atoms can't expand into the hole without breaking the crystal structure of the metal. If you heat the metal up to the point where the atoms are rearranging themselves within the molecular structure, you're essentially talking about a different material at that point, and beyond the scope of the thought experiment.

You kind of address this later: "One very important trait of metal is that they bend. They undergo plastic deformation. As the Disk expands under heat, the atoms on the outer edge of edge would see a greater increase of distance between them than the atoms on the inner edge of the hole, which I think necessarily will cause some kind of deformation."

Plastic deformation is a response to stress, not heat. You can get fancy metals that react is all kinds of ways to heat, like smart metals that can "remember" their original shape and snap back to it when heated. But that's not what's going on in the thought experiment. In this though experiment, there's no driving impetus, no potential energy, to cause the sort of deformation you're talking about. The atoms will tend to stay in their molecular formation, and as the atoms take up more space due to the heat, that means the shape of the object expands in all directions roughly evenly, which includes the spaces in that shape.

Easiest way to think of it, for me anyway, is recreating a Lego structure with offbrand building bricks that are about 1% too large. Everything in that structure will be 1% larger, including the openings and the spaces.

2) Sure you can, it's actually really common to abstract things away like that. I mean, you could posit some sort of deformation where the atoms bunch up on top of each other to stay in the middle, but a) that's going to take potential energy that just isn't there, the atoms have no driving force to rearrange themselves, and b) that would be more likely in a 2 dimensional structure, since there's nothing in that 3rd dimension to stop them from moving. In the 3D disk, most of the atoms except those on the surface, aren't actually free to move in any direction. Since they can't really move up or down, it makes sense to abstract the problem to a 2D analogue. You're essentially considering the cross section of an idealized crystal lattice. Which, granted, doesn't exist, since reality is messy, but the reasoning is sound and barring unusual phenomena or major flaws in the metal, it accurately predicts what will happen to a heated disk for about the right reasons.

9. Re: Expansion of metal rings puzzle

Originally Posted by Khedrac
And the difference with both of these cases is that they are multiple unconnected objects, not one object made of connected atoms - and that is the reason why the hole gets bigger, do it with anything not connected and the hole will shrink. (Subject to the reolution of the size of the objects - it they are too big to move into the hole then they won't.)
With the segments you only need to allow to push on each other. however at that point it shows we have to be a bit careful when adding connections (especially when the connections are to imaginary material).

10. Re: Expansion of metal rings puzzle

Originally Posted by Yora
1. If you don't have a hole but simply a circle drawn on the surface, then the material inside the circle expands and pushes the atoms outside the circle away from the center. But if you have a hole instead, then atoms on the edge of the hole would have free room to expand into the hole.
2. Even before looking at the answer I was thinking about this: Metal disks are not two dimensional object. You can not simply abstract away the third dimension and expect it to still behave the same way.
Think about it this way - when you heat a metal, it gets closer to its melting point. One of the differences between a a kilo of solid iron and a kilo of molten iron an increase in volume (it goes from 127 cm3 to 143 cm3), because that the gaps between the atoms are wider. Heat it up even more and the atoms separately entirely to form a gas.

From an anecdotal point of view - I find it hard to put my wedding ring on when it's cold. However if it's been in my pocket, then I can get it on easily - true while my finger expands and contracts, the bone generally doesn't.
There's also the old Canadian \$2 coins which were made out of 2 metals. Due to differences in the expansion rate of metals, the inner core could drop out of the coin due to the outer ring expanding faster than the inner one.

Of course the key point in all of this is that the ring is heated evenly, which is what factotum mentioned. Uneven heating can cause weird effects - like the inner edge of the circle compressing to make the hole smaller, which is where I think you're getting lost as Khedrac said.

11. Re: Expansion of metal rings puzzle

A simple, somewhat correct thought experiment/actual experiment would be to partially inflate four balloons and arrange them in a square. Not the size of the empty area in the middle, themblow them up some more. The hole in the center will, perforce, be larger.

The trick of heating a metal band to expand it has a number of useful applications. It's a traditional way to put on iron wagon tires, since when the band cools it shrinks onto the wheel, giving you one hell of a friction fit.

This was also used in the late nineteenth/early twentieth centuries to make large caliber artillery. You make the barrel liner, then heat the jacket up, and drive the liner into it. As the jacket cools, it squeezes the liner with immense force. That inwards pressure works against the outwards pressure of the propellant gases when you fire the gun.

12. Re: Expansion of metal rings puzzle

Originally Posted by warty goblin
A simple, somewhat correct thought experiment/actual experiment would be to partially inflate four balloons and arrange them in a square. Not the size of the empty area in the middle, themblow them up some more. The hole in the center will, perforce, be larger.

The trick of heating a metal band to expand it has a number of useful applications. It's a traditional way to put on iron wagon tires, since when the band cools it shrinks onto the wheel, giving you one hell of a friction fit.

This was also used in the late nineteenth/early twentieth centuries to make large caliber artillery. You make the barrel liner, then heat the jacket up, and drive the liner into it. As the jacket cools, it squeezes the liner with immense force. That inwards pressure works against the outwards pressure of the propellant gases when you fire the gun.
I always wondered why wagon wheels were made hot. Now I know. I did know that lots of things work better if they are stressed (concrete behaves very differently when it has forces on it than when at rest), but I never thought to apply it to guns.

13. Re: Expansion of metal rings puzzle

Originally Posted by Rockphed
I always wondered why wagon wheels were made hot. Now I know. I did know that lots of things work better if they are stressed (concrete behaves very differently when it has forces on it than when at rest), but I never thought to apply it to guns.
I believe it's one of the more important developments in terms of big gun battleships, since it vastly reduces the amount of metal (hence weight) needed for genuinely huge artillery. It also means that when the barrel liner gets too worn down from firing, you can replace just the liner, instead of needing a whole new gun. Unfortunately I haven't found a source that describes how you get the liner out of the jacket.

(Interestingly through WW1 the Germans were better at this than the British, who didn't have the industrial know how to make fault-free steel tubes large enough, and so instead used wire wrappings. This in turn meant their guns were generally shorter than a German gun of the same barrel diameter. Of course the Germans usually lagged behind in terms of barrel diameter, so overall British guns were more powerful. )

Nowadays they apparently don't use the jacket/liner system anymore, probably I'm part because nobody makes 15 inch gu s these days. Now they pit the inside of the barrel under enormous pressure, past the point of plastic deformation. When the pressure is released, the outside of the barrel wants to shrink back to its original size, but cannot because of the permanently expanded interior. Hence you get the same effect of the outside squeezing the inside

14. Re: Expansion of metal rings puzzle

Loosening Jam Jars too, though that's arguably a bit more disk like.
Hoses you also heat up, though I think most of that might just be making it more elastic.

15. Re: Expansion of metal rings puzzle

First things first - it expands, and you can easily find time stop video of exactly that for any common metal. As for why, consider a straight wire. If you heat it it'll expand in two directions, both in radius and length. We'll call these R1 and L1. For metals you can assume the same proportionality constant for either dimension (which holds for most materials, though you do run into weird edge cases sometimes), which we'll call k. So R2 is kR1, and L2 is kL2.

Now, take that wire and bend it into a circle. The circumference of the middle of the wire has length L1, the circumference of the interior of the wire has length L1-(2*pi*R1). Heat them up, and you get L2-(2*pi*R2), which is kL1-k(2*pi*R1), which is k(L1-2*pi*R1).

k is above 1 by definition (because we're doing thermal expansion here), and as long as L1>R1 the inner perimeter clearly also has to increase.

So, let's talk L1 and R1. If the wire is so thick there is no hole in the middle L1 = 2*pi*r, and R1 = r. That'll get a 0 in the parenthesis, which when multiplied stays 0 - which just means that the middle of the solid object stays in the same place (no surprise there). Critically though, we have an annulus of some sort. R1 has to be less than r, which means that 2*pi*R1 has to be less than L1, which means that a positive number is being multiplied by a number above 1 for the change in interior circumference. Thus, it gets bigger.

16. Re: Expansion of metal rings puzzle

Originally Posted by Brother Oni
I find it hard to put my wedding ring on when it's cold. However if it's been in my pocket, then I can get it on easily
Likewise, if your finger gets swolen and you can't take the ring off, run the ring under hot water for a minute. Much easier than trying to coat your finger in butter!

17. Re: Expansion of metal rings puzzle

Originally Posted by Ninja_Prawn
Likewise, if your finger gets swolen and you can't take the ring off, run the ring under hot water for a minute. Much easier than trying to coat your finger in butter!
Maybe its the type of ring but mine is way harder to get off when its hot in the summer time. My fingers swell far faster than my ring expands clearly.

18. Re: Expansion of metal rings puzzle

Originally Posted by Yora
I saw a logic puzzle and I am having doubt about the explanation that was given as the correct answer.

Say you have a coin with a hole in the center, or a washer that goes on a screw or a bolt. When it is heating up, the metal is expanding, like all metal object do. The question is: Does the diameter of the hole get bigger or smaller as the metal expands?

The answer that was given was that the diameter of the hole gets larger, with two reasons why that is the case:
1. Imagine instead of a hole, you simply draw a circle on the metal disk. When the disk expands, the circle of course has to get larger.
2. Propose the disk would be virtually two-dimensional and consist of only a single layer of atoms. The atoms that form the edge of the hole are arranged in a ring, and as the space between atoms increases, this ring has to expand.

I wouldn't bet my life on it (though I probably would 5€), but I feel that this seems wrong. And both explanations that are given for why the diameter has to expand seem to be based on false premises.
1. If you don't have a hole but simply a circle drawn on the surface, then the material inside the circle expands and pushes the atoms outside the circle away from the center. But if you have a hole instead, then atoms on the edge of the hole would have free room to expand into the hole.
2. Even before looking at the answer I was thinking about this: Metal disks are not two dimensional object. You can not simply abstract away the third dimension and expect it to still behave the same way.

One very important trait of metal is that they bend. They undergo plastic deformation. The atoms are not existing in a rigid crystal lattice. As the Disk expands under heat, the atoms on the outer edge of edge would see a greater increase of distance between them than the atoms on the inner edge of the hole, which I think necessarily will cause some kind of deformation.

It could still be the case that the hole gets actually smaller or it might be the same, but I think these explanations are completely wrong.
A hole is just four arches put together. When the material tries to expand it cannot expand inward without collapsing, as the arches push against each other and keep their shape. So it expands outward instead. Think about if you had an arched doorway and the wood got wet and expanded; the resistance of the arch would make it bow out, not in.

19. Re: Expansion of metal rings puzzle

Here's a website with an explanation and video: Dr. Norman Herr's site

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