When a Metal Ring is Heated, Does the Hole Expand, Contract, or Stay the Same Size?

Spoiler alert: The puzzle is answered in the update at the end of this post.

Most solids expand when heated and contract when cooled. Water/ice is anomalous in that it expands when cooled, at least near its freezing point. If you’ve been unfortunate to forget a bottle of water in the freezer, only to find that it has broken once its contents have frozen, then you’ve seen this effect first-hand. Similarly, if we don’t clear the water from our outdoor pipes in winter, we run the risk that they might burst should the water within freeze. You can also observe this effect by carefully filling an ice-tray up to a mark, then noticing that upon freezing the ice has expanded so that it is now higher than the mark.

Why does water have this unusual behaviour? Its molecular structure and hydrogen bonding tell the story.

Ordinary solids (that is, virtually everything except water/ice) expand upon heating and contract upon cooling. Molecules jiggle about, and when a substance is heated the jiggling becomes more vigorous on average; just as dancers move apart on a dance floor when the dance becomes more energetic, molecules tend to move further apart when their average energy increases due to heating. The attractive forces between the molecules keep them from flying apart unless the jiggling becomes truly enormous, which requires correspondingly enormous heating; we call this vaporization.

The rate at which solids expand when heated depends on the substance. Metals tend to have higher rates of expansion (per degree change in temperature) than non-metal solids, but there is variation even among metals. A table of expansion coefficients can be found here or here. As you might expect, liquids have a much greater rate of expansion than solids (because the attractive inter-molecular forces are smaller for liquids than solids), and very hard solids (such as diamond) tend to have smaller expansion rates, thanks to their very strong inter-molecular forces.

This phenomenon has practical implications. Engineers design large structures with expansion joints to prevent buckling and cracking during summer heating. You can see such expansion joints in railroad tracks, bridges, highways, and even sidewalks. You can imagine that in situations where temperature changes are extreme (for example, in spacecraft, or supersonic jets (see also here)), research into alloys that have minimal temperature expansion is important.

So here is the puzzle: Consider a washer, or some other metal ring or disk with a hole in it. When the ring is heated, we expect the ring to expand, and experiments confirm that it does expand. But does the hole in the ring expand, contract, or stay the same size?

This is meant as a training exercise, so give the puzzle serious thought before looking up the discussion (which is all over the internet). Just knowing the answer to the question is useful, of course, but the struggle to think through the puzzle is far more useful, in the same way that physical exercise is much more useful than watching others do it!

PS: The puzzle can be found on Page 221 of a delightful book by Lewis Carroll Epstein, called Thinking Physics. Full of insight, it is highly recommended for young physics students.

PPS: The dancing analogy is meant to be taken loosely; molecules do not move apart because of a conscious decision to avoid striking each other, the way polite dancers would. Rather, their average speeds increase thanks to their increase in energy, which allows them to move further away from each other before they are reeled  back in by the attractive forces acting between them.

Update: I’ve had so many hits on this post that it occurs to me that the solution to the puzzle might be useful, so here it is.

The most important point that should be made is this: The only scientifically valid solution to the puzzle is to go out and do the experiment! No amount of reasoning can ever convince us of anything in science; of course, we value reasoning, and use it to guide our thinking; but ultimately, one must do the experiment.

So go, do the experiment!

But don’t throw your wedding ring into the fire! Rather, think about what you do when you are trying to open a Mason jar, and the screw-top metal lid is stuck. You either tap on the lid with a spoon (to try to jar loose any part of the lid that is stuck), or you place the lid under hot water. You do the latter because you know the metal lid will expand more than the glass jar, and so it will be easier to get the lid off.

And by saying the metal lid will expand more than the glass jar, what we really mean is that the hole in the lid will expand.

And that is the end of the story. You do the experiment, repeat it in many different circumstances, and you draw your conclusion.

But nevertheless, it is helpful to try to use reason to understand the situation, as this will help us understand such phenomena. There are a number of ways to reason your way through to the correct conclusion, but Epstein’s answer from page 222 of Thinking Physics is a good one, and I paraphrase it in the following paragraph.

He suggests taking a square piece of metal plate, dividing it with a 3-by-3 grid into nine equal smaller squares. Then heat the entire plate. Each of the smaller squares expands. But if the central square were missing from the start (a hole), then the same expansion would take place in the other 8 squares, leaving a bigger hole. Alternatively, if you heated the entire plate and then removed the central square at the end, after it has expanded, the remaining hole is larger than the original size of a small square.

Similar reasoning applies no matter what the shape of the original metal ring is.

If you feel you’ve understood this, test your reasoning on Epstein’s following puzzle, taken from page 222 of his book: A nut is very tight on a screw. Which of the following is most likely to free it? Cooling it, heating it, either, or neither?

And, once again, the best test of your reasoning is to go try it!

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 25 January 2021.)