Too much philosophy, let’s get some data!
Part of the motivation for the recent posts on modeling is that I’m writing a paper for a friend. It takes off from the hexnut weight data and cubic model that appear in The (still unpublished, sigh) Model Shop. More on that soon, because I’ve found some interesting features in hexnut data. But first, another friend sent me a link to this post on doghousediaries I now share with you. Check it out.
Because I had just been doing hexnuts, I immediately thought about coins as an alternative: how do you suppose the weight of coins (Physics people: I will be reporting weight in grams. We both know that’s mass, not weight. We can handle that, right?) is related to their size?
I mean, hexnuts look more or less geometrically similar (they’re not, but that’s the other story), so you’d think their mass is cubic in the linear size: a nut that fits a half-inch bolt ought to be eight times the weight of one that fits a quarter-inch bolt. And that model fits pretty well.
So as teachers, we look for situations that stretch that understanding. EGADs has you cut out cardboard squares or circles and weigh them; since the difference in size is two-dimensional, the weight-size relationship is quadratic.
But what about coins? They look kind of two-dimensional, but the big ones tend to be thicker. So are they cubic or quadratic or something in between? Well. Whenever I come back from some other country, I usually have leftover coinage, which gets saved in that jar in the top drawer. And I have a good scale and (I know, I’m a measurement geek) a decent micrometer. So it’s an empirical question.
Yet we expect trouble—or at least more departures from any model—from the outset:
- The coins are made of different metal, so should have different densities.
- It’s obvious, looking at a bunch of coins from different countries, that although the big ones are generally thicker, the relationship between thickness and diamater is not directly proportional.
I would ordinarily just let the noble reader measure his or her own coins, but I have just done this and know that it’s a pain in the patootie. So without further ado, some data!
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| Cubic model | Quadratic model |
I did record the overall color of the coins (silver-ish, gold-ish, mixed) as a poor proxy for density, and there’s no obvious pattern. In the graphs, the four red points are Mexican coins (1, 2, 10, and 20 pesos). I selected them because their overall design is so unified and because they are generally so impressive and heavy. The 20-peso coin, man, you could wear it around you neck and feel like you’ve won a race or something.
Anyhow, it looks as if—despite all the variation—the cubic is plausible. The quadratic, however, is not. The trend is just too steep; the coins are not fundamentally two-dimensional objects.
Well. We have to look at a linearized plot, don’t we?
Sure we do. But I’ve deprived you of enough fun. Here are the data so far. I am still measuring thicknesses, so I do not include it yet. Mass is in grams, diameter in inches. It’s tab-delimited. When I copy from the page and paste into a spreadsheet (or a Fathom collection) it works right:
number unit country mass diameter color 25 cent USA 5.68 0.9526 silver 25 cent USA 5.61 0.9538 silver 25 cent USA 5.66 0.9518 silver 1 dollar USA 8.03 1.0444 gold 1 dollar USA 7.99 1.0446 gold 1 dollar USA 7.95 1.0432 gold 10 peso MEX 10.23 1.08268 mixed 1 krone SWE 6.97 0.9843 silver 2 pence GBR 7.03 1.0221 gold 20 peso MEX 15.83 1.25984 mixed 5 rand RSA 6.93 1.0203 silver 1 real BRA 6.83 1.0613 mixed 50 centavo BRA 3.96 0.9036 silver 2 peso MEX 5.17 0.9092 mixed 5 centavo BRA 4.11 0.869 gold 5 cent USA 4.95 0.8339 silver 5 cent USA 4.92 0.8362 silver 5 cent USA 4.98 0.834 silver 2 euro FRA 8.55 1.00394 mixed 1 euro GER 7.51 0.9144 mixed 1 pound GBR 9.55 0.9871 gold 10 centavos BRA 3.6 0.8656 silver 1 peso MEX 3.92 0.8293 mixed 5 euro-cent EUR 3.94 0.8367 gold 1 cent USA 2.46 0.7507 gold 1 cent USA 2.49 0.7487 gold 10 cent USA 2.25 0.7041 silver 1 krone DEN 3.68 0.7974 silver 10 öre SWE 1.45 0.5904 silver 1 dollar CAN 6.96 1.02362 gold 10 krone NOR 8.97 0.9472 silver
A Best-Case Scenario 
Did you have any aluminum coins in that jar? They’re so light that I imagine they would seriously throw things off.
Good point! But no, I don’t. I now have the thicknesses, and although they vary a lot, they let me at least get a slighty-better-than-ballpark (perhaps infield?) estimate of the densities of the coins. Those numbers came out between about 6 and 9 (g/cc), not the 2.7 we would get for Al.
Interestingly, I got a better fit using a mass=A*diameter^3 than A*diameter^B. That is, the improvement in fit from letting B vary was less than the cost of increasing the number of degrees of freedom.
I suspect that putting in Al coins or Au coins would through things off a bit. Just about all the standard coins these days are mostly zinc, with small amounts of copper or nickel. Your denser ones probably have more copper (silver is possible, but unlikely these days).