Have you read the article? The point is that the definition of the metre, which is used in g, originates from the length of a pendulum that swings once per second in the gravity field around Paris. So it is a matter of definitions, and the length of the metre originates from the duration of the second and the Earth's gravity field. The definitions of 1/40.000 of the Earth's circumference or ~1/300.000.000 of a light second came later.
My intuitive assumption, then, is that on Mars they would have come up with a different meter such that π² ≈ 10 "mars meters" / s².
Or alternatively stated, that the Mars meter would be much shorter than Earth's meter if they used the same approach to defining it (pendulums and seconds).
A Martian meter defined by martians should relate their average size, the number of fingers they have on their hands and some basic measure of the planet.
I mean, one meter is defined as 1/10^7 of the distance between the equator and the poles which leads to a round number in base 10.
A unit system is not just something that matches objective reality but something that has some cognitive ergonomy.
> A unit system is not just something that matches objective reality but something that has some cognitive ergonomy.
Beautifully stated!
And that's one reason why I like the US units of measurement better than SI. I mean, the divide-by-ten thing is nice and all. But _within a project_ how often are you converting between units of the same measurement (e.g, meters to centimeters)? You pick the right "size" unit for your work and then tend to stay there. So you don't get much benefit from the easy conversion in practice.
But if you're doing real hands-on work, you often need to divide by 2, 3, 4, and so on. So, for example, having a foot easily divisible by those numbers works well. And even the silly fractional stuff make sense when you're subdividing while working and measuring.
Of course it all finally breaks down when you get to super high precision (and that's probably why machinists go back to thousands of an inch and no longer fractions).
I think there's a little bit of academic snobbery with the SI units (though, it is a good idea for cross-country collaboration), but for everyday hand-on work the US system works really well. I always love the meme: There are two kinds of countries in the world, those who use the metric system and those who've gone to the moon.
I'm an AMO physicist by training and my choice of units are the "Atomic Units" where hbar, mass of the electron, charge of the electron, and permittivity are all 1. That makes writing many of the formulae really simple. Which is what you say: it has cognitive ergonomy (and makes all of the floating point calculations around the same magnitude). Then when we're all done we convert back to SI for reporting.
One example where picking units within a project is still not saving you from cognitive load is e.g. when doing woodworking. Ymmv, but I can add decimals way faster than I can add 7 9/16" + 13 23/32" (numbers picked arbitrarily but close to a precision of 1mm so if you are ok w/ that precision, you don't even need fractions in SI).
I have to admit I only read half of the article. Even if there is some historical fact there (but it was not mentioned at the beginning of the article), from a physical standpoint this comparison is already dimensionally wrong and also coincidentally only correct if you choose appropriate units. That was the point I was trying to make. There is not anything "deep" here.
"I only ran the first half of the program, but it didn't seem to give the correct answer, so it's obviously broken."
"I only read the first half of the proof, but the answer wasn't contained there, so I'm forced to conclude the proof is worthless."
You simply gave up before encountering the mathematical reason the relationship exists, why the units are different, and so on. You just ran with your incorrect initial assumption.
