201–210 of 352 posts
`T = 2π√(L/g)`
substitute T = 2 s and L = 1 m:
`2 s = 2π√(1 m / g)`
solve for g:
`g = π² m/s²`.
This holds up in any strength of gravity, but the length of a meter would be different depending on it.
[1]. This actually was proposed by Talleyrand in 1790. Imagine the world if this were true!
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A standard free measure for distance? Sounds dubious.
No. The other way around. Two seconds is the period of a pendulum with a length of two Sumerian cubits. (One meter is thus two Sumerian cubits, but that's an artifact due to us still using Sumerian time measurements.) P.S. I don't know why Sumerians used a factor of two. Americans still divide the day into two 12 hour spans, according to Sumerian fashion. P.P.S. One second is 1/(2*12*60*60) of a solar day. 12 and 60…
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Usefully, the speed of light is extremely close to one foot per nanosecond. This makes reasoning about things like light propagation delays in circuits much easier.
I really wish we had known this back before it was way too late to seriously change our units around. It would mean that our SI length units wouldn't have to have some absolutely ridiculous denominator to derive them from physical constants, and also the term "metric foot" is pretty fun.
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No, the equality requires the length of a 2 second period pendulum be g / pi^2. Change your definition of length - that no longer holds true. g in imperial units is 32 after all. g has units; pi does not
The equation holds in imperial units as well. The length of the 2 second pendulum needs to be in feet AND the value of g in ft/sec2.
If the length of a meter were defined as the length of a seconds pendulum [1], then g would equal exactly π². From the pendulum equation: `T = 2π√(L/g)` substitute T = 2 s and L = 1 m: `2 s = 2π√(1 m / g)` solve for g: `g = π² m/s²`. This holds up in any strength of gravity, but the length of a meter would be different depending on it. [1]. This actually was proposed by Talleyrand in 1790. Imagine the world if this w…
why not?
Another interesting coincidence (or perhaps a decades-long dad-joke troll perpetrated by German-speaking scientists) is that 1 hertz is roughly equal to the frequency at which a human heart (“Herz” in german, with a nearly indistinguishable pronunciation to “Hertz”) beats.
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Another bad way to check for non-coincidences is to use a value like g which changes depending on your location. Pi is the same everywhere in the universe. g on Earth: 9.8 m/s² g on Earth's moon: 1.62 m/s² g on Mars: 3.71 m/s² g on Jupiter: 24.79 m/s² g on Pluto: 0.62 m/s² g on the Sun: 274 m/s² (Jupiter's estimate for g is at the cloud tops, and the Sun's is for the photosphere, as neither body has a solid surface.)
Fun fact: pi is both the same, and not the same, in all of those places, too. Because geometry. If you consider pi to just be a convenient name for a fixed numerical constant based on a particular identity found in Euclidean space, then yes: by definition it's the same everywhere because pi is just an alias for a very specific number. And that sentence already tells us it's not really a "universal" constant: it's a m…
That isn’t a different pi. That’s a different ratio. Your hint is that there are ways to calculate pi besides the ratio of a circle’s circumference to its diameter. This constant folks have named pi shows up in situations besides Euclidean space.
This is neat, but still something if a coincidence. It appears the first definition of a metre is in fact around 1/4e10 the circumference of Earth, and the further coincidence is that a 1m mathematical pendulum has a period of almost exactly 2 seconds. So there's still a neat relationship between mass/radius of Earth, its diurnal rotation period and the Babylonian division of it into 86,400 seconds.
According to the article the 1/4e10 circumference definition came second
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Aside from the fact that the post already explained what the actual historical connection is, your explanation requires some serious hand-waving about the mass of the Earth and the gravitational constant, neither of which were known when the meter was first defined.
Reasonably accurate values for both M_earth and G were known at the time the SI meter was defined. Also it's not too hard to extend this. M_earth is a function of Earth's radius which goes into the definition of the meter. G is a function of earth's orbital period, which goes into the definition of the second. Further our definition of mass is based on the density of water, which is chosen because it is a stable liqu…
But even that doesn’t matter, because the mass of the Earth didn’t play a direct role in the definition of the meter. If you take out the whole thing about the meter’s definition targeting half a toise, then all you have is “related to the circumference of the Earth”, and it would be a monumental coincidence if the mass of the earth and gravitational constant just conspired to somehow drop an unadulterated pi^2 out of the math.