Fill a bath tub with six inches of water is the water completely flat or level on top?
Ponder this: Is a lake 'flat or level on top'?? How about an ocean??
Oops, I stopped pondering. What now?
A bath tub is not "flat or level" because surface tension causes a slight rise at the edges.
As for a lake or ocean, exactly what do you mean by "flat or level" at this scale?
that it isn't...it's curved by gravity....so must be the bath tub water affected.
Over short distances however, 'level' water works great..as with a water level....a long hose filled with water, plexiglass tubes equally marked at each end....stretch it out, adjust the marks for equal levels at each end...voila....two 'level' points......
What exactly does "level" mean on scales where tidal forces are significant?
has meaning only locally...based on our usual interpretation.....over larger scales I guess all we can say is that the locally flat space is perpendicular to the tidal force.
I was thinking in terms of surveying if you are on one end of two points reading would be off if you are in the middle of two points reading would be ok.
You posted this thread 11 months ago. Could you please get to the point a little faster.
Note that for small distances the curvature of the earth is insignificant and can in some circumstances be included in the definition of "level". In either case, water is used for leveling:
It is believed that water-filled trenches were used to level the Egyptian pyramids.
The tub's water surface approximates a single side of a polygon, with respect to the Earth and gravity...
Wikipedia says, "A megagon is a polygon with 1 million sides...for a circle the size of the Earth, with a circumference of 40,075 kilometers, the difference between the perimeter of the megagon and the circumference of the circle comes to less than 1/16 millimeters."
So this is a comparison between an ideally flat side segment and the corresponding curved piece of circumference that subtend the same angle... equivalent to the ideally flat water vs the gravitationally curved water.
That total difference of 1/16mm (0.0625mm) is distributed across one million sides of length 40.075m each.
So each 40.075m length has an individual contribution to the total error of 0.0000000625mm.
If your bath tub water surface is on the order of 1m, that's 40 times smaller, so a flat tub water surface would deviate from curved the amount corresponding to whatever the 40 million side polygon would be called, with an individual side's part of the error being 0.0000000016mm, or 1.6^-9mm.
So, the difference between the flat and curved water surfaces in the tub is 1.6^-9mm.
The distance between two hydrogen atoms in a molecule of water is about 1.5 angstroms, which is 1.5^-7mm.
So the difference between an ideally gravitationally curved vs true flat water surface for the size of a 1m tub is about 100 water molecules with their H to H hypotenuses aligned end to end.
I'm not sure what the surface of still (unfrozen) water is like at this close inspection, but I suspect that the movement going on in the top layer may be well active enough to exceed the precision of the 1.6^-9mm error.
Maybe someone with knowledge of the work of Einstein, Nernst, and Stoke could confirm...?
You mean with http://www.surveyantiques.com/image-viewer.htm?old_survey_photos/1920s_Surveyor_Gurley_Transit.jpg [Broken]
How wide is this bathtub, anyway?
That is a 1/16 millimeter discrepancy in circumference.
That is a calculation of the discrepancy in the width of the bathtub surface, not in its height. But, it is an incorrect calculation. The discrepancy in circumference scales with the the difference between 1 and the cosine of a small angle. That means that the discrepancy will tend to scale down in inverse proportion with the square of the number of sides. The discrepancy per side will tend to scale down in inverse proportion with the cube of the number of sides.
The discrepancy in height scales with the difference between 0 and the sine of a small angle. On a per unit distance basis, this scales down in inverse proportion with the number of sides. On a per-side basis, this scales down in inverse proportion with the square of the number of sides.
The radius of the earth is around 6,000,000 meters. The angle subtended by a one meter bathtub is accordingly, about one six-millionth of a radian. The half angle is one twelve-millionth of a radian. Multiply one half meter by one twelve-millionth of a radian and you are at about 0.00004 mm.
That's how much the center of the chord formed by a flat 1 meter bathtub surface would dip below an earth-sized perfect circle. Yes, it's small. But it's not atomic radius small.
Thanks, that's much better; I kind of began suspecting my indirect approach of using the circumfirence error as a "proxie" for the true error when the resulting values were so small.
I did some searching to get a sense of the magnitude of molecular motion excursions for liquid water to gauge the limit of measuring the flatness of a water surface. I found lots of info about Brownian motion but no particular numbers, except that the collision frequency is about ^20 and the particle (not the water) can have velocities from 10 to 1000 cm/s.
I'm still wondering if your 0.00004mm would be "under" or "within" the range of motion at the water surface.
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