Can a glass of water be filled to its edge?

[davidjoe]

TL;DR Summary

Effect of gravity emanating from a spherical source on contained fluids

Looking out the window on recent flights I have given some thought to the implications of the fact the bodies of water we see are never actually flat, anywhere, at any size.

The earth beneath the bodies of water on its surface takes on the shape of a sphere of course, and one could say that water resting upon it must conform to its shape.

But that is not fully the correct explanation, is it? If one were to envision an enormous, flat bottomed, shallow pan, on the order of scale of scores of miles wide, and a couple of miles deep, and this gigantic flat pan of thin steel is placed in the middle of the ocean and filled with water, would the water in this gigantic pan take on the curved contour of the ocean surrounding it?

In other words, does it matter that the top and bottom of the gigantic pan is perfectly flat, or is going to fill up such that it creates the same domed “over the horizon” feature of the adjacent water all around it? I could also intuit that with a big enough pan, thousands of miles wide, its sides are further from the source of gravity than its center, so gravity is stronger in its closer center, depressing that water and raising the waterline along the wall of the pan.

If the pan were filled such that its rim is flush with the surface of the ocean, and the water had NOT formed a dome, then a passerby in a small craft would, on the edge of the pan, literally be looking at water that appears to slope down toward the middle of that pan, but has no impetus to try to move in the direction it is sloping. To me, this is nearly inconceivable. It would mean that if the pan were slightly lowered, water would rush in, even though it was already level with the pan before it was lowered, quite disturbing to think about.

So, which is correct, flat, convex or concave? Is every glass of water by analogy, domed by reason of the effects of gravity at the same curvature of the earth, or the opposite, or flat? Let’s ignore surface tension for the time being. Does gravity therefore allow, or perhaps require that a cup, pool or anything else that is open, to hold a different volume of water that what its dimensions suggest, and what is the shape of the surface at that opening as a result of gravity?

 

[russ_watters]

The earth beneath the bodies of water on its surface takes on the shape of a sphere of course, and one could say that water resting upon it must conform to its shape.

But that is not fully the correct explanation, is it? If one were to envision an enormous, flat bottomed, shallow pan, on the order of scale of scores of miles wide, and a couple of miles deep, and this gigantic flat pan of thin steel is placed in the middle of the ocean and filled with water, would the water in this gigantic pan take on the curved contour of the ocean surrounding it?

In other words, does it matter that the top and bottom of the gigantic pan is perfectly flat, or is going to fill up such that it creates the same domed “over the horizon” feature of the adjacent water all around it? I could also intuit that with a big enough pan, thousands of miles wide, its sides are further from the source of gravity than its center, so gravity is stronger in its closer center, depressing that water and raising the waterline along the wall of the pan.

Lakes are sitting in giant flat-bottomed pans, up to tens or hundreds of miles wide. For larger pans (the oceans), gravity acting on the water and pan bends the pan into a curve.

Your intuition here is wrong about what causes the oceans to find the elevation of least gravitational potential. The easiest way to understand it, IMO, is with pressure. If the water is "piled-up" higher in one area, the pressure at the bottom is higher, and therefore the water will move away from that area, to areas of lower pressure. Water seeks to equalize pressure.

If the pan were filled such that its rim is flush with the surface of the ocean, and the water had NOT formed a dome, then a passerby in a small craft would, on the edge of the pan, literally be looking at water that appears to slope down toward the middle of that pan, but has no impetus to try to move in the direction it is sloping. To me, this is nearly inconceivable. It would mean that if the pan were slightly lowered, water would rush in, even though it was already level with the pan before it was lowered, quite disturbing to think about.

The surface of the ocean is not a dome, it is a portion of the spherical surface of the Earth. A dome would rise above that. The rest of that sounds like gibberish, though seeming to point away from the prior "logic".

So, which is correct, flat, convex or concave? Is every glass of water by analogy, domed by reason of the effects of gravity at the same curvature of the earth, or the opposite, or flat? Let’s ignore surface tension for the time being. Does gravity therefore allow, or perhaps require that a cup, pool or anything else that is open, to hold a different volume of water that what its dimensions suggest, and what is the shape of the surface at that opening as a result of gravity?

Very small containers are ruled by surface tension. You could if you wanted to calculate the cutoff size between when a container is ruled by surface tension or gravity.

 

[davidjoe]

Would the water surface in the large pan be flat, though, since the flat pan does not conform to the curvature of the earth?

 

[DaveC426913]

Would the water surface in the large pan be flat, though, since the flat pan does not conform to the curvature of the earth?

No, it the water surface would not be flat; it would follow the curvature of the Earth.

Note that, if you puttered around this giant pan in a boat, you would observe that its edges are not horizontal and not parallel with the ocean's surface; they will seem to be pointing up at all its edges.

 

[davidjoe]

No, it the water surface would not be flat; it would follow the curvature of the Earth.

Note that, if you puttered around this giant pan in a boat, you would observe that its edges are not horizontal and not parallel with the ocean's surface; they will seem to be pointing up at all its edges.

View attachment 345644

The illustration is on point. As pictured, the pan is moot because it’s below the surface, but raising it up to flush with the surface, and stopping right there at that exact spot, the pan would be holding “more” than its volumetric dimensions, and in fact even if you raised it higher, there would still be that “dome”, or would there?

It’s not meant to be a trick question. When the pan’s rim is flush with the surface, the pressure of the ocean water is no longer maintaining that dome, “above” the rim. But if water is going to be forced out of the rising pan at flush, then it’s probably going to have to be forced out of the pan before it even is flush.

Starting at flush, envisioning water rushing out of the rising pan isn’t difficult. But envisioning it rushing into the slightly lowered pan, to take a much higher position than the distance lowered from level, is.

 

[DaveC426913]

The illustration is on point. As pictured, the pan is moot because it’s below the surface, but raising it up to flush with the surface, and stopping right there at that exact spot, the pan would be holding “more” than its volumetric dimensions, and in fact even if you raised it higher, there would still be that “dome”, or would there?

It’s not meant to be a trick question. When the pan’s rim is flush with the surface, the pressure of the ocean water is no longer maintaining that dome, “above” the rim. But if water is going to be forced out of the rising pan at flush, then it’s probably going to have to be forced out of the pan before it even is flush.

Starting at flush, envisioning water rushing out of the rising pan isn’t difficult. But envisioning it rushing into the slightly lowered pan, to take a much higher position is.

The pan's centre is closer to the centre of the Earth than its edges.

As far as the water is concerned (being that all it "knows" is gravity) the ocean's surface is flat (so that it goes nowhere on the ocean's surface) - but that the pan is an inverted dome - a sinkhole - into which it is obliged to flow.

(Let's see how good my Photoshop "Spherize" Filter Fu is...)

I have taken the original diagram and "unbent" it to illustrate what the water "sees" (and I lifted the pan to the surface).

Can you see now how the water wants to flow into the pan, which is - to it - a sinkhole?

 

[russ_watters]

Would the water surface in the large pan be flat, though, since the flat pan does not conform to the curvature of the earth?

No, it will be round, as gravity demands. Again, lake basins are largely flat, though pretty irregular, and their shape has nothing whatsoever to do with the shape of the water surface.

The illustration is on point. As pictured, the pan is moot because it’s below the surface, but raising it up to flush with the surface, and stopping right there at that exact spot, the pan would be holding “more” than its volumetric dimensions, and in fact even if you raised it higher, there would still be that “dome”, or would there?

The edges of every lake are at the surface of the water. And most lakes are above sea level. Yes, they have curved surfaces, as gravity demands. the shape of the bottom and sides has nothing to do with the curvature of the surface of the water.

 

[davidjoe]

No, it will be round, as gravity demands. Again, lake basins are largely flat, though pretty irregular, and their shape has nothing whatsoever to do with the shape of the water surface.


The edges of every lake are at the surface of the water. And most lakes are above sea level. Yes, they have curved surfaces, as gravity demands. the shape of the bottom and sides has nothing to do with the curvature of the surface of the water.

I follow you guys, both, that the surface is flat in the sense that it is parallel with the sea floor, generally, and in all directions. I also follow that gravity demands spheres on the macro scale, be they solids, liquids or gas.

If you “slice” a flat spot on a pear with a blade, the small piece displaced is a small “dome”, which is what I’m envisioning here, above the round pan, which I’m understanding above, is the shape maintained at the surface, even though it means the depth of water over flat pan’s center will be greater than at the pan’s wall.

Also, this has if unnoticeable, general applicability across fluids in open containers I’m assuming. The surface is never truly flat.

 

[Rive]

Also, this has if unnoticeable, general applicability across fluids in open containers I’m assuming. The surface is never truly flat.

Please define 'flat' with some sense of practicality involved.
At some level of the required accuracy there bound to be irregularities at molecular/atomic level, so indeed, nothing is 'flat'... but that kind of accuracy has no value without tolerances/definitions given.

I also follow that gravity demands spheres on the macro scale, be they solids, liquids or gas.

Again, a definition of 'flat' is required: with practical sense.
Ultimately, gravity means the involvement of GR and then we are really screwed with the meaning of 'flat'...

 

[davidjoe]

Please define 'flat' with some sense of practicality involved.
At some level of the required accuracy there bound to be irregularities at molecular/atomic level, so indeed, nothing is 'flat'... but that kind of accuracy has no value without tolerances/definitions given.


Again, a definition of 'flat' is required: with practical sense.
Ultimately, gravity means the involvement of GR and then we are really screwed with the meaning of 'flat'...

My sense of surface “flatness” would be that it has no variation in the vertical dimension. Captured fully in only a two dimensional plane/axis.

A very interesting implication here on the thinking shared above is if you put very sensitive measuring equipment in a windowless lab room, with water or a variety of liquids to experiment with, one would be able to discern properties about the unseeable outside environment, such as the size of the planet the lab sits on.

 

[DaveC426913]

A very interesting implication here on the thinking shared above is if you put very sensitive measuring equipment in a windowless lab room, with water or a variety of liquids to experiment with, one would be able to discern properties about the unseeable outside environment, such as the size of the planet the lab sits on.

Indeed you would.

Of course, you don't need water for that. A sufficiently long, sufficiently rigid rod, lying on your laboratory floor would not touch the floor at the ends.

 

[hutchphd]

unseeable outside environment, such as the size of the planet the lab sits on.

Of course a quick high jump will tell you something about the size of your new planet immediately. There are very sensitive gravity meters that can measure nearby mass perturbations. So yes this is possible.

 

[davidjoe]

Indeed you would.

Of course, you don't need water for that. A sufficiently long, sufficiently rigid rod, lying on your laboratory floor would not touch the floor at the ends.

I’m assuming the rationale is that the lab is on the ground floor which was made from poured concrete, that was, when more viscous, induced by gravity to dry along shape of the curvature of the earth.

This does raise a related question, if the pan is wider than the earth, with its midpoint tangent to earth, does the shape of the curved surface of water in the pan invert due to the relative strength of gravity in the center?

 

[davidjoe]

Of course a quick high jump will tell you something about the size of your new planet immediately. There are very sensitive gravity meters that can measure nearby mass perturbations. So yes this is possible.

It would tell you about its gravity, but planets vary greatly in density, so will the jump tell you much about its size? The thinking, as I understand it, with water or a substituted more viscous liquid, is that gravity will force it into uniformity with the shape of its greater, surrounding curvature, even though it is in no other way connected to it, if I am understanding the reasoning.

edit, I read your post too quickly. I though you were talking about gravity’s resistance to your jump.

 

[DaveC426913]

I’m assuming the rationale is that the lab is on the ground floor which was made from poured concrete, that was, when more viscous, induced by gravity to dry along shape of the curvature of the earth.

Right. When you get into the weeds of precise measurements of very small values in enclosed rooms, every detail counts.

This does raise a related question, if the pan is wider than the earth, with its midpoint tangent to earth, does the shape of the curved surface of water in the pan invert due to the relative strength of gravity in the center?

I'm not sure what you mean by inverting, but it's true we have been treating the mass - and therefore self-generated gravity - of the water as negligible.

 

[hutchphd]

edit, I read your post too quickly. I though you were talking about gravity’s resistance to your jump.

No its OK that is what I meant and the rest was an expansion to include more nuance. Of course the use of the verticle angle of a pendulum (or compare several) would also work. The field of a sphere can be "easilly" distinguished from uniform field in many ways without" looking out the window" . A flat dish of fluid (mrcury?) is not a bad start......lasers could be reflected from the surface for indstance, and interfernce fringes used for very smal changes in shape to be evidenced. Better folks than I have given this thought I am certtain.

 

[Rive]

My sense of surface “flatness” would be that it has no variation in the vertical dimension.

If you make it into a mathematical/theoretical problem with no tolerance, then indeed, there is no 'flat'. Even the effect of a minuscule difference in the density of the material of the stand will matter, so - no flats.
Even with the atomic stuff not considered.

However, for example for any practical purpose the surface of a puddle of mercury can be kept/made flat well within the accuracy of the wavelength of visible light.
Though tiptoeing around that puddle might make that already ripple ...

 

[russ_watters]

This does raise a related question, if the pan is wider than the earth, with its midpoint tangent to earth, does the shape of the curved surface of water in the pan invert due to the relative strength of gravity in the center?

Do you realize you're asking if water flows uphill?

 

[Vanadium 50]

My sense of surface “flatness” would be that it has no variation in the vertical dimension. Captured fully in only a two dimensional plane/axis.

Then nothing made of atoms is flat. Can we close this thread as answered now? Or do you want to rethink this?

PS Anyone want to open up the meniscus can of worms?

 

[berkeman]

PS Anyone want to open up the meniscus can of worms?

Not me, nope, not gonna happen...

 

[davidjoe]

Do you realize you're asking if water flows uphill?

High tide as a response to the moon’s gravity is more the thought, which does pull water upriver. I’m just wondering if water pulled downward onto a flat surface by stronger gravity in its middle, would have the effect of raising the water level where gravity wasn’t as strong.

 

[DaveC426913]

I’m just wondering if water pulled downward onto a flat surface by stronger gravity in its middle, would have the effect of raising the water level where gravity wasn’t as strong.

Has this not been covered by the previous material? What parameter are you changing, that would result in a different outcome?

 

[russ_watters]

I’m just wondering if water pulled downward onto a flat surface by stronger gravity in its middle, would have the effect of raising the water level where gravity wasn’t as strong.

You're asking if gravity pulling water downwards will push it upwards. The answer, obviously, is no.

It's also the same as asking why when pouring water into a bowl doesn't all of it rise over the edges of the bowl and spill onto the ground instead of staying in the bowl.

Water flows down, not up.

 

[davidjoe]

Possibly Dave, but I wasn’t sure. On your diagrams, my understanding is:

When the pan rim is flush with the surface, the water in the pan assumes the contour of the sphere, meaning that the water level is deepest over the center of the pan.

If the pan were to be elevated, still full of water, such that only the center point of its bottom touched the surface, would the profile of the water in the pan change, possibly being inverted from convex, following the earth’s contour, to concave, for the reason that gravity pulling the water downward more forcibly in the middle of the pan, displaces water not being pulled down as strongly toward the pan’s wall, because that part of the pan is further from the source of gravity, and experiences weaker gravity.

 

[russ_watters]

If the pan were to be elevated, still full of water, such that only the center point of its bottom touched the surface, would the profile of the water in the pan change

Why would you think changing what is outside the pan could impact what happens inside?

Drawing a diagram and labeling slopes/gravity vectors might help, if you're not messing with us.

 

[davidjoe]

You're asking if gravity pulling water downwards will push it upwards. The answer, obviously, is no.

It's also the same as asking why when pouring water into a bowl doesn't all of it rise over the edges of the bowl and spill onto the ground instead of staying in the bowl.

Water flows down, not up.

Lemme try expressing through another example, if we blow into a bowl of hot soup, it creates a low point forcing the soup to rise up the sides.

If the pan is huge, and gravity is strongest right under its middle, does that pull the water level down at that point, such that it rises elsewhere in the pan?

 

[russ_watters]

If the pan is huge, and gravity is strongest right under its middle, does that pull the water level down at that point, such that it rises elsewhere in the pan?

Repeating this obviously not true statement(phrased as a question...) won't make it true. You need to change your explanation to describe reality. Try this: explain in your own words why water flows downhill instead of uphill.

 

[davidjoe]

Why would you think changing what is outside the pan could impact what happens inside?

Drawing a diagram and labeling slopes/gravity vectors might help, if you're not messing with us.

I’m not messing with anyone… the reason that I’d think raising the pan up, over the earth would change things is that there is water above the rim, and at some point it seems intuitive that this water above the rim that is in the convex shape of the earth’s sphere, is going to spill out over the rim as the pan is raised.

 

[russ_watters]

the reason that I’d think raising the pan up, over the earth would change things is that there is water above the rim

No there isn't*. This is why you need to draw a diagram.

*for negligible change in elevation/this isn't a trick.

 

[davidjoe]

No there isn't. This is why you need to draw a diagram.

Dave’s first diagram captures my thought. There is water “over” the rim, when the rim is exactly flush with the ocean’s surface.


https://www.physicsforums.com/attachments/1716258259557-png.345644/

 

[russ_watters]

Dave’s diagram first diagram capturs my thought. There is water “over” the rim, when the rim is exactly flush with the ocean’s surface.

https://www.physicsforums.com/attachments/1716258259557-png.345644/

That diagram does not show elevations. Draw the rest of the Earth, with lines tracing the elevations.

 

[davidjoe]

I can’t do a better job than Dave’s, Russ. That pan’s rim would define a “flat spot” on a sphere, unless water does assume the contour of the earth, which I thought we were all in agreement on.

And if we raised that pan up out of the ocean, it’s hard not to envision water running over the rim.

 

[hutchphd]

I thought we were all in agreement on.

I, too, assumed that this was stipulated by all. Now I am lost in the arguement. Can we just agree that there are local methods to differentiate a radial field from an absolutely uniform one? What else is important here (in fact what is the question??). The OP never indicated the original motivation for the question.......

 

[russ_watters]

I can’t do a better job than Dave’s, Russ.

I told you what it's missing and you didn't even try to add it. This does not help convince me you are serious.
[Edit]
Heck, you can also tell me the numbers: what is the elevation at each edge and the center? After you lift the pan 1m, what is the new elevation of each side and the center?

That pan’s rim would define a “flat spot” on a sphere, unless water does assume the contour of the earth, which I thought we were all in agreement on.

The Earth's surface is 70% water. Obviously the contour of the ocean is the contour of the earth.

 

[davidjoe]

I told you what it's missing and you didn't even try to add it. This does not help convince me you are serious.
[Edit]
Heck, you can also tell me the numbers: what is the elevation at each edge and the center? After you lift the pan 1m, what is the new elevation of each side and the center?


The Earth's surface is 70% water. Obviously the contour of the ocean is the contour of the earth.

Usually I think pictures are worth a 1,000 words. But here I think words might do a better job. It’s agreeable, right, that because the earth is a sphere, that even if we used a small wire cheese cutter to cut a “slice” off of the surface of the ocean, it would be incredibly thin but it would be “dome shaped”, an invariable trait of spheres.

The “pan” simply replicates this on a larger scale. The water over the center of a pan will be deeper as shown in Dave’s drawing, just like the “tallest” part of that “dome” will be the “middle”.