# Glass - paper: Stevin's Law

In my understanding the well-known experiment of the glass full of water in equilibrium with a piece of paper, the atmospheric pressure acts on a small layer inside the glass (on the top) and under the paper (outside the glass), the hydrostatic pressure (basically the weight of the water) acts downward, so in term of forces I initially have a net force $$\boldsymbol F = (p_A -p_A+\rho g h)A\hat z$$ where $A$ is the section of the glass, $\hat z$ is the vertical direction, and $h$ the height of the layer of water.

Now, when the paper bends under the weight of the water, and the air layer on the top increases in volume, so I can apply (with good approximation) $$pV=nRT,\ V\ \uparrow\ \Rightarrow\ p\downarrow\ \Rightarrow\ p'<p_A$$

So we have: $$\boldsymbol F = A(p'-p_A+\rho g h)\hat z$$ and since $p'-p_A<0$, it is possible to have $\boldsymbol F$ upward (clearly, depending on $h$ and $\rho$)

Now, I made the experiment with water, and the bending of the paper was upward. Can I say that the only reason is the presence of the surface tension of water? Or my reasoning is lacking somewhere else too?

Moreover, if instead of the paper I put a strongly stiff material, I can't have the same effect, no matter the weight, the geometry, etc?

Thanks

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What is this "well known experiment"? Can you link something? I have no idea what you're talking about. – Ron Maimon Jul 11 '12 at 3:08
Sorry, voilà here it is! – usumdelphini Jul 11 '12 at 5:18
What is surprising? The air pressure went down, you expected the paper to go upward, and it did. Where's the puzzle? – Ron Maimon Jul 12 '12 at 9:50
My read is that when the air pressure goes down, the volume goes up and so the paper should go down. To see the paper go up was a surprise. – Carl Brannen Jul 12 '12 at 17:12

## 1 Answer

You expect that the paper will bend downwards due to the decreased pressure applied to the air gap causing an increase in its volume, but your observation is that instead the paper is bent upwards.

I think that this is probably caused by water leakage. Try the experiment again with a plastic seal over the glass, I expect you to see the film bend down. As an estimate of how much down, air pressure is around 30 feet of water. Your glass is holding a few inches. So I would think this fraction (2/12)/30 = 1/180 would be about the amount of movement, on average. For 2 inches of water this would be about 1/90 of an inch, which is detectable (look to see how reflections in the film are distorted).

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Can I also ask you how much does the surface tension matters? Actually, I expect that with a big tank this same experiment doesn't work. Is that true? – usumdelphini Jul 14 '12 at 4:15
@usumdelphini Surface tension keeps the water from leaking sideways. It also holds against the remaining water weight since the air pressure difference doesn't fully support the water. If we have a longer water column, the water pressure will be higher. It would be more difficult to retain and the air gap also needs to expand more. We know that the water pressure force is proportional to $r^2$ and the surface tension force to $r$. Thus if we increase the circumference of the glass, the pressure force will increases faster and win the battle more easily. That's why it's difficult for big tank. – Emitabsorb Jul 14 '12 at 15:49
I agree with Carl, I did an experiment to test it. If I let some water out, the paper bents slightly upward. I just need to slightly tilt the glass with a paper attached below it. It makes the water pressure more concentrated in a smaller region, thus in this region the surface tension must retain a very strong pressure. Eventually it won't be able to hold and it breaks. If I let more out, again the paper bents upward more. – Emitabsorb Jul 14 '12 at 15:51
Yes, I figured this. So, now I am wondering if the effect I described in the question is the main responsible of the non-fall of the paper, or if the surface tension (for small radius of the glass) is much more important as some people say explaining this experiment, and as confirmed by the fail with a bigger surface. In fact, with a bigger surface, the pressure effect should be the same (actually even better because of the less importance of border effects) but the surface tension becomes irrelevant. – usumdelphini Jul 14 '12 at 16:01
They should work together. The surface tension force itself is extremely weak, it loses against water's total weight even for small radius cases. You can see how little amount of water an open straw can hold (even if it is made of glass). The air pressure difference holds most of the water's weight, and the surface tension takes care of the rest which is very small. So in terms of supporting the water's weight, the surface tension is negligible. But it does an important job to prevents water leaking from sideways and it also prevents water to penetrate through little holes in the paper – Emitabsorb Jul 14 '12 at 16:24