# 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

@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