# What is the math behind 'Inverted cup of water experiment'?

1. Feb 9, 2012

### Mr.Bajjgas

It's really a known experiment, I've googled it intensively to find the mathematical explanation but none were found. All I found was a simple explanation of the physical laws enabling this to happen.

The experiment:
Fill a cup of water (not to the top, just a little), place a solid piece of paper covering the opening of the bottle. Flip it and the water won't fall out and the piece of paper will stay on it's position as if it was stopping the water from coming out.

I know the air pressure + water pressure in the cup is lower than the outside pressure (atmosphere?), and that the surface tension is preventing air seeping in between the paper and the cup. This is the explanation I've got from my teacher and the internet.

Now comes my question:
How can you use math to explain how much pressure inside the cup will make the water fall out? How can you calculate it(which formulas)?I don't know many symbols in formulas so please explain the symbols you are using.

EDIT: More specifically, I want to know the formulas needed for this so that I can try to calculate this myself :)

Sincerly, Mr.Bajjgas

Last edited: Feb 9, 2012
2. Feb 9, 2012

### LawrenceC

This can get a bit complicated as it depends to an extent on the stiffness of the paper and the weight of the paper. If the paper is thin and very flexible, it will sag in the middle. That creates a little extra volume inside the cup. When that happens the pressure in the air space drops below atmospheric. When the pressure in the air space plus the pressure due to the head of water in the cup equals atmospheric pressure, the paper remains in place.

An inch of water is only 0.036 psi so we are dealing with very small pressures as compared to atmospheric pressure.