About an upside down cup of water against atmosphere pressure There is an experiment we learned from high school that demonstrated how atmosphere pressure worked.
Fill a cup of water and put a cardboard on top of it, then turn it upside-down, the water will not fall out. The explanation said this was because the atmosphere pressure was greater than the water pressure, which holds the water up.
I believed this explanation once, until I found some points that confused me:
1. Is the water pressure in the cup really smaller than the atmosphere pressure?
This is what we have been taught through our life. However, consider an object in the water under sea level, it experiences the pressure from water plus atmosphere pressure. So the water under sea level must be greater than the atmosphere pressure. Even it is contained in a cup, the pressure wouldn't change. Is this True?
I read the webpage which gave the explanation excluding the reason of pressure. If the cup is fully filled, the compressibility of water is much greater than that of the air, also the surface tension of the water keeps the air out of the cup. So the water is held in the cup. This explains the problem. But I still want to ask if the water pressure smaller or greater than the atmosphere pressure in this situation.
2. When the cup is half filled with water, why it still holds?
I saw most of articles or opinions against this. They all agree that the water would not fall only if the cup is entirely filled. But I did the experiment myself, the water stayed still in the cup even it is not fully filled. Actually even with little amount of water, as long as it covers the open of the cup and the cardboard on it, the water stays in the cup.
Even the compressibility of water is much smaller, the air inside the cup provide enough compressibility, how come it still holds?
 A: I assume you meant "cardboard", not "cupboard".
When the cup is full of water, it is empty of air, and water is relatively incompressible.
So when you turn it over, in order for the water to leak out, the cardboard would have to move a small amount away from the edge of the cup, which it cannot do without expanding the water slightly, which the incompressibility of the water does not allow.
So if the seal around the edge of the cup is good, you cannot move the cardboard without reducing the pressure in the cup, and the air pressure outside is not being reduced, so the air pressure outside holds it in place.
If you did this in a vacuum (ignoring that the water would boil) it would not work. The cardboard would just fall off.
It is essentially no different that pressing the cardboard against a wet plate of glass. where it sticks unless somehow you can inject some air into the space, say by inserting a needle.
A: Actually surface tension has to play an important role in this. Ever seen when two glass plates are kept one above the other with a layer of water in the middle, it's so difficult to separate them away? Why so? See, when a layer of water gets in the middle of the two plates, there is actually a tiny air-water interface near the edges. That's where surface tension works. The contact angle of water in water-glass interface being acute, the surface of water(in contact with the air)is bulged a bit inwards. As a result, the surface film is pulled in the outward direction due to the reaction forces from the glass plates due to the surface tension of water pulling down on them(much like in a capillary). To maintain a balance there's the need for a excess pressure on the outside which amounts to $2T/r$ where $T$ is the surface tension and $r$ is the radius of curvature of the surface. And therefore the pressure on the inside(the water layer) is less than the atmospheric pressure by an amount $2T/r$. Therefore an excess of pressure pushes down on the upper plate from the up which makes the separation difficult. Plus the adhesive forces between the water and glass molecules make them stick to the water layer, which further adds to the diffculty.
(Point to be noted* The things I have mentioned here only come into existence when we try to lift the upper plate. In case the plates are left undisturbed, the water surface at the air-water interface actually bulges outward for obvious reasons, well, actually this time the excess pressure from water balances the upper plate's weight... Ok, so don't get confused)
Ok, so now what happens in the water and cardboard case is that the cardboard slightly moves down allowing for the same tiny air-water interface near the edges. To maintain an equilibrium the pressure on the inside drops down below atmospheric pressure therefore creating a difference of pressures sufficient enough to hold the cardboard's weight. Plus the adhesive forces also add to the stability.
Ok so now you might ask how, because the pressure on the inside must be more than the atmospheric pressure because of the air already present in the empty space above the half filled glass. What happens in this case is that because the cardboard goes a bit down the volume of the empty space increases. Initially the pressure of air in the empty space was equal to the atmospheric pressure, but now an increase in volume lowers it down. In some cases you can also see a bit of the water getting out of the system during your upside down turn. This is to ensure that the volume of the empty space above gets high enough and therefore the pressure gets low enough  so that the balance can be maintained.
Might be a bit long answer. But I hope it helps...
A: Atmospheric pressure is caused by air gravity and air molecules movement, air pressure in the half filled upside-down cup is lower than the air pressure outside due to less gravity, so the air pressure can still hold the water in half filled cup.
A: *

*Is the water pressure in the cup really smaller than the atmosphere pressure? 


The answer is: NO. 
The simple hydrostatic says,
The pressure along the depth of the water cup will increase. 
So, net pressure in the bowl is more than the atmospheric pressure. !!!
Then, How the card board is held ?
The $ \bf{Surface~tension}$ holds it. 


*When the cup is half filled with water, why it still holds? 


Still, $ \bf{Surface~tension}$  holds it. 
The plate will be held as long as we are breaking the force, which is provided by the surafce energy of water (with contacting medium - air in the current case).
As Mike said, (in the answers) In vaccuum (assuming water not boilling !!!),  there is No interface -> No surface energy -> Plate will not be held.
A: In a simple words, I think everything takes up space, so if you prevent the water from falling out, you succeed in preventing the air from getting in. By this way, nothing will change, everything will stay the same.
