Why does water not spill out of a cup when the cup is upside down on a smooth surface? Note: This is for a physics project, so please don't just give me the answer... 
Say we have a cup that is 3/4 full with water. We place that cup (with the water in it) upside down on a smooth table. Why doesn't the water fall out?
This is my understanding... The cup on the table has created an air tight seal that does not allow water to leak out nor any air to enter the cup. Also, as water gets deeper, the pressure increases. Therefore, there is a greater force of water pushing down on the table which the cup contains. But, wouldn't this cause the water to escape the cup?
Also, there would be an area of low pressure in the "top" of the cup, but I'm unsure how that has anything to do with the water not escaping the cup...
Any insight would be great! 
 A: Consider a couple of cases:
If the glass was completely full, no air, there is nothing inside the glass to balance the air pressure outside (the water can only exert its own vapour pressure, which at room temperature is much less than typical ambient air pressures). So, the air outside the glass keeps everything in place.
If the glass is partly filled with air, some amount of water may escape; if air is not allowed in to replace the escaped water, the pressure inside the glass will drop. The height of the water inside the glass, because of its weight, equates to a pressure, which can manifest as a pressure difference between the inside and outside of the glass. At the point the water column pressure plus the air pressure inside the glass balances the air pressure outside the glass, there is no longer any net force to push more water out of the glass.
In both cases, what keeps the water in the glass is ambient air pressure. What keeps air from entering the glass (which would allow water to escape) is surface tension. If you very carefully lift your inverted glass off the table, you can create a tiny but noticeable gap between the table and the rim of the glass, and everything still stays in place. But if you lift it too far, surface tension is no longer sufficient to keep everything in place, air starts to bubble in and water starts to run out.
A: I would imagine this has to do with surface tension. If the space between the table and the glass is small enough, you get the same effect that makes water not able to escape a small hole (the force due to the surface tension is higher than that due to the pressure).
I guess a good experiment to test if this may be the case is to leave the cup upside down and check if after a while some of it has evaporated. If it indeed has a small gap on the bottom but not enough to escape, it will still evaporate (although very slowly!). If the glass were completely sealing the water, you'll see no evaporation.
Also (although probably beyond your experimenting equipment), liquid Helium, being superfluid would probably escape.
A: The low pressure at the top of the inverted cup is everything. In order for water to flow out, a partial vacuum would have to be created at the top of the cup. Nature abhors a vacuum, and this acts as suction pulling the water back up into the cup. 
A: The pressure is not great enough to allow the water to spill. If the cup was more flexible the pressure would overcome the seal.
