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During my children's bath time we were playing with an upturned plastic jug filled with water and making some plastic divers float up inside the jug - see image.

water in upturned glass

If one of the divers swam from A to B to C, would the water pressure on them remain constant? Or would it increase at B because of the higher column of water above?

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up vote 8 down vote accepted

The pressure at A, B, and C are the same.

The pressure in the water in the jug above the outside water surface is below your ambient air pressure. If you had a long pipe instead of a jug and you kept pulling it out of the water, eventually the pressure at the top of the pipe would reach 0 and the water column would stay in place even if more pipe was raised. There would be a near vacuum in the pipe above the water (it would be at the vapor pressure of the water at whatever temperature it was at the top of the column).

Think of this another way. If the pressure at B were higher than at A and C, then the water at B would flow to A and C. Since the water would be still if you waited for the waves to die down, all the water at the same height is at the same pressure.

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As an extreme example, if the surfaces at A and C are at pressure $P$ and the height of B is more than $h=P/\rho g$ (about ten meters for water at one atmosphere) you'd pull a vacuum at the top surface of B. – rob Jun 9 '14 at 12:31
Another thing: the water level at A and C is slightly lower than it would be if you dumped the jug. This means the air pressure is slightly ( $\epsilon$ :-) ) greater now than it will be when the jug is dumped. The mean pressure over the entire bathtub-jug top surface will remain the same, I believe. – Carl Witthoft Jun 9 '14 at 13:34
Thanks for that! Some really good things to think about – Paddy Jun 9 '14 at 15:56
The pressure of the water at the top of the glass is lower than atmospheric ambient, but is higher than the pressure would be if the bottom of the glass were at atmospheric ambient (e.g. there were a membrane across it with holes large enough for air or water to flow through, but too small to allow an air bubble to flow beside the water). – supercat Jun 9 '14 at 20:08

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