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I have quite a basic question in hydrostatics that's bugging me and would be happy to get some help.

Suppose I have a tube inside a water tank (the tube's opening is inside the water) and say the water inside of the tube raises to some height $h$ above the water surface inside the tank. Thus, the pressure of the water surface inside the tube would be $P=P_{atm}+\rho_w gh$, where $\rho_w$ is the density of water. Now I have the following two cases:

(1) Suppose I add, say, $5cm$ of water to the tank. I expect the water level inside the tube to also raise by $5cm$.

(2) Suppose I now add, say, $5cm$ of oil (which has a lower density than water) to the tank. I expect (might be wrong here) that the water level inside the tube won't raise.

I'm unable to convince myself through the simple pressure dependence formula on the height on why either of these cases I described is true.

I'd be happy to get some help here and know what I'm missing.

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2 Answers 2

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Suppose I have a tube inside a water tank (the tube's opening is inside the water) and say the water inside of the tube raises to some height $h$ above the water surface inside the tank.

That's impossible: Law of Communicating Vessels.

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If you are referring to the capillary effect and using a tube thin enough, h is given by Jurin's law, supposition one is correct. For supposition two, assuming the tube opening remains in the water below the oil to water interface, the water will not quite rise 5cm when 5cm of oil is added. It will rise by the amount it would rise if water equaling the weight of the oil were added, slightly less than 5cm as water is slightly heavier than oil by volume.

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