# Pressure inside when a container is closed

Let us suppose that there is a container with its lid open sitting at the bottom of an ocean. The pressure at the bottom of the container will depend on the water column above it assuming only static pressure. Now if I close the lid of the container will the pressure at the bottom of the container reduce? I think it will because the height of the water column has been reduced. The lid supports almost all the weight of the water column above it. I know it's a stupid question but I don't know where I am wrong. Am I missing something?! Sorry for the bad drawing. P is the required pressure near the bottom of the container after closing the lid. Assume that the pressure is only due to the weight of the water column.

• When you say that lid supports all the weight you forget that now that whole weight acts on the column below so the wait does not change and not the pressure. – Lost Dec 19 '20 at 18:33
• The molecules of the water above the kid are bombarding it and the lid communicates that pressure to the water inside the container which then communicates it ti the bottom and the situation is just like the original. – Lost Dec 19 '20 at 18:34
• Ok so the whole weight now acts on the lid so how does the pressure gets transferred inside as there are no holes are gaps. – shahrOZe Dec 19 '20 at 18:36
• If you keep a book on top of a table ...is the combined weight of the table and the book not increased on the floor? – Lost Dec 19 '20 at 18:37
• The lid is not cancelling out the weight as you think. – Lost Dec 19 '20 at 18:37

We often simplify the pressure at the bottom of a (dense) fluid to be $$P = \rho g h$$. But this is only the additional pressure at the bottom due to the fluid. If there is pressure at the top of the fluid, that is added in as well.

So the pressure at the bottom is $$P_{bottom} = \rho gh + P_{top}$$

When you close the lid, the column of fluid in the box is shorter, but the pressure applied to the top of the column is increased due to the rigidity of the lid. The pressure at the bottom remains unchanged.

• Ok now i got it. Can the spring model for the water molecules described here (physics.stackexchange.com/questions/312119/…) explain this too? – shahrOZe Dec 22 '20 at 9:40
• Sure. At the beginning the springs are compressed by the weight of the water. When you close the lid, the lid keeps them compressed. If you take the water off the top (lift the container out of the water), then the pressure inside remains the same due to the lid. – BowlOfRed Dec 22 '20 at 20:47
• Thanks for the amazing answer. – shahrOZe Dec 23 '20 at 11:16
• @BowlofRed. I was thinking on the same lines as your answer but I was confused regarding something. If the lid is completely rigid and supported by the walls connected to the ocean bed then the weight gets transferred to the ocean bed and hence the pressure. How does it add to the inside bottom of the container?. Note here that I do agree that the pressure will get added but I am disagreeing on the point where it gets added. – Lost Dec 25 '20 at 14:35
• If the container is closed, there is no path to the ocean bed, just to the other walls (including the floor). – BowlOfRed Dec 25 '20 at 16:55

When you attach the lid the pressure above and below it is the same, it's in equilibrium. The pressure inside the container doesn't change when the lid is attached. Yes, the water column above the base of the container has been reduced in size but the top of that small container-sized column is at high pressure, it's not at surface pressure.

The answer depends on the type of lid.

For an ideally rigid lid, the weight of all the water above it will be transferred to the floor, through the container. So, the pressure at the bottom of the container will be the same as what it would have been, if the container were outside water.

For an ideally elastic lid, the pressure will completely be transferred into the container, and the pressure at the bottom of the container will be same as that outside the container, at the same depth.

For real container in between the above two cases, the pressure will be intermediate of the above two, based on how much is transferred through the lid, and how much is tranferred throught the container.