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I was currently reading about Archimedes' principle and saw following picture on the wikipedia page:

enter image description here

It says that the medium is water. I know from Archimedes' principle that we can calculate the buoyancy force by calculating the hydrostatic pressure on the top of the object and below the object, and subtract the forces resulting in the buoyancy force.

However, why does this work? As far as I know, the hydrostatic pressure depends on depth ($p=ρ*g*h$) because there are water layers on top of each other which increase the pressure with depth. However,under the cube (where it says 2943 Pa) there is no water above it, but we can still calculate the water hydrostatic pressure like we're used to. Why?

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  • $\begingroup$ Wikipedia page translated into English $\endgroup$
    – Farcher
    Commented Aug 28, 2023 at 9:21
  • $\begingroup$ Interesting that the English version of the Wikipedia article doesn’t have this figure. It should also be noted that the pressures shown are guage, not absolute $\endgroup$
    – Bob D
    Commented Aug 28, 2023 at 9:22

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Pressure is a scalar quantity and so the pressure immediately below the cube is $2943\,\rm Pa$ at every point along the bottom of the cube and along a horizontal line extending to the right and left of the base of the cube.
If that were not the case there would be net horizontal forces acting on water elements along that horizontal line and those elements would not be in static equilibrium.

Given that the pressure immediately below the cube is $2943\,\rm Pa$, the water exerts an upward (normal} force equal to $2943 \,A\,\rm N$ on the base of the cube where $A$ is the base area of the cube.

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  • $\begingroup$ Makes sense, but why exactly is the pressure the same under the cube as well as left and right from it (as you put along a horizontal line)? Having trouble understanding because the water under the cube isn't in contact with water above it which could transfer hydrostatic pressure $\endgroup$
    – 冰淇淋
    Commented Aug 30, 2023 at 9:46
  • $\begingroup$ Along a horizontal plane the pressure must be the same otherwise the fluid would not be in equilibrium. The pressure is determined by the vertical height of fluid at the sides of the block, again otherwise it would not be in equilibrium. The situation would be radically different if the block had a cross section exactly the same size as the container and there was no liquid at the vertical parts of the block and fluid could not squeeze through. The pressure at the bottom of the block would then be dictated by the weight of the block and the weight of fluid above the block. $\endgroup$
    – Farcher
    Commented Aug 30, 2023 at 12:00
  • $\begingroup$ Thank you it makes sense now $\endgroup$
    – 冰淇淋
    Commented Aug 30, 2023 at 15:05

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