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If I consider a system give like this

enter image description here

where W represents water and O represents any other liquid denser than water and the black object is a sphere of unknown density. according to a question i have to calculate density of sphere on basis of the part of sphere immersed in liquid and water respectively. when i look at the solution it considered the buoyant force from both liquid and Water. I get the buoyant force by liquid part but what i don't understand why water would apply any buoyant force because there is no water below the sphere to exert force on it , only the surface in contact with water would exert force that too in downward direction. So why did the solution had both the buoyant force acting on the sphere.

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Usually, oil floats on top of water :).

It is true that the pressure exerted by the water exerts a downward force on the sphere. But the water indirectly pushes upward on the sphere as well: the fact that it's there increases the pressure in the oil, so the force exerted by the oil is greater than it would be if the water wasn't there.

Archimedes' principle allows you to skip thinking about the actual pressure on the surface of the sphere that is the origin of the buoyancy force: the total buoyancy is just equal to the weight of the fluid displaced.

So if you were imagine the oil-water interface extending through the sphere (the way it would be if the sphere wasn't there) and then calculate the total weight of the water that it would take to fill the part of the sphere above this surface, and that of the oil it would take to fill the part below it, that's your total buoyancy.

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