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I would like to know if Archimedes law take in equation the lack of matter. Fluid density at 1. An object with density of 0 (or near 0) is put in liquid, it's fixed. The external sphere where liquid is inside is fixed too. The lack of matter change the attraction in all the sphere itself, no ? Density change with depth, no ? How forces can be calculate ?

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I don't understand the question. What is $a$? And why are there lines coming from the black point? What makes the black point special? Is this a vertical cross-section of the geometry? And what do you mean "change the attraction in all the sphere"? There is no attraction in Archimedes principle. –  DumpsterDoofus Oct 24 '13 at 22:20
    
I changed my image –  Ludovic Oct 25 '13 at 5:19
    
@Ludovic have a look at bosons, fermions, bose-einstien condensates, superfluids. And then Superfluid Vacuum Theory –  Waqar Ahmad Oct 25 '13 at 6:14
    
It's quite unclear HOW the floating object is kept fixed. The water on top of it is pulled down by gravity, creating a pressure underneath the object pushing it up against its (unseen) fixture. –  MSalters Oct 26 '13 at 16:01
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Archimedes' Principle states that a body in a fluid experiences a buoyancy force equal to the weight of the fluid it displaces.

In this case, the weight of the fluid will be the gravitational attraction it feels due to everything else present, i.e. the rest of the fluid and the heavy mass. It doesn't matter if you consider the gravitational attraction it feels due to itself, because it will cancel out.

Note that the gravitational attraction of the rest of the fluid is not affected by the presence or absence of that displaced fluid part. The net gravitational field will be affected, though. But we do not need to consider that.

So all you need to do is compute the gravitational attraction the displaced fluid would feel, and that will be equal to the buoyancy force on the mass-less object.

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"Note that the gravitational attraction of the rest of the fluid is not affected by the presence or absence of that displaced fluid part. " I don't understand this point. If sphere is fixed and the gas object fixed too, for me the pressure in water has changed by the presence of a hole in it. If sphere can move, all fluid will move but it can't. –  Ludovic Oct 25 '13 at 5:22
    
@Ludovic: Are you assuming that the water is massive in comparison to the external gravitational field? Are you even assuming an external gravitational field? Archimedes was working on the surface of a planet many times heavier than his experiments. –  MSalters Oct 26 '13 at 16:05
    
Imagine object big enough like a planet. But there is a sphere around outer of the planet that prevent to move (theoretical study). Put in the planet the white object and fixed it too. How is Archimedes in this case ? –  Ludovic Oct 26 '13 at 17:16
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