I don’t have to. The equations say the net force is zero. If the net force is zero and the object is initially at rest it won’t move. If you object that the real world is glorious with a complexity beyond mere Newtonian mechanics great, but that has nothing to do with the Greek guy in the tub.
Never thought of that … was still thinking of the previous experiment in a hot tub with bottlles Perrier and wine. 
SettingSun
You have been provided with many examples and have ignored them.
Also, just because you haven’t observed something, doesn’t mean it is false or impossible.
I have never seen a total solar eclipse, therefore they don’t exist? Hell no.
This will be my last posting in this thread.
$4.95 for this?
Umm, the floating basketball? The little floating gummy things in the pop? A floating diver? How many do you want?
It is obviously something that is very difficult to achieve perfectly under normal conditions. But in a lab (or a basketball exhibit) where you can control the fluid conditions very precisely, you can certainly do it.
Try this mental experiment: Assume that you have a perfectly mixed tank of water at thermal equilibrium with the room. Take a box filled with air. Submerge it and hold it in place with a stick until waves subside completely. Remove stick. Box float? Open the box and add a little lead. Try again. Ball sink? Remove some lead. Eventually you will reach the point where you satisfy Archimedes.
I’m thinking that perhaps you have not fully thought this principle through. This priciple has NOTHING to do with whether an object will float or not.
What this is telling us is that if you submerge an object with 1 cu in of volume, the fluid pushes up on the object with as much force as 1 cu in of the fluid weighs. That is it. Nothing about floating.
To get into floating you must compare the force up from bouyancy, with the force down, due to the weight of the object. If they are exactly equal, the object will be neutrally bouyant. So claiming that this is impossible is essentially making the claim that we cannot tweak the weight inside a sealed box to make it’s weight equal to the weight of water. Which we can.
Have you ever attempted to achieve a perfectly still tank? Taken the time to shave tiny bits of material off or add air back in? I have never lifted 300 lbs. Have you? No? It must be impossible then.
Sorry, this is Abuse. If you want an argument, it’s 2 doors down on your left.
Wow, I am truly sorry 
I could the present day position, don’t read it if you don’t like it, but I will not post again to this thread …
Come on kids, let’s slow down and think a minute.
“neutral” bouyancy is just that; a razor fine distinction which exists between a smidgen positive and a smidgen negative on the “bouyancy line” which extends from helium balloons at one extreme to lead blocks at the other. Therefore it exists as an ideal perfection, like a true crircle. Any mechanism you could build, and any liquid in which you could immerse it, would only approximate that perfection to some degree of precision, maybe only a little, or maybe to 27,000 decimal places. But their will be error between the unachieavble perfection of the Form and the reality.
So before we debate too much further we should decide whether we’re seeking a scientific answer (yes, it exists), an engineering answer (it can be done for some finite duration in ideal conditions given a really big budget), or a philosophical answer (what is the meaning of things which exist and yet can’t be readily realized on our human scales of time and size & perception)
The OP can decide which question he/she’s really asking and then you’ll notice the assembled folks have already given pretty much unanimous (and IMHO, correct) verdicts (three of them).
As an engineer, I naturally favor that perspective. Assuming you start out with an object which is both less compressible than water and has the same specific gravity to within a tiny tolerance, say 1 in 10,000 (a WAG), then you’d get a positive feedback system wherein if it was pushed above the start point it’d tend to get more dense than the surrounding water & hence sink, and if it sank a bit it’d get less dense than the surrounding water and hence rise.
Given the miniscule degree of compressibility in water, particularly near the surface, I can see how an object meeting my design parameters and starting at say 10 feet of depth which was pushed gently downwards might have to sink two or three miles into the water before the bouyancy forces got big enough to reverse its direction. So the fact that none of us have experienced this phenomenon in everyday life is hardly proof of its non-existance.
I’ve also never seen the Lorentz contraction while driving, but I do believe it exists. Archimedes was silent on that topic, but I’m sure he’d have declared it foolish had someone tried to sell him on it.
Considering the folks with the bottles in the hot tub, I wonder if the non-rigid bottles filled mostly with air are reacting to the tub’s temperature gradient by expanding and contracting enough to drive a not-real tight feedback loop?
OK this is MY last reply on this subject:
SettingSun; “I have never held an object under water that didn’t float to the surface or sink to the bottom. Have you?” Yes I have and I have given you examples. I have even given you a link so that you yourself can perform the experiment with only bits of foam, empty soda cans, washers and an aquarium.
LSLGUY: “neutral” bouyancy is just that; a razor fine distinction which exists between a smidgen positive and a smidgen negative on the “bouyancy line” which extends from helium balloons at one extreme to lead blocks at the other. Therefore it exists as an ideal perfection, like a true crircle. Any mechanism you could build, and any liquid in which you could immerse it, would only approximate that perfection to some degree of precision, maybe only a little, or maybe to 27,000 decimal places. But their will be error between the unachieavble perfection of the Form and the reality."
No it’s not a razor fine distinction. Nor do you have to take it to 27,000 decimal places which as an engineer I find ludicrous, mostly 3 significant figures works fine. You can do it at home in a hot tub or an aquarium. Please see the link I have posted. Again, I reiterate, this is not Rocket Science. This is a Statics problem. The sum of the forces = ZERO. It is no more complicated that that.
Perhaps this should be posted in the PIT as in I refuse to believe in Newton’s Laws because I personally haven’t ever experienced them in my own little life. Prove to me they exist but I won’t except anything I can’t see with my own eyes and I’m keeping them shut.
It seems pretty obvious.
I agree, however, there will come a point where that error is so small that its effect is swamped by things like Brownian motion, the pressure exerted by photons hitting your object, the gravitational force exerted upon your apparatus by Sedna etc - it probably is possible to set up a real-world experiment where practical neutral buoyancy is observed and that any failure of the object to remain in position is caused by the other variables in the mix. Convection is the biggest problem, I think.
As a thought experiment, we could suggest that a spherical body (with imaginary bounds) of unmoving pure water was neutrally buoyant when immersed in a tank of unmoving pure water - sure, it might eventually become mixed with the rest of the water, but not because of any buoyancy effect.
Thing is, we’re not talking about some immutable law of physics here, we’re talking about something that is just practically very difficult - like cutting a cake exactly in half.
Oh that’s easy: Just have the guy who doesn’t cut the cake pick his piece first.
Jeez, I gave you a link to a picture showing a basketball doing exactly that!
How much more proof do you want?
I’m not sure whether you’re joking or missed the point (I do understand the I-cut-you-choose sharing solution) - it’s practically impossible to cut a cake exactly in half because we lack tools of sufficient precision to a)do the cutting and b)measure that the two pieces are exactly equal, but it is nevertheless theoretically possible to divide something like a cake into two equal portions (assuming that there is an even number of cake molecules).
[sup](Yes, I know there’s no such thing as a ‘cake molecule’).[/sup]
Ok I don’t believe I’m doing this. Tell me you’re just woooooshing us here.
It’s Newton’s First Law: The sum of the forces = the mass x the acceleration = 0
when you have static equilibrium, which is exactly what you have when you are in a neutrally bouyant condition.
Oh and since heret:
LSLGuy: Perrier bottles are made of glass. There are no compressibility effects. And if you wait a couple of minutes the air, the water and the glass bottle are in thermal equilibrium.
[Monty Python]What about really little rocks?[/MontyPython]
Sorry, I should have been more explicit - the OP implies there is some kind of law which prevents neutral buoyancy and my comment was made against this idea, not against Newton’s laws.
Whew, that makes me feel better. 
Come to think of it, I never saw any object, not in motion, hover in air above the ground either 
It can happen - if you have a room full of still air and a helium balloon with just enough gas in it to make it neutrally buoyant (In fact there’s nothing at all different about that scenario and the underwater one - air and water are both fluids in which bouyancy is possible.