| Hi Joe. Not sure if I am reading your questions correctly, 
but I will try to help and hopefully be corect for what I think you are 
asking. Remember that 1 atm is the air pressure at the 
surface and is the 14.7 psi we have pressing against our bodies (GENERALLY 
speaking)  ALL the time.  If you want to submerge something to say 33 feet, 
that would be 2 atm. Remember to count the single atmosphere we already have pressing on us 
all the time. So 33 ft is 2 atm, 66 ft is 3 atm, etc.  You asked.....#1 One submerges a 
bubble of air (enclosed in any material) and open to ambient pressure to 
1atm . Equal forces of pressure are applied to either side of the material. I 
assume that the lifting force of the air in pounds of buoyancy is not cancelled 
out by the equalization of that pressure. That the material is subjected to a 
stress in psi equal to that force. Then if one were to apply an opposing force 
,as in ballast, that the material is then subjected to both forces. Is this a 
correct assumption? The first part of your question I assume you mean't 
submerging a bubble of air at ambient pressure at greater than 1 
atm. The second part of your question states about 
applying equal force to either side of the material. I assume you mean the water 
pressure force going into the main soft ballast tanks 
or even part of the subs ambient hull (depending 
on design) and then having equal force applied by releasing air into the same space and therefore the 
material (hull or tanks) has equal force applied onto it from both sides which 
cancels each other's force out, remembering of course that there has to be a 
slight pressure difference inside, in that the inside air pressure force holding 
out the water has to be slightly greater than the water's 
pressure trying to get in. That sounds correct if that is what you mean't. I 
wasn't quite sure what your third part of the question mean't  when you asked whether the lifting force of the air 
was not cancelled out by the equalization of that pressure. Are you asking if by 
pressurizing the vessel, (material, bubble, whatever) if  the lifting force of the air is decreased by being 
compressed against the outside water pressure? If that is what you were asking, 
I am not the best to answer this, but I do know that the 
deeper you go and the more you compress air, it does 
have an effect on its buoyancy I believe. I'll leave that one for someone a bit 
more knowledgeable in math to compute buoyancy loss due to air 
molecule compression. But I hope I helped 
you some, and remember we are always at approximately 1 atm all the 
time. Bill. 
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