🚀 go-pugleaf

My goodness!  If we cannot do sophomore physics, how can we settle questions
about manifolds and Higgs bosons?

  >>  Thus, consider this: If the air surrounding the balloon had exactly 
  >>  the same density on all sides, then the balloon would not move (I 
  >>  think).
  > 
  >   No, actually it would sink, I think, because there is still the effect
  >   of gravity.

Double nonsense.  A gas cannot have uniform pressure in a gravitational 
field, not in hydrostatic equilibrium.  If it did, the gas would move 
around until it developed pressure differentials such that each volume 
element would have a pressure gradient between top and bottom equal to its 
weight.  

In a perfect gas the density is proportional to pressure.  In a liquid like
water or oil the density is almost independent of pressure, but the above
requirements of hydrostatic equilibrium are preserved in either case.

Now for balloons.  We can think about an infinitely stretchy balloon, which
will always maintain the same pressure inside as the air outside, and fill it 
with helium or hydrogen, which are less dense than air at the same pressure.
Or we can imagine a perfectly rigid balloon, the volume of which doesn't 
change with outside pressure, either empty (vacuum) or containing any mass
(composition irrelevant).  Or you could take an intermediate case. 

In any case it's the outside pressure gradients that count.  If the net 
pressure gradient is greater than the weight of the balloon, it will rise.  
If the pressure gradients are less than the weight of the balloon, it will
sink.  Of course if there is no gravity and no pressure gradients, it will go
nowhere except for small random forces like Brownian motion. 

As someone pointed out, you can come to the same conclusions by thinking about
gravitational potential, which is really just force integrated over distance. 

  > If you had a very long and very skinny baloon, would it rise faster
  > floating vertically than horizontally, e.g. imagine a baloon the
  > shape of a meter stick. 

Now that's a more interesting question.  You can answer it the hard way, by
integrating pressure over the surface area, or you can just think about 
gravitational potential ...

Ben