Re: buoyancy puzzle (long!)

[Brian Whatcott <inet@INTELLISYS.NET>]

At 11:00 PM 4/30/02, Carl wrote:
> ...
> Two uniform cubes have sides of length L. Cube 1 has volume mass
> density d1, and cube 2 has d2 > d1. Their average density,
> d=(d1+d2)/2, is equal to that of an incompressible fluid filling a
> beaker. The two cubes are glued face-to-face with the lighter cube 1
> positioned directly above cube 2. (For simplicity, suppose that the
> glue has a density equal to that of the fluid, both when it is liquid
> and when it is solid.) The glued combination is thus neutrally
> buoyant in the fluid.
...
> Carl E. Mungan, Asst. Prof. of Physics  410-293-6680 (O) -3729 (F)
>        U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5026
> mungan@usna.edu    http://physics.usna.edu/physics/faculty/mungan/


Though this particular puzzle was ultimately less satisfying than Carl
 had hoped, there is no question that buoyancy puzzles are of interest
 and importance to the denizens of picturesque Annapolis.

I imagine one case of interest features a structure whose bulk modulus
is less than that of water, and which is neutrally dense at some depth.
 A displacement downwards can effectively increase the structure's
average density, as it shrinks in the increased pressure.
This is an unstable situation, and a structure of this kind will soon be
diving past some crushing depth.

A worthy design objective for man rated structures would be to provide
some automatic positive stability with depth. Perhaps a pressurized double
 shell to swell the outer skin at increasing depth could be helpful?

It would be even better to provide passive positive stabilty with depth.
One could use a shell whose bulk modulus is greater than sea water.

On the face of it, this need not be difficult: even some plastics are
stiffer than water after all.    But I suspect that a hollow structure of this
kind is rather more difficult to arrange.

But a focus of Carl's interest was the glue interface. And if one introduces
more of the real world, and dispenses with more of the physicist's
idealizations, it is easy to see that increasing compression on a glue line
can stress the glue to compression failure, while differential movement  of
the glue and substrate under compression stresses the glue line to shear
failure.
I suppose that a compression induced failure  might have little prompt
effect, but at the next tension phase, the failure would become manifest.

Concerns for submariners, no doubt.




Brian Whatcott
  Altus OK                      Eureka!