Re: buoyancy puzzle (long!)

["John S. Denker" <jsd@MONMOUTH.COM>]

"Carl E. Mungan" wrote:
> The key disagreement is this. If I hold a block of wood underwater
> and let it go, it will rise. The reason is that the upward force on
> the bottom, d*g*A*H, exceeds the two downward forces, namely
> d*g*A*(H-L) on the top and the weight W. But if I glue that block to
> a piece of steel, there is no longer water under it. There is now
> solid glue under it. Hence we no longer have the upward force on the
> bottom due to water pressure because there ain't no water below it.
> So others argue and that doesn't sound completely unreasonable.

OK, I (belatedly) see where they are coming from.

When preparing my first reply, I briefly subconsciously considered
this issue, but ruled it out because it involves a sign error.  If
the blocks are glued together in the normal way, so that they stick
together before immersion, then the "missing" pressure on the unwetted
faces means the blocks are driven together.  You can view it as a
Magdeburg hemispheres situation and/or a Cartesian diver situation.
The greater the depth (i.e. pressure), the more tightly the blocks are
driven together.  The problem-statement explicitly asked for the
"maximum" depth at which the blocks would stick together, which is
diametrically wrong physics;  the only way you can get a depth-
dependence would be to construct a Magdeburg system at great depth
and raise it, asking what is the !!minimum!! depth at which it will
hold together.  Because of the sign of the effect, and because it
is inconsistent with the way normal glue works, I discarded this
way of looking at the problem.

> But if I glue that block to
> a piece of steel, there is no longer water under it.

That hides a subtle fallacy, much more subtle than the
aforementioned sign error.

Saying there is no water on the glue-covered face does
not mean that there is no pressure there!

This is subtle, and hard to visualize, because it involves
(to one way of looking at it) a "zero divided by zero"
indeterminate expression.  That is, we imagine the glue
as being incompressible.  That means:
 -- it doesn't change its volume, and/or
 -- if it did change its volume infinitesimally, there
    would be a more-than-infinitesimal upsurge in pressure.

As always, when dealing with indeterminate ratios or
infinite quantities, I recommend figuring out the !!finite!!
case;  don't pass to the limit until the last possible
moment.

In this case, imagine that the glue is a form of rubber cement.
Sticky, but somewhat compressible.  When you subject the glued
blocks to great pressure, the glue compresses a bit (decreases
in volume) and exerts a pressure on the glued faces.

Non-rubbery glue will behave the same, but the process is a
bit harder to visualize.  But the point remains:  Pressure
is a scalar.  All liquids in equilibrium feel the same
hydrostatic pressure.  Homogenous solids feel the same
hydrostatic pressure, too.  The only way to decouple
something from the hydrostatic pressure is to have a markedly
inhomogeneous geometry, such as the Magdeburg hemisphere
(there is a big discontinuity between the hard metal shell
and the squishy air inside).

If you imagine a really extreme case where the glue is rigid
and brittle, and the blocks are rigid and brittle with !!unequal!!
compressibility, then you can conjure up a scenario where the
glue-joint will shatter.  But this scenario is many, many steps
removed from the spirit of the original question.