Re: buoyancy puzzle (long!)

["Carl E. Mungan" <mungan@USNA.EDU>]

Brian Whatcott wrote:

> I spent a minute or so reading this puzzle in part at least.
> It appears to be indeterminate - in that the means of placing the conjoint
> blocks at depth is unspecified: is there a down push, a down pull, a
> combination?  The submersion method affects the stress at the glue joint.

I really wasn't worried about the stress *during* the immersion
process. So assume the following. Hold the blocks in their final
position in the empty beaker using a hand underneath the combination.
Now slowly fill the blocks with the fluid, before careful not to
slosh fluid around and create any shear stresses. Okay?

> This is an amazingly odd question.  Suppose the two cubes weren't
> glued.  Then the only forces on the cubes would be weight (which is
> obviously independent of depth) an buoyancy (which is obviously
> independent of depth).  The glue has to overcome the difference in
> weight -- so the burden on the glue is so obviously independent of
> depth that I can't imagine why anybody would ask the question.  I
> can't imagine why any calculation is necessary.

John, you have a remarkably sharp mind. Which is to say that what is
obvious for you is often painfully slow for the rest of us to see.
Clearly you think that solutions A (published in TPT) and B (proposed
by a senior colleague) are so obviously wrong, that you can't imagine
anyone proposing them. And yet the challenges editor of TPT has
written to me privately that *no one else* has challenged the
published solution. (Okay, maybe this just means no one is actually
*reading* them!)

I take it you agree with my solution. The crux of the matter is how
to convince others. A decisive experiment would be ideal if you can
suggest one.

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.

To put it another way. If I press a block of wood to the bottom of a
beaker of water so hard that I squeeze every drop of water out from
underneath it and then let go, will the block rise? Asking the
question this way knots my mind up in what appear to be peripheral
issues, such as the possibilities of surface tension and cold
welding. That's why I prefer the glue version of the question.

> You can perfectly well go to the extreme of replacing the glue with
> a string made of a tiny filament of glue-stuff.

I already tried that argument. My friends objected that if you have
water between the blocks, then they agree the upper block will be
buoyed upward. They even concede this will happen if you have
(nonsticky) liquid glue. So I really have to convince them the same
is true for solid glue completely filling the interface. I agree that
seems intuitive to me, but not to them apparently.

>  > I agree with A that the blocks can be torn apart by the fluid.
>
> I wouldn't say that.  The expression
>         F = (W2-W1)/2
> has no dependence on fluid properties, so if/when the blocks
> are ripped apart, it's hard to interpret it as being done "by
> the fluid".

I hear you. And yet the fluid is playing a key role in the process
obviously. One can rewrite the above expression as F=(d-d1)*V*g. This
holds even if we slightly decrease the fluid density (while holding
the cube densities constant) so that the blocks sink. This form of F
manifestly depends on the fluid density. Would it be okay if I
instead state that the fluid *mediates* the ripping apart? Carl
--
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/