[Phys-L] Re: Buoyancy question

[betwys1 at SBCGLOBAL.NET (Brian Whatcott)]

After I wrote the following note earlier today:

> If I place a sewing needle on a piece of tissue paper and
> float it on clean water, the paper waterlogs and sinks,
> and the needle floats.
> If I examine the surface of the water I see that it is
> depressed under the needle. The depression curves up
> smoothly to the surface.
> Though the water surface layer is supportive, I wonder if the total
> depression displaces as much water weight as the needle weighs?

Gary Turner <turner@MORNINGSIDE.EDU>,
you responded in this way:

> No, it has to occur before there is that much water displaced.
>
> When the surface is stretched, its PE increases, at an increasing rate
> (w.r.t. displacement), going until it reaches a maximum level - at which
> point it breaks.  Let's assume we don't get there - otherwise we don't have
> a problem to consider.
> As the needle settles onto the water, its PE decreases, but that of the
> displaced water increases, at an increasing rate (displaced water is moving
> over a greater vertical distance).  The net effect is one of decrease, at a
> decreasing rate.
>
> When the two rates match (so add to zero), we have a local equilibrium and
> that is where the needle floats.

I found this to be an interesting description of the skin effect-assisted
buoyancy effect.  I had difficulty justifying the assertion that the rate
of surface PE increases with increasing displacement.

If I take a model used elsewhere in this thread, of a long thin cylinder,
vertically plunged into the water, the surface's rate of PE increase seems
to be constant with displacement increase. Or did I miss something?

> Suppose though that the combined grav PE fell more rapidly than the PE of
> the surface rose, even up to the point where the needle was entirely below
> the water "level".  Suppose further that the  surface does not close in on
> itself - like dropping a cannonball onto a loose sheet of polythene - that
> is trivial, it's going to sink!.

Hmmm...I seem to be finding Devil's Advocate counter-models:
A cannonball sinks a loose sheet of polythene floating on the surface:
but a cannonball does not sink a circular sheet of polythene fixed at its
perimeter to a wooden hoop. Or at least I can visualize circumstances where
the cannonball does not sink for this arrangement.
This would be comparable to an inflatable whose bottom skin is attached to
an inflated perimeter ring.
I would enjoy more on this balanced rate model that Gary laid out.

Brian Whatcott    Altus OK    Eureka!