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[Phys-L] Re: Buoyancy challenge activity



At 04:50 PM 10/11/2005, Hassan, you wrote:
Hi all!

ALUMINUM FOIL BOAT

Someone asked me for help with the following challenge pertaining to
buoyancy:

"You are given a square piece of aluminum kitchen foil. You have to
make a boat out of it. The boat that holds most number of marbles before
sinking wins."

I made one suggestion to the person but haven't got the feedback
yet...(may be he lost... hehe)

Any thoughts about it?; anyone tried it?

~ Hasan Fakhruddin

This is an interesting puzzle - one that sorely tempts physicists to
wield their expertise in using the calculus to find maximal
included volume and so come up with absolutely impractical
solutions - ones that are unstable to any slight shift of the cargo.

So let us set this analytical approach aside, and instead, start with
a plainly non optimal shape - one that does however have the virtue of
carrying marbles stably.
We fold a crease into the square in order to make a cone which will
captivate any load of marbles against roll instability until an edge
submerges. This allows a forced roll to be withstood to a fair fraction
of the cone's half angle at the apex.

We next desire to exploit some of the foil that is out of the water
by the largest amount, and which makes no useful contribution.
For this purpose, we try for a shape with four way rotational symmetry.
We fold a four sided pyramid. Again, we see that this has desirable
stability against rolling, but if laden too low, the permissible roll
angle is still limited by swamping.

We next fold the "floor" to a four side squat pyramid, and fold up
the sides nearly vertical to achieve a higher lading.

At this point we might be tempted to recall that a hemisphere is
a maximal volume, but quickly retreat, knowing that a slight displacement
of the hemisphere, where the marbles can roll about, will be disastrous.

But hold on: if we stack the marbles in a regular array, will they not
show some self stability from friction at the contact points?
Possibly so! Then by all means let us optimize on a shape which is
hemi-cylindrical in two orthogonal axes. It involves waste material at
four corner gussets, but that may well be optimal for a square of foil.
Or is it? :-)


Brian Whatcott Altus OK Eureka!