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comments: toroidal perpetual motion machine



Miscellaneous comments:

1) Draw a grid on the rubber if you need help visualizing how areas grow
and shrink as it rolls.

2) If you want a full understanding of the meniscus you must consider three
quantities:
a) water/rubber interfacial energy
b) rubber/air interfacial energy
c) air/water interfacial energy
but let's simplify the discussion by concentrating on (a) for now.

3) Suppose the tube rolls by an amount dTheta in the direction
outside->bottom->inside->top->outside

Then we can write down an area budget as follows:
-- A lot of wet area created at boundary at outer radius: R dTheta
-- A little wet area destroyed at boundary at inner radius: r dTheta
-- Balance of wet area destroyed by shrinkage underwater: (R-r) dTheta

Area corresponds to energy. The area balances. The energy balances. No
perpetual motion machine.

4) The work done by the meniscus is real. It is just the boundary term
necessary to balance the area shrinkage.

Surface tension (force per unit length) is related to interfacial energy
(energy per unit area) in the same way that gas pressure (force per unit
area) is related to energy density (energy per unit volume). It's a
boundary term.

To say it another way: surface tension is the Flatland equivalent of pressure.

Being able to see the connection between a boundary term and the
corresponding integral quantity comes in handy in lots of situations. For
instance, Bernoulli's principle is easier to understand in terms of energy
than pressure.

5) This analysis is manifestly dimensionally correct. It scales correctly
if we consider tubes of differing R and r. It scales correctly if we
consider fluids with different amounts of surface tension.