Capillary Wheels
We ignore the obvious problem of the time it would take the capillary to
respond to motion of the wheel. (Let the wheel move very slowly.)
There's a much more important mistake
in the claim made for this machine. The forces due to adhesion between
belts and water
which supposedly make this work act not only in the narrow capillary
between the belts but also
anywhere that the water surface contacts the belts.
The diagram shows what's happening. The key principle of capillary action
is that adhesion forces along a meniscus line exert a net adhesion force proportional
to the length of that line. Therefore, each length of meniscus can lift only
an amount of water whose weight equals that adhesion force. In the diagram,
the water outside the closely spaced plates is lifted until a volume
of liquid V is above the reservoir water level. But inside the meniscus,
each surface lifts that same volume of water, but in the confined space,
lifts it higher. That's the conceptual explanation of capillary tubes which
many textbooks fail to reveal. If the plates are brought closer together,
the meniscus column will rise higher, but will still contain the same volume
of water, independent of height.
So in our belt engine, the net downward force on the inside surface of a belt
(in the narrow space between the belts) is the same as the net downward force
on the outside of the belt. The net downward force on the belt sections which
form the capillary is equal to the net downward force on the remaining two
belt sections. There's zero net force on the belts even when there is a liquid
capillary column between them.
But what if...
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Forces on belts with different coatings
on inside and outside.
Not to scale.
The capillary width is exaggerated.
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John Patterson suggests to me that some true believers may complain
that my explanation isn't complete.
What if you coated the two sides of each belt with different materials?
One side of the belt could have an upward meniscus and the other side
could have a downward meniscus. Would this achieve perpetual motion?
No, it would not. Geometry thwarts you again. For a cyclic process
of this sort, the belt passing into the water must pass out again.
When all four segments of both belts are considered,
and both sides of each,
the net force due to the water acting on each belt is still zero.
Try variations. Try Möbius belts if you wish. Whatever you
try, geometry stands in your way every time. Try such things over
and over, and eventually you may discover that the geometry of
the universe, and the physical laws constrained by that geometry
are all conspiring to do one thing: absolutely prevent the
possibility of a perpetual motion machine. This may be the most
profound statement one can make about the way our universe works.
Additional discussion
Adhesion forces come into balance as the
liquid rises in the capillary, responding by changing their size to
achieve an equilibrium condition much as other elastic material
contact forces do.
Our conceptual diagram at the top of the page can be used as a starting point
for a derivation of the equation for the height of rise in a circular bore
capillary tube. The lifting force around the circular meniscus at the top is
proportional to the length of that circle, which is proportional to the radius,
R, of the tube. The weight of water lifted is proportional to the volume of
the water column, which is proportional to R2. Therefore the
height of this cylinder is proportional to R/R2 which is 1/R.
Textbooks seldom discuss capillarity these days. When they do, they have
misleading diagrams such as the ones shown here. The relative size of the
capillary column and the meniscus are greatly out of proportion. This is
yet another example of the fact that textbooks are one of the greatest obstacles
to physics education. I'll admit that for a long while I failed to see the
relative volume relations concept of this puzzle because this sort of
picture was stuck in my mind, biasing all of my thinking on the matter.
At the very least, textbooks which present such diagrams should include
a disclaimer such as "Not to relative scale," or "This diagram is
schematic only."
In 1864 Johann Ernst Friederich Lüdeke and Daniel Wilckens applied
for a British patent on a device with two partially immersed wheels whose
axes made a small angle so that on one side of the axles the wheels were
close enough together that liquid could rise there by capillary action.
Now you know why it couldn't work.
The reader is invited to consider what would happen if the inside and
outside surfaces of
the belts were coated with a material with a very different adhesion
with water. Then consider what would happen if the outside of the
wheels were coated with a material of different adhesion from the
inside. This exercise should convince you of the importance of geometry
in determining what is possible and what is impossible. In this case,
if the wheel or belt intersects the surface and moves in a closed
cycle, then what goes into the water must later come out of the water.
DES, 1 May 2003.
