Hmmm well not to push this thread "over the edge" like so many have gone
before....
Imagine you are sitting on a seat behind the unicycle wheel, with a
counterbalancing weight in front. Just by pedaling you could still go
up the hill, even though your feet are completely behind the contact
patch.
Your point of view is difficult to work with. This in physics is called
"reference frame". It is easier to find insight in some reference
frames than in others. Many times the skill lies in finding the right
way to look at things. For example, in rectangular coordinates, a
circle is hard to describe; in circular, easy (r=4.5, say).
Here let's split the notions. First, the only thing pushing the wheel
forward along the ground is pressure along the surface of the ground.
The only way to do this is to apply pressure on the pedal perpendicular
to the crank (torque). Any pressure parallel to the crank doesn't help.
Second, the only way the rider is going to stay on the unicycle is for
the center of balance of the unicycle to be vertically ahead of the
contact patch, otherwise when he applies torque the wheel will move
ahead of him and he will fall off (oh, the memories).
By examining these concepts carefully, we can see that, if we had a way
to apply torque without shifting our weight forward, all we have to do
is keep the balance point of the uni slightly ahead of the contact
patch, so that we move forward. Assume that we have infinite friction at
the ground so the wheel won't slip.
However, when the hill gets steep we don't have enough strength in our
legs to apply enough torque, unless we shift our weight forward to put
all our weight on the pedal. This is actually more than we need, but
because we can't do that at all positions of the pedals we need to
invest a little bit in the momentum of the wheel to get through the
vertical pedal position.
However, if we had, say, a motor on the unicycle that could apply torque
uniformly around the circle, we wouldn't have to do that, and could
actually creep up the hill as easily as riding slowly on a flat smooth
surface.
So although a longer crank would help us move our weight a little
farther out ahead of the contact patch, that's not the real issue, and
the contact patch explanation is a little misleading. The real issue is
that shorter cranks make the strength/weight limitations on the _torque_
(F*d) more severe, not that they make it harder to reach ahead of the
contact patch.
To see this, think of our motor-driven uni with perfect 360 degree
torque. With shorter cranks, the motor would have to push harder, but
our balancing act, just barely ahead of the contact patch, would be
exactly the same. With longer cranks, the motor would not have to push
as hard, but our balancing act is unchanged.
< I'm ignoring the diagram because it may just make the explanation more
complex. To make the diagram useful would take more elements and
vectors. >
I hope that helps, but it may not be clear enough.
--
U-Turn - Member of Generation XO
Weep in the dojo... laugh on the battlefield.
'29er Tire Study' (http://u-turn.unicyclist.com/29erTireStudy/)
'Strongest Coker Wheel in the World'
(http://www.unicyclist.com/gallery/albup39)
'New York Unicycle Club' (http://www.newyorkunicycle.com)
-- Dave Stockton
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