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In the case of the bicycle one must neglect the mass of the bicycle
itself.
...
Pursuant to making your example more nearly analogous to the bicycle
you should shorten the seesaw to a length equal to the pedal
separation on the bike,...
One feature of this model which connects to my experience, and therefore
convinces me that it's basically correct, is what happens to the board.
Note that, considering the (board + mechanism) alone, it experiences a
torque out of the page, so that it initially tips down on the *left*.
However, angular momentum of the (board + mechanism + mass) must remain
zero, so that once the extension has occured everything will tip down on
the right as described.
That's a motorcycle see-saw. with only two feet on it you can't exert
a torque on an inertialess see-saw.
Replacing the pivot with the road is unrealistic for a bicycle. A
bicycle has an adjustable pivot. One can move the front of that pivot
from side to side by turning the handlebars. That is how one balances
a stationary bike.
Instead, I suggest an experiment. Ride a bicycle standing on the pedals
(no seat contact) and only lightly touching the handle bars. Now move all
your weight onto one foot. This morning I found that this had a very
significant effect on the bicycle, leaning it over and tending to make it
turn.
You will find that the results of this experiment depend sensitively on
the initial condition, the lean (or displacement of COM from line of
support) of the rider just before the weight is transferred. Try it
*both* with the bike in a very gentle turn to the left and to the right,
both times transferring weight to the left foot. I think you may be
surprised at the different results.
PS. While riding fast in a tight cicle (to check whether the front wheel
is actually turned a bit), I also found out that it is possible to break a
rear axle by skidding severely! No damage to myself, happily.
On the flat?! You must be even heavier than I*. I used to break rear
spokes while laden with camping gear (always chainside, of course -
Murphy rides a bike), but I have never broken an axle.