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Re: sailing upwind



You guys seem to like this problem. Recall the simpler one which, I
think, will be easier to analyze*. You will see that John Denker's
point about mechanical advantage being important is clearly
illustrated here.

Take a child's bicycle with training wheels attached (and with the
child's permission, of course). Orient the cranks vertically. Now
push the lower pedal toward the rear of the bicycle with your hand.
Which direction does the bicycle move? Toward the rear, of course;
how else *could* it move? After all, that's the direction in which
you were pushing. [But some people will guess the opposite!]

Two objections must be raised to this answer. First, note that when
you reattach the child to the bicycle, and the child pushes the
lower pedal in the same direction without pushing on the upper pedal
at all, the bicycle moves forward! Aha, you say, you can't fool me;
the kid is pushing forward on the bike's handlebars and seat, and
she's pushing harder in the forward direction than in the backward
direction.

You'd be right, of course, *so long as she and the bike are
accelerating*, but you must be careful about such glib explanations.
If she is moving at a constant speed the forces she exerts on the
bicycle in the direction of its motion must exactly balance. If
they did not she would accelerate with respect to the machine.

Oh, yes, the second objection I mentioned. Now the bike I've been
talking about is the one I bought my granddaughter last week at Toys
R Us. Being a perverse sort I once made a special bike with training
wheels which I demonstrated at an AAPT meeting here in Vancouver.
That bike moved *forward* when I pushed backward on the lower pedal!
The reason? The mechanical advantage of the system was greater than
one**. I used my touring bike in its lowest gear (24 inches) and an
extension on the crank to lengthen it sufficiently. If one does this
the pedal will move backward when the bike moves forward.

It follows that there is a point on the crank at which one can push
backward - and the bike won't move either way. That works.

Leigh

*If you want my analysis of the boat version I can do that sometime
in the new year.

**It should be pointed out that a bicycle is a device which achieves
proper impedance matching for horizontal transport by providing a
mechanical advantage considerably less than one. On a penny-farthing
bicycle the mechanical advantage is just the ratio of the crank
length to the radius of the driving wheel; on a modern bicyle one
can provide an adjustable mechanical advantage using a derailleur
transmission. I have never encountered a bicycle which had a
mechanical advantage greater than one achieved in this manner, but
there is no reason, in principle, that it cannot be done.