Pedals on a Bicycle

Balance an ordinary bicycle upright without restraining its motion in any way. Position the pedals so that they are top-and-bottom. Standing on the ground alongside the bike without mounting it, grasp the top pedal and push it forwards (in the normal direction of travel). The pedal moves forward and so does the bike of course. Now, reposition the pedals as before, but this time grasp the lower pedal and push backwards. Three questions:
  1. Which way does the bike move?
  2. Which way does the pedal move?
  3. Why?
(Try to solve this puzzle in your head before referring to an actual bicycle.)

Source: Reader Peter Bentley, citing someone named "Jellio".

Solutions were receive from Peter Bentley, David Madfes, Jimmy Hu, and Dick Saunders, Jr. If anyone would like to add clarity in some way, I'm happy to post more solutions.
Here is David's solution:

The bike would move backwards, while the pedal moves forwards. The force acting on the bike as a whole imparted by the hand pushing against the pedal is larger than the forward force exerted by the drive wheel against the ground. Bike are geared in such a manner as to trade distance for force. That is the force exerted on the pedal is greater than the force exerted by the wheel against the ground as a trade off for the bike wheel moving a greater distance than the pedal. The law of conservation of energy (Right law?) requires that force x distance be constant.

It may be possible with some weird gearing to reverse this - Ill leave this analysis to others -- in which case assuming that the bike wheel did ot lose traction and spin freely against the ground, the bike would either not move at all (forces equal), or would move forward, with the pedal doing something strange, It would move in the "drive" direction with respect to the bike, but actually begin to move "backward" with respect to the ground, then as it continued "around" the force pushing the bike backwards would diminish and the bike would continue to move forward that mush easier.

Here is Jimmy's solution:

My answer is that the bike will move backwards and the pedal will move forwards. As for why, I think that since the radius of the pedal is smaller than the radius of the bicycle wheel, the backwards force of pushing against the pedal is stronger than the forwards force (torque?) of the wheel rolling. If the pedal's radius were equal to the radius of the bicycle wheel then the wheel would stay stationary. If the pedal's radius were larger than the radius of the bicycle wheel (it would have to be a very strange bike mounted on a test platform), then the bike would move forwards.

Actually, I think the exact physics calculation goes something like this:

Let Rp be the radius of the pedal and Rw be the radius of the wheel. Let the pedal be at angle A with the ground (in the problem specified, A = 90).

If sin(A) = Rp/Rw, the bike will remain stationary.
If sin(A) < Rp/Rw, the bike will go forwards.
If sin(A) > Rp/Rw, the bike will go backwards.

In the problem given, sin(A) = 1, and Rp/Rw is less than 1 on a normal bike, so therefore the bike will go backwards. Notice that if the pedal is at angle 0 or less (i.e. you are pushing the top pedal), or if the pedal radius is bigger than the wheel radius (the strange bike), then the bicycle will always move forwards.

Update 9/04/01: Benedetti Renzo clarifies that the bottom pedal actually moves BACKWARD in space, but it moves forward in relationship to the bicycle.
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