The Bicycle Problem

If you pull straight back on a pedal of a bicycle when it is at its lowest position, will the bicycle move forward or backward?

The surprising answer is that (usually) the bicycle will move backward.

When a bicycle moves forward, the trajectory its pedal traces with respect to the ground is called a trochoid. Depending on the selected gear of the bicycle, that trochoid could be:

  1. Curtate trochoid (for almost all gears of most bicycles)
  2. Prolate trochoid (if the gear is very low and the bicycle moves slowly)
  3. Common trochoid, a.k.a. cycloid (if the wheels of the bicycle and the pedal spin at identical speeds, practically never happens)
curtate trochoid
prolate trochoid
common trochoid (cycloid)

Since we are fixed with respect to the ground, by pulling the pedal backward, we are causing it to move leftward along the trochoid and therefore the bicycle will be moving backward. We note that despite that, the pedal will be moving forward with respect to the bicycle (but not with respect to the ground).

You can see a visual explanation of this puzzle in the video below.

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