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Scene 1 – A hallway

There I was, in a corporate hallway 110 shaftments long (approximately 18 m or 60 ft) and unspooling a 100 -shaftment length of cable from a reel. All too soon, due the following constraints, this attempt was foiled by the hallway’s being too short.
•  I could kick the reel and let it roll along the floor without resistance or slippage.
•  I was to keep any unspooled cable straight at all times by holding its loose end.
•  An extra 10 shaftments of hall length was needed for elbow room, including the reel, so it seemed at first that a hallway 110 shaftments long should have been just right.

  1. Approximately what length of hallway was actually needed?
  1. How did I rewind the already-unspooled cable, while respecting the given constraints, after having begun to unroll it in the original hallway?


Scene 2 – A bike shop

And there I was, some other day, fixing a bicycle and intending to advance its chain by pushing its bottomed-out pedal backward. The bicycle was standing on the shop floor and this was meant to pedal it forward but . . . foiled again!   How puzzling.

  1. How did the bicycle actually respond?
  1. What relationship between dimensions A, B, C and D allow a bicycle to be pedaled like that without lifting it off the floor?
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Q1:

The reel's outer circumference is twice its spindle's circumference, so to unspool 100 length of cable it will roll a distance of 200. With the elbow room, you need the hallway to be 210 long. To keep holding the end of the cable during the unspooling, you will have to walk forward half the speed of the reel, travelling about half the length of the hallway.

Q2:

Just pull the cable, walking back to the start of the hallway. The cable was being pulled forward during unrolling, since it was being unspooled at half the speed of the reel. So by pulling it back, the process reverses, with the reel moving towards you at twice the speed at which you pull the cable. When you reach the start of the hallway, the reel reaches you and the situation is as it was at the beginning.

Q3:

The bike moved back. The pedal is like the reel's inner spindle in Q2. Imagine you standing behind the bike with a cable attached to the bottom pedal, pulling the cable. Just as with the reel, the cable unspools slower than the bike itself (but even more so due to the gearing) so you pull the bike towards you.

Q4:

You need the pedal to move back relative to the bike faster than this moves the bike relative to the ground. So the pedal needs to move back faster than the bottom of the wheel moves back, both relative to bike. This needs A/B < D/C. One way to make this happen is to swap the two gear wheels, swapping the values of C and D.

It may seem strange that the bike can move in the opposite direction to the push applied to the pedal. One way to explain it is as follows. Imagine the pedal being fixed onto the rear wheel itself, without gearing. If the pedal is at the bottom, then the spoke going from the hub of the wheel past the pedal to the bottom of the wheel is like a lever. The point of the wheel on the ground is the fulcrum, and when you push the pedal which is halfway up the lever, the other end of the lever pushes the bike faster in the same direction. When you introduce gearing, you change how much effect the movement of the pedal has. Normally you add higher gears, which is like moving the pedal up, closer to the hub so that less pedal movement leads to more bike movement.

In this case we add an extremely low gear, which is equivalent to moving the pedal down so far that the spoke that acts as the lever would have to be extended below ground. It is obvious that pushing one end of a lever makes the end on the other side of the fulcrum go in the opposite direction. In this case of the bike it is a bit counter-intuitive that the gearing allows the same thing to happen without actually needing to be on the other side of the fulcrum.

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  • $\begingroup$ @humn: I've expanded my answer a bit. $\endgroup$ – Jaap Scherphuis Sep 4 '17 at 8:29
  • $\begingroup$ Thank you for the elaboration and demystification, Jaap S. If you figured all this out without using an actual reel or bicycle, wow! Also, again, how you relate Scene 2 to Scene 1 is much cleaner than it was in my mind. A bounty will be on its way, once allowed, and will get to you within a week. $\endgroup$ – humn Sep 4 '17 at 13:48
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    $\begingroup$ @humn: If you like this stuff, have a look at this old thread on the XKCD forum about going downwind faster than the wind. $\endgroup$ – Jaap Scherphuis Sep 4 '17 at 14:09
  • $\begingroup$ Great examples in that thread, Jaap S., completely unexpected and bewildering. I had to imagine a vehicle going upwind, along the lines of your Q4 explanation, in order to understand one outpacing the wind. $\endgroup$ – humn Sep 4 '17 at 15:21

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