Solution to Riddle of the Week

Solution to Riddle of the Week: The Bicycle Problem

Difficulty level: Moderate

Michael Stillwell
 
By Jay Bennett

As you might have guessed, the solution to this problem is not obvious. The bike will not stay still, and it will not skid on its rear wheel. If you want to take one more guess you can check the original problem here.

SOLUTION

What is counter intuitive about this problem is that the bike's crank will spin the opposite way from the direction you are pulling it.

The bike moves backward, and the crank also spins backward. (On a standard freewheel or cassette driven bike, the crank needs to spin backward to allow the rear wheel to spin backward). The crank spins the opposite direction of the direction you are pulling it, and the bike rolls backward. It looks something like this:

The reason this happens is because when you pull the bike as a whole backward, the rear wheel wants to spin backward from friction with the ground, and as I pointed out earlier, the crank of a bicycle must always be spinning backward if the rear wheel is spinning backward. The crank therefore spins the opposite way of the direction you are pulling it.

(Bicycle mechanic's note: The freewheel or cassette on a bicycle rear wheel, which is what the gear cogs are mounted on, is designed to only spin freely in one direction. Pedaling forward will cause rear wheel to spin forward, and consequently, rolling the rear wheel backward will cause the crank to spin backward. On the flip side, the rear wheel can roll forward freely without the crank moving, and the crank can spin backward freely without moving the wheel. Except, of course, on a fixed gear bicycle, or fixie, that has no bearings to allow the rear cog to spin freely in any direction. This riddle works on standard bikes and fixies.)

This is not the end of the story, however. The rear wheel wants to move backward and spin the crank backward, and it does. But it can only do that because, relative to the ground, the point that the string is tied to on the crank also moves backwards even though the crank arm itself is moving forward. This will happen on any standard bike, but as George Hart shows us in his Mathematical Impressions video series, if you have a small enough gear, the bike will initially move forward. This is because a small enough gear changes the rate at which the crank is rotating relative to the bike's movement—the crank has to rotate more per wheel revolution. As a result, with a small enough gear, the point the string is tied to can no longer move backward relative to the ground, but will instead move forward along with the bike.

It can be a bit tricky to visualize that last part, so make sure to check out Hart's video to see it illustrated beautifully. He even had a custom gear put on his bike so the gear ratio would be small enough for the demonstration.

Still don't believe it will move backward? Get your own bike out and give it a try.