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BIKE PEDAL-FORCE SOLUTION: WHICH WAY DOES THE BIKE GO? A Ruina, April 20, 2009 Question. You stand next to a normal bike on flat ground. You delicately balance the bike with the steering straight. Your friend stands next to the bike and pushes backwards (to the left on the figure) on the bottom pedal. Which way does the bike accelerate? Assumptions. You apply negligible force. Your friend applies Fp O{ to the pedal only. The rear wheel rolls. The only negligible mass is of the bike as a whole, m. All the bearings have negligible friction. The gearing of the bike is normal. The tension in the bottom part of the chain is negligible (not needed, but simplifies calculation). The key dimensions are ` = the crank length. The crank is the piece from the middle of the bottom bracket to the pedal. rc = the radius of the chain wheel. The chain wheel is the gear that is in the front of the bike. It chain wheel is rigidly attached to the crank and rotates with the crank. rf = radius of the free-wheel sprocket. This is also called the cluster. It is the gear at the back of the bike. When there is tension on the upper chain, and the rear ratchet is thus engaged, the rear sprocket rotates with the rear wheel. rw = radius of the rear wheel. Correct Answers. (1) The bike accelerates to the left. Try it. It always goes backwards. The next answer is different and is also correct. (2) The answer depends on the bike. Despite (1) above, the answer actually depends on what bike you try it on. For any bike you have ever seen or can buy, in any gear, answer (1) is correct. However, if the gearing is sufficiently low (how low? read solution below) then the bike will go forwards. Reasoning. A. The bike starts with no kinetic energy. It ends with some kinetic energy. Therefor some force had to do some positive work. The only force that has non-zero dot product with the velocity of the point it is touching is Fp O{. Therefore the pedal has to go backwards. Does this mean the bike goes forwards or backwards? When a bike goes forwards the bottom pedal goes backwards relative to the bike. But relative to the ground it goes forwards. Otherwise people wouldn’t ride bikes. The whole idea of a bike is that you go faster with a given leg motion than you do when you walk or run. So the bottom pedal goes the same direction as the bike. Since the pedal goes backwards, so does the bike. B. Same as above, but lets calculate the pedal velocity and see if it really is in the same direction as the bike. Lets define !w and !c as the counterclockwise angular velocities of the rear wheel and crank,
