why is it easier to keep your balance on a moving bicycle than on at rest
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Actually it is easier to balance a bicycle at rest then when it is in motion. The reason for this is the conservation of the angular momentum. A bicycle has wheels that once in motion, they rotate. Each rotating wheel is generating a non zero angular momentum
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Newton said that bodies in motion tend to stay in motion. Add to the the momentum of the wheel rotation and you have a more stable platform to balance. There are upper and limits of course, but they allow for an easy to balance bike at a wide range of speeds.
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