How is the bicycle able to maintain its equilibrium while in motion but not while at rest?
Let’s consider a disc instead of a bicycle. The disc has no person on it but still it’s in “equilibrium” while in state of motion! Instead of giving you mathematical formula, allow me to explain on the basis of one of Newton’s Law: That an object in motion shall remain in motion unless an external force act upon it. It the disk is “rolled” in a forward motion, and then it will continue on rolling until it is overcome by friction and gravity. The type of friction between the disk’s edge and the table or pavement it rolls upon is one factor, another factor is the thickness and roundness of the disk and still another factor is the angle of the disk relative to the pavement it would roll upon. Once the disk starts to roll it would not stop right away as if an invisible external force subdued it. Instead, it would roll in forward motion – the rolling force prevent it from tipping. Hope this explains!