The common explanation is that the angular momentum of the wheels provides enough stabilization so that the Bicycle Cranky Vintage Shirt. Conservation of angular momentum is a pretty cool phenomenon, anyone that has ever played with a gyroscope has experienced this first hand. However, the mass and rotational speed of the wheels on a bike are insufficient for this effect to be the primary mechanism of keeping the bike upright against the weight of a human being. To find that we need to look at the geometry of the bicycle.
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The red line is the axis about which the handlebar and Bicycle Cranky Vintage Shirt. Notice that it is not vertical, it is tilted backward at the top and forward at the bottom. The blue dot is the contact point of the wheel on the ground and is the spot where a force will be exerted on the wheel by the ground. It is important that the blue dot is behind the red line, this is what stabilizes the bike. Now, imagine you are riding the bicycle and you begin to fall over to your right side. The bike is tilted right so that the force exerted on the wheel at the blue dot is no longer up and parallel to the plane of the bike, but rather up and slightly to your left (or up and slightly out of the page in the above diagram). This force acts to rotate the wheel about the red axis because the blue dot is not coincident with the red axis. In the right-leaning case, the force will act to steer the wheel to the right. If you were leaning left the wheel, the wheel would steer left. This keeps the bike upright because as you begin to fall in one direction, the bike steers in that same direction, moving the vehicle back underneath you, which keeps you from hitting the ground.