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There are a number of mechanisms that can contribute to a bicycle’s stability. They all have one thing in common: Whenever the bike tilts, they turn the bike toward the direction of lean, returning the center of mass (of the rider and bike) over the wheels, hence restoring balance. So, even if we think we're going in a straight line, we’re constantly tilting right then turning right or tilting left and turning left .
Two important means of stabilization are:
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Researchers discovered that neither of the above mechanisms is universally necessary for bike stabilization, however . Without these two, a third means can stabilize the bike if the front wheel assembly possesses a lower center of gravity than the rear portion of the frame. In the event of tilt, then, the front wheel will fall more quickly than the rest of the bike, allowing the bike to turn in the direction of the lean.
* By convention, the direction of the angular momentum is determined by the so-called right-hand screw law. If you cup your right hand such that the fingers are curled in the direction of the wheel rotation, your extended right thumb should be pointing left, parallel to the ground. If your bike tilts left, this vector angles down toward the ground. But, conservation of angular momentum requires that some other vector must point upward to compensate. To that end, the bike turns left, going in a counter-clockwise direction—note the screw for this rotation points upward, as required.
 DEH Jones, The stability of the bicycle, Phys. Today, April 1970, p. 34
 JDK Kooijman et al., Science 15, April 2011, Vol. 332, No. 6027, pp. 339-342, http://www.sciencemag.org/content/332/6027/339.full