A “wheelie” is when a bicycle is ridden with its front wheel lifted from the ground. Riding the bicycle in a way to lift the rear wheel off the ground might lead to the cyclist tumbling over the handlebars, called a “header.” Other colorful terms for this situation are “endo” (short for “end-over-end”) and “face plant” (for landing face first on the ground). Let's determine the conditions required in order to keep both tires touching the ground at all times.

Frank Rowland Whitt and David Gordon Wilson, Bicycling Science, 2nd ed. (MIT Press, Cambridge, MA, 1982), p. 196.
Since the coordinate system is fixed to the bicycle, it is no longer inertial when the bicycle is accelerating, producing a pseudoforce. But since this pseudoforce acts at the center of mass of the bike-and-rider system, it produces no torque about that point.
Daniel P.
Timothy J.
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