So I've been trying to create a mathematical model for an electric motorcycle and began to wonder about the maximum possible torque that could be supplied to the rear driven wheel without having the bike begin to lift up on one wheel. I found ways to calculate this value online; however, the basic concept as to how the bike actually lifts escapes me.
My problem started when I began to think about what axis the bike will rotate about when it is doing the wheelie. My first intuition was that the frame and front wheel, together, rotate about the rear axle. But when I drew a free body diagram of the frame/front wheel system just at liftoff (see below) I made note that the only forces acting about the rear axle, point O, is the force of weight. This means that an increase in applied torque and subsequently, the applied force Fa, should not effect the rotation about the rear axle.
I know that the applied force on the back wheel is indeed correlated with the propensity for a bike to wheelie, so I considered that the axis of rotation I was choosing was wrong. If we take the free body diagram above and sum the moments about the center of mass, we would find that an increased applied force would in fact cause the solid body to rotate. The problem with this understanding is that during rotation, the center of mass of the drawn system should actually rises relative to the surface the bike is moving across. If the bike were to be truly rotating about its center of mass, then the back wheel would begin to dip below the surface of the road like you might see in a glitchy video game.
So I suspect that the bike is in fact rotating about the back axle, but I don't understand why, please help!
edit: I added the external torque from the back wheel to the frame, which would allow the bike to rotate about the back axle
edit 2: I suppose that the torque acting on the frame via the engine should not effect the rotation as the movement of a motorcycle can be perfectly replicated by applying a force at the back axle. A third possible axis of rotation might be the lowest point of the back wheel.