My Motorcycle and Me

Unfortunately it's that time of year in Central New York that the summer playthings really need to be put away. After coming home for Thanksgiving Break I found time for one more short ride on my motorcycle and it was there that I started thinking about physics. We all know that turning quickly on a bike or in a car or even ice skating causes a shift in weight and the feeling of being pulled out of the turn so I took it upon myself to figure out what was going on.

I started with the angular kinematics of a smooth turn and used rough values for my velocity, a turning radius of 6 m, and a 90 degree (1.57 radians) turn.

v0 = 5 m/s
w0 = 0.83 rad/s
vf = 7 m/s
wf = 1.17 rad/s
∆Theta = 1.57 radians

Ideally one wants a constant angular acceleration and these values result in a value of 0.22 rad/s^2. However, the real crux of the matter comes in the conceptual understanding of torque.

Any object traveling in a circular path must necessarily have a force acting on it in the direction of the center of the circle because, as we know from Newton's first law, any object wants to continue in straight line motion. If the friction force from the tires acts in the direction of the center of the turning circle then the center of our mass is "left behind" due to the nature of objects proceeding in said straight line. Because of this we experience the "phantom force" known as centrifugal force which is the sensation of being pulled out of any circular path. Torque comes into play when we think about where these forces are acting.

(This is easier to see and understand with a diagram but imagine the bike and rider as a lever)
Because the axis of rotation is where the wheels meet the ground the friction force has no torque. If the center of mass is directly above the axis of rotation (no lean) then the force of gravity results in zero torque because the angle is 180 degrees. This leaves the phantom centrifugal force which acts away from the center of the circle at whatever distance above the axis of rotation that the center of mass is at. By not leaning into the turn a biker experiences a net torque away from the center of the circle which results in a crash.

So what happens when we lean? When we lean we alter the angle that the centrifugal and, more importantly, the gravitational forces act on the center of mass in relation to the axis of rotation. A biker riding away from us (i.e. into the board) making a left turn would experience a lessened clockwise torque and introduce a counterclockwise torque thereby canceling each other out and allowing the rider to proceed through the turn without incident.

Now comes the part that proves professional racing riders are master physicists. If, upon exiting the turn, a rider is still leaning to one side they will inevitably fall over because the centrifugal torque disappears. As a result when learning to ride a motorcycle one is taught to accelerate out of turns to increase the centrifugal torque and bring the bike upright. The trick is that if you don't speed up enough you're not upright when the turn ends and if you speed up too much you will be upright before the turn ends and end up crashing. The ideal scenario is to have the perfect acceleration that results in the rider being upright the moment the turn ends and they can continue in the straight line that Newton's first law describes.