Riding a motorcycle is just like riding a bike except it's much faster—oh, and you don't have to pedal. In both cases, the two wheeled vehicle can lean significantly while taking a turn. Why? Two reasons: fake forces and torque.

Fake Forces

In your introductory physics course, one of the biggest ideas is that a force changes the motion of an object. One way to mathematically write this is with Newton's second law:

If a net force acts on an object, it accelerates. If you hold a ball in front of you and let go, only one force acts on it—the gravitational force. The ball has an acceleration in the same direction as the gravitational force so that it starts speeding up in the downward direction and falls straight down.

Now for a quick example. Suppose I hang a pair of fuzzy dice from the rear view mirror of my car. Now I accelerate the car, and the dice swing back. Don't try to pretend you don't have fuzzy dice hanging in your car. I know you do.

OK, let's say it's a ball instead. But why does the ball (fuzzy dice) swing back? If you looked at the forces on the ball, you have gravity pulling down and the tension in the string pulling both up and forward. If the ball is at rest, what force pushes it back to balance the horizontal component of the tension? The answer: nothing. There is not a force pushing the ball back because the ball is accelerating forward.

Here is the key point: Newton's second law really only works in a non-accelerating reference frame. When a human is in an accelerating car, we want Newton's law to work like always. The only way to fix this problem is to add a fake force, like this.

This fake force is in the opposite direction as the acceleration of the car. It is this fake force that "pushes the ball back" in the accelerating reference frame and this fake force would have a value of:

Most introductory physics courses don't cover fake forces. Why not? Because students already have some difficulty identifying the forces on an object. Add in fake forces and it just gets crazy. This means that for all of the situations in an intro physics class, an object will be observed from an inertial reference frame (which means non-accelerating).

What about a motorcycle turning in a circle? Since the velocity vector of the motorcycle is changing, it has an acceleration (even if it is at a constant speed). This means that a fake force pushes the rider in the opposite direction of the acceleration. The acceleration for an object moving in a circle points towards the center of the circle and has a magnitude of:

Where r is the radius of the circle and v is the speed of the motorcycle. Of course, you can probably guess that we have a special name for this fake force—we call it the centrifugal force which literally means "center fleeing force". Don't confuse this with centripetal force which is the force that causes an object to move in a circle.

Torque

When a car or motorcycle takes a turn, some external force pushes on the vehicle in the direction of the center of the circle. This force is almost always the friction force between the tires and the road. This frictional force will be important when looking at a turning motorcycle.

Now we can get to the leaning motorcycle. Suppose that I have a motorcycle that is going around a curve and NOT leaning. Since the motorcycle is turning, it is accelerating towards the center of the circle. It turns out that this is easiest to explore in the accelerating frame of the rider such that there will be a fake force pushing away from the center of the circle.

Here is a front view of the motorcycle along with the forces acting on it. The motorcycle is turning to the left (as seen from the viewer).

In this reference frame, all the forces add up to zero. However, all of the torques do not add up to zero. Try this. Put a pencil flat on the table and then push with two fingers in opposite directions on the pencil. If these two forces are at the same location on the pencil, the pencil remains stationary. If you push at the top and bottom of the pencil, the pencil turns.

Just as a force can change the velocity of an object, the torque can change the angular velocity. With zero torque, you would have no change in angular motion. The torque from a force depends on the magnitude of the force, the distance from the location of the force to some rotation point and the angle the force is applied. If you wanted to write this as an equation, it would be:

Where θ is the angle between F and r. Technically, torque is a vector, but let's just leave it like this for now.

Going back to the diagram of the non-leaning and turning motorcycle, you can see the problem. Just like the pencil, the friction force and the fake force are not at the same location. If you don't lean, the net torque is not zero and you would "fall over." In a motorcycle race, this would be a bad thing.

What changes if you the motorcycle leans? Here is the same motorcycle, but now leaning.

The net force is still zero in this accelerating reference frame—and now, the net torque is also zero. Let's look at the torque as calculated about the point where the wheel touches the ground. The frictional force and the normal force (from the ground pushing up) have zero torque since they are both applied at the point that torque is calculated. That leaves just the torque from the fake force and the torque from the gravitational force. They are in opposite directions and so they can cancel. In the non-leaning bike, the gravitational force was pushing right through the torque point so that it produced a zero torque and could not cancel the torque from the fake force.

In short, leaning the bike allows there to be a gravitational torque to balance the torque from the fake force. Leaning prevents you from falling over. I know that seems strange, but it's true.

Why Doesn't a Turning Car Lean?

Well, a turning car does actually lean. However, it doesn't have to. Here is a force diagram that is just like the turning motorcycle except that I have replaced it with a car.

Cars have 4 wheels (usually). If I take the right front wheel (seen on the left in the diagram) as the point to calculate the torque, the gravitational force does indeed have a non-zero torque since the center of gravity is not directly above the point of the tire. Also, the normal force from the other tire would also exert a non-zero torque. With this many forces, it's easy to see that you could have a net torque of zero. Cars don't have to lean to turn—but motorcycles do.