### Lateral Dynamics and Inertial Motion

From Newton’s Laws, we know that moving objects possess inertial motion and move at constant velocity and in straight lines, unless acted upon by external forces. In our discussions of cycling against resistive forces, we have been able to assume straight line or more formally longitudinal motion.

But when we are cornering, we are looking for forces to turn the cycle rather than change its speed. Different problems have different underlying Physics which is why longitudinal and lateral riding are described differently.

Cornering involves two distinct physical aspects: Centripetal and Centrifugal. While interrelated, Centripetal discussed here is related to the physical force that is turning the cycle, while in the next section we Centrifugal is discussed as a factor that must be accounted for in order to maintain balance.

### What is the force to move an object in a circle?

When dealing with resistive forces, the Physics approach is to start with the force equation, and then use the techniques of Integral Calculus to first derive the velocity equation and from that the position equation describing the objects location over time.

With lateral motion, the reverse approach is taken. We already know the equations of motion, and from that, we want to derive the force equation describing what is “pulling” the object away from its inertial tendency to move into a straight line into a circular path. This force is sometimes called the Centripetal Force.

For circular motion, the Centripetal Force equation is computed as:

**F = mA _{c} = mv^{2}/R**

where m is the object mass, v is the velocity as it moves around the circle, and R is the circle radius. Notice this result is derived from the circular motion requirement and so is applicable to any object moving in a circle. This can be a weight on a string, a satellite circling the earth, or a cyclist making a turn.

### Where is the force turning a cycle?

From this result, we know some thing must be “pulling” the cycle from its straightline tendency into a circular motion. But we also know nothing is pulling the cycle toward the center of the circular motion. You may be tempted to think it is the lean, but when we discuss that later, you will see the lean is involved in balance not turning.

Forces are called “pushes and pulls.” If the cycle is not being pulled, our alternative is that something is pushing it into its circular motion. The only part of the cycle touching the ground are the tires. So when you turn your wheel, the tires are not only rolling but also “biting” into the road.

This then is the source of what is causing you to turn. While your tires are continuing to roll, what is enabling your tires to be a part of the turning process is that they are resisting the tendency to slide which is described by its own coefficient. The implication is that you can aggressively turn as much as you want as long as you are not putting more force on the tire so as to cause them to slide out from under you.

### What are the Centripetal Forces needed to Execute various turns?

We have the formula as to how much force needs to be applied to push a CyclistCycle in a cornering motion. Let’s look at the numbers for turns as a function of radius and speed.

We discuss tire traction next to get a feel how tires “bite” into the road. For the time being, be aware that forces greater than the CyclistCycle weight are pushing the envelope of what the tires can handle.

This chart is for an Elite Cyclist of a 150 lb riding a 16 lb cycle. The numbers have been color coded. Green implies the tire force is approximately their weight or less. Yellow is in the double range. Red is in the NoGo range. You can see the tradeoffs between range and velocity that a cyclist must make when turning. You can also see that a a 5 ft. radius turn is ridiculously impossible at 50 mph.

Next Topic: Centrifugal Counterbalancing