Exercise
When I am out riding my bike, wide turns feel different from tight ones. At a fixed speed, a tighter turn requires more sideways acceleration. How can we describe the tightness of a turn? Under a simple friction model, what is the smallest turning radius available at two different speeds?
Pre-requisites
This exercise uses speed, centripetal acceleration, friction, and the radius of a circle. Assume a steady turn on a level, unbanked surface, with no braking and enough lean and clearance to follow the turn. Use a constant maximum static-friction coefficient of 0.4 between the tires and road, and gravitational acceleration of approximately 32 feet per second per second. Compare speeds of 8 and 16 feet per second, approximately 5.5 and 10.9 mph. These are assumptions for the calculation, not a measurement of the grip available on a particular road.
Facts
A circular turn is described by its radius. At a given speed, a larger radius requires less sideways acceleration and a smaller radius requires more.
The required centripetal acceleration is a = v²/r, where v is speed and r is turning radius. Under our assumptions, the maximum lateral force is μmg, so the maximum available lateral acceleration is μg = 0.4 × 32 = 12.8 feet per second per second. This available acceleration is constant in the model; the required acceleration changes with speed and radius.
Setting the required acceleration equal to the available maximum gives the limiting radius: r = v²/(μg).
At 8 feet per second, the limiting radius is 8²/12.8 = 5 feet. At 16 feet per second, it is 16²/12.8 = 20 feet. Doubling the speed quadruples the limiting radius.
These values mark the model’s traction boundary, with no extra margin. They are not recommended riding radii. Real tires, surfaces, balance, lean clearance, and changing conditions can impose other limits.
Thoughts and opinions
A limit can be explored on paper before it is approached on a bicycle. What interests me here is how quickly the required room changes with speed.
At 8 feet per second, a 4-foot radius would require more sideways acceleration than our assumed friction can provide. A radius greater than 5 feet would require less than the model’s maximum. At 16 feet per second, the corresponding boundary moves out to 20 feet. The calculation describes the available force; it does not predict every detail of a real loss of control.
We held the friction coefficient constant to make the comparison. In practice, tire condition, the surface, and the weather can change the available grip. Skill can help a rider choose a path and respond to conditions, but it does not remove the need for enough force to follow that path.
Curve warning signs and advisory speeds also involve road geometry, banking, visibility, and engineering judgment. They are not exact labels for the speed at which every vehicle would lose traction. Our simple model helps explain why speed and curvature matter without replacing those practical considerations.
I have learned about turning limits the hard way on a real bicycle, as well as through calculations. Working through a model can spare us some of the cost of learning by losing grip. Ordinary practice at comfortable speeds, with room and traction to spare, can help connect the calculation to experience.
The relationship worth carrying away is that doubling speed quadruples the limiting radius in this model. What would that change look like on a turn you know?
Feel free to email me with your thoughts and opinions on this matter at [email protected].