# Motorcycle: Lean Angle, Speed, and Turn Radius

1. Jan 22, 2004

### Julian Solos

Suppose you are riding a motorcycle at a constant speed v and about to enter a level, circular, curve of radius r. You intend to lean with the bike and want to make negotiating the curve as smooth and effortless as possible.

Then the lean angle (from the vertical) of the motorcycle $$\theta$$ to which the motorcycle will settle into is approximated by the following relationship

$$tan (\theta) = \frac {v^2} {rg}$$

where g is the acceleration due to gravity.

If the widths of the tires of the motorcycle are narrow, like those of bicycle tires, the above relationship should give a good approximation. However, many of today's motorcycles have wide tires, i.e. 19 cm. Can we refine the above relationship with the addition of a variable or variables related to the widths of the tires of the motorcycle?

(For simplicity's sake, shall we assume the widths of the front and rear tires are the same so hat we need to use only one width and the sections of the tires which make contact with the ground are semicircles of the radius equal to the tire width and their shape remains the same throughout the range of possible lean angles?)

Last edited: Jan 22, 2004