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Bicycle Physics

picture of bicycle

Bicycle physics is a broad and complex subject, perhaps more so than one can imagine. Although the number of components of a bicycle is small, the interaction between them and the dynamic principles involved, is complicated. This is especially true with regards to bicycle stability, which is the result of a complex dynamic interaction within the bike-rider system.

On this page I will explain some of the main aspects of the physics of bicycles, which should give the reader a greater appreciation of how bicycles work, from a physics perspective.

Bicycle Physics – Stability

Bicycles are inherently stable when riding. Even riderless bicycles are stable if given enough forward velocity. Much effort has gone into analyzing the factors which make a bicycle stable. It has been determined that “trail” (shown below) is often an important contributor to bicycle stability. For the traditional bicycle design, if trail is positive, meaning the projection of the steering axis with the ground is in front of the contact point of front wheel and ground, then the bicycle is more stable when riding (i.e. it’s less likely to fall over when riding it). If this projection is behind the contact point (negative trail) then the bicycle is less stable and the bicycle is more likely to fall down when riding it.

front wheel schematic of bicycle

Based on the geometric parameters shown, the mathematical formula for trail is:

trail equation for front bike wheel

where Rw is the wheel radius, Ah is the head angle as shown, and Of is the rake, as shown, also known as the fork offset.

When analyzing bicycle stability it is common to use two parameters; the lean angle and steering angle of the bike. The lean angle is the left and right angle the bike frame makes with a vertical plane, and the steering angle is the angle the front wheel makes with the plane of the bike (containing the bike frame). The figure below illustrates the lean and steering angle.

bicycle schematic showing lean and steer angle

where θ is the lean angle and α is the steering angle. The sign convention for these angles is typically as follows, and is with respect to a rider sitting on the bike: Right lean is positive θ and left lean is negative θ. Right steer is positive α and left steer is negative α. For stability analysis both of these angles are the only independent variables needed to mathematically analyze bicycle stability. They completely describe the orientation of a bicycle as it travels in the forward direction. For a bicycle to be stable the lean and steering angle must have a tendency to “die out”, which means that these angles will fluctuate around zero with small positive and negative values. This in turn means that the bicycle tends to stay upright with little turning, while moving in the forward direction. It is interesting that locking the front steering will always result in a bicycle falling over. The physics of stability requires that the front wheel can freely steer.

As mentioned, analyzing bicycle stability


KSU Physics Education Bike Project

KSU Physics Education Bike Project

Scientific and Cultural Aspects of the Bicycle:

An International Pedagogical Project


This project is a multi-national effort to collaborate on the adaptation
and creation of pedagogical materials.  The bicycle, a highly developed
yet simple device, is the focus of this effort.  Students and faculty
are using materials developed in a variety of countries and creating new
materials using contemporary multimedia.  This effort began almost
15 years ago when Robert Fuller and Dean Zollman created the videodisc
Transformations featuring the Bicycle
at about the same time that the
PLON Project in The Netherlands developed the teaching module Traffic
and the British Open University developed a course on Materials and Structures
which featured the bicycle.  These efforts were independent of each
other.  Since that time we have worked to combine instructional materials
from these and other countries.


Web Site at the Unversity of Amsterdam

Contents of KSU Bicycle Project Web Site

•  Description
and Application of 2000-2001 International Exchange Program

•  International
Study & Exchange Program

    European Community. and United States students enrolled in one of the
    partner institutions will become part of an international team of students
    who will investigate various scientific and cultural aspects of the bicycle,
    and create multimedia instructional materials about their activities. The
    students will become part of a three-year effort that will link international
    students by computer and bring them together periodically to work face-to-face.

• Workshops
on the Bicycle in Science, Technology and Culture, 1995

    These workshop, held in Great Britain and The Netherlands, brought
    together science and technology educators  and multimedia experts
    from the U.S., Australia, and several European countries.  Together
    they developed plans for pedagogical, multimedia materials for teaching
    about the bicycle.  This effort led to the International Study and
    Exchange Program.  U.S. participation in these workshops was supported
    by the National Science Foundation.

• International
Conference on the Bicycle in Science Pedagogy

    This conference, held in Lincoln, Nebraska, was jointly hosted by the
    University of Nebraska – Lincoln and Kansas State University.  Multimedia
    specialists, researchers on the science and technology of the bicycle and
    physics educators worked to gather to lay the basic ground work for a series
    of lessons on the science and technology of the bicycle and their cultural
    adaptations in different cultures.  The conference was supported by
    the Association of Big 8  (now Big 12) Universities.

• Resources

• Bicycle

Principal Investigator at Kansas State University is Dean
  email: dzollman@phys.ksu.edu.

The project has received funding from the Association of Big 8
(now Big 12) Universities, the U.S. National Science Foundation, the European
Commission, and the U.S. Department of Education.

This page last updated on February 19, 1999

Copyright © 1999 Physics Education Group, Kansas
State University

Source Article


Are Expensive Bicycle Wheels Worth the Money? Let’s Check the Physics

In this video, you see a cyclist testing new aerodynamic wheels from Zipp. Swapping your wheels may seem like a small change, but can make a big difference. From his tests, the rider discovers:

  • With conventional wheels, he can ride 20 minutes at an average speed of 41.12 kph with an average power of 379 watts.
  • With the Zipp 808 NSW aero wheels he rides 51 minutes at an average speed of 41.13 kph and average power of 344 watts.

Before looking at power and energy, I should go over two small details.

First, how do you measure power? Cyclists can measure power by installing a small computer, called a power meter, that measures the input torque at the pedals or crankshaft and records the rotation angle at timed intervals. If you know the torque and angle, you can calculate the input energy. Dividing this energy by time gives you power.

La te xi t 1

Second, this isn’t a perfect test of aerodynamics. If you really want to examine the effect of the new wheels, you probably would have to put a bike with a dummy in a wind tunnel. When the reviewer takes his second ride, many things could have changed—wind, body position, amount of sweat on the body—and impacted performance. Let’s assume the only thing that changed was the wheels.

Air Drag and Power

What happens when you ride a bike? If you are moving at a constant speed, then the net force on the bike-human system must be zero. In a slightly simplified view, I can draw the following force diagram:

Spring 2016 Sketches key

The vertical forces (gravity pulling down and the ground pushing up) don’t really matter here. Just forget about them and pay attention to the horizontal forces. First, let’s look at the air drag. Air acts in complicated ways when an object passes through it. But who cares when we can make a simple model of air drag force? Here’s an expression for the magnitude of this force:

La te xi t 1

In this model, the air force is proportional to the square of the bike’s speed (v). For the other terms, we have:

  • ρ is the density of air (around 1.0 kg/m3).
  • A is the cross sectional area of the bike plus the rider (how much of the object interacts with the air).
  • Finally, C is the drag coefficient. This parameter depends upon the shape of the object. If you change the wheels, it is the value of C that should change.

The second horizontal force is the frictional force. An interaction between the road and the tires propels the bike. I know what you’re thinking: Doesn’t the human propel the bike? In a sense, yes. But the reality is sort of complicated. The rider’s power goes through the pedals and chain to the wheel, which turns. But the force comes from the tire pushing against the road. So for our energy perspective on this problem let’s just say the human provides the friction force.

Clearly the faster the biker