The Physics of Cycling: Understanding Bicycle Motion and Stability

Bicycles are a fundamental mode of transportation, recreation, and sport, yet the physics behind their motion is often overlooked. Understanding the forces at play when riding a bicycle reveals the intricate balance, stability, and mechanics that make cycling possible. This document explores the fundamental principles behind the physics of cycling. This includes bicycle motion, the role of gravitational, centrifugal, and frictional forces.

We begin by analyzing how a rider maintains stability, examining the effects of leaning and counteracting forces. The study extends to the influence of centrifugal force during turns, the mechanics of braking, and the significance of front and rear wheel dynamics. Additionally, we discuss gear switching and its impact on torque and acceleration, as well as the mathematical principles governing a bicycle’s circular path when turning.

By breaking down these forces and interactions, this document provides a deeper insight into the physics of riding a bicycle, helping both enthusiasts and engineers appreciate the science behind an everyday activity.

Table of Contents

Leaning – Situation Along Primary Axis

Now, we will consider the forces acting on the bike. We will view them from behind along the primary axis.

Figure 1. Ideally positioned bike with the rider perpendicular to the ground.

In an ideally positioned bike, pictured by Figure 1, there is no leaning to the left or to the right.

Gravitational Force Fg

We all know that situation shown in Figure 1 is impossible. The center of mass of the rider will always shift to the left or to the right of a vertical line. Hence, the rider will always be leaning or falling off to either side.

Figure 2 shows the situation where the rider leans to the right. There is a gravitational force Fg acting from the center of mass down perpendicular to the ground. There are two components derived from the gravitational force: Force applied with torque F torque and radial force F radial. Torque force rotates the rider counter-clockwise. Radial force acts along the body of a bike and rider.

Figure 2. Leaning of rider to the right.

We can calculate the torque force and radial force as follows:

\[ F_{radial} = F_g \sin{\theta} \] \[ F_{torque} = F_g \cos{\theta} \]

Radial Force F radial

Next, the radial force is divided into two further components: contact force F contact and force F move of movement to the left. See Figure 3. Contact force Fc is equal to Normal force N. Force of movement F move is parallel to the ground.

Figure 3. Radial force is further divided into two forces: one parallel and one perpendicular to the ground.

We find two forces as follows:

\[ F_{contact} = F_{radial} \sin{\theta} = N \] \[ F_{move} = F_{radial} \cos{\theta} \]

Also, force of friction prevents the bicycle from skidding to the left as long as it is greater than force of movement F move:

\[ F_{friction} = \mu N = \mu F_{radial} \sin{\theta} \] \[ F_{friction} \geq F_{move} \]

When force of friction becomes less than force of movement, the rider and bicycle start skidding to the left.

Centrifugal Force Fc

Now we will consider what happens when the bike is turning to the right as well as leans to the right. When turning, there is centrifugal force Fc that pushes and twists the bicycle to the left as it turns to the right, see Figure 4.

Centrifugal force Fc has components for torque and radial forces:

\[ F_{torque2} = F_{centrifugal} \sin{\theta} \] \[ F_{radial2} = F_{centrifugal} \cos{\theta} \]

Second radial force Fr2 further divides into force along the ground and force perpendicular to the ground:

\[ F_{c move} = F_{radial2} \cos{\theta} \] \[ F_{c contact} = F_{radial2} \sin{\theta} \]

Centrifugal force is equal to the centripetal acceleration times mass:

\[ F_{centrifugal} = m a_{c} = m \frac{v^2}{r} \]

You drive a bike such that torque from centrifugal force equals torque from force of gravity:

\[ F_{torque2} = m \frac{v^2}{r} \sin{\theta} = F_{g} \cos{\theta} \]

This way you do not fall over either side. If you increase speed, it becomes harder to compensate torque from centrifugal force with torque from gravity. As a result, you tend to increase the radius of traveling by straightening the handlebar. As a result, you tend to go off in a straight line.

Reducing speed helps you decrease torque from centrifugal force. It also allows you to turn left or right. The torque from gravitational force compensates torque from centrifugal force during the turn. This is crucial in the physics of cycling.

You also decide intuitively which angle theta to use. You must have force of friction be large enough to counteract force of moving along the ground:

\[ F_{net move} = F_g \sin{\theta} \cos{\theta} + m \frac{v^2}{r} \cos^2{\theta} \] \[ F_{friction} = \mu (F_g \sin^2{\theta} + m \frac{v^2}{r} \cos{\theta} \sin{\theta}) \]

Force of friction F friction must be greater than net force of movement for the bike to not skid.

Next let us consider braking.

Braking Forces – Situation along Side Axis

Every bicycle has two brakes: one for the front wheel and one for the back wheel. The two brakes are not equal and produce different efficiencies. Figure 5 shows bicycle when viewed from side.

Back Wheel Braking

When you apply brakes on the handlebar, you squeeze the braking pads on the corresponding wheel. The pads squeeze the wheel and apply friction to it. Your hand brake applies friction on the wheel. This is the same amount that the wheel will experience by ground friction, see Figure 6.

The amount of friction depends on how hard you squeeze the hand brake. The brake will resist rotational movement of the wheel. When rotational movement is decreased, the wheel resist translational movement along the ground. This, in turn, slows the whole bike down:

Rate of slowing down:

\[ a = \frac{F_{friction back wheel}}{m} \]

The maximum amount of friction by braking for the back wheel is no greater than static friction on the back wheel:

Figure 7. Force of gravity on bike and on each wheel.

Static friction depends on normal force and mu coefficient. When the force of gravity is centered in the middle of two wheels, it gets distributed along two wheels equally. The force per wheel is twice as less as the force of gravity.

\[ F_{static friction} = \mu N = \mu \frac{F_g}{2} \]

Greatest deceleration is achieved when “friction back wheel” by squeeze pads equals static friction:

\[ a_{max} = \frac{F_{friction back wheel}}{m} = \frac{\mu F_g}{2m} \]

Braking back wheel will not affect speed in any way more than what half force of gravity allows. Braking front wheel is different as the weight shifts from back wheel to front wheel (Figures 8-10):

Figure 8. Braking is applied to the front wheel.

Front Wheel Braking

Braking front wheel with pads causes front wheel to slow down. It causes ground friction to decrease translational speed of the bike. Breaking force and force of friction are not the same thing. Breaking force is the force applied by braking lever on the handlebar to the braking pads on the front wheel. Ground friction is the force causing bicycle to decelerate.

Figure 9. Model of braking with front wheel. Weight is distributed evenly resulting in forces on top of the wheels equal to half bicycle weight.

In equilibrium, vertical force is balanced:

\[ N_f + N_r = mg \]

Where Nf and Nr are normal forces at the front and rear wheels respectively.

Next, we write the moment balance equations about rear wheel ground contact:

Nf produces a moment Nf * L
weight mg produces mg L/2 (half wheelbase distance)
Horizontal braking force F at height h produces counter-clockwise moment Fh

Equilibrium gives:

\[ N_f L = \frac{mg}{2} + Fh \] \[ N_f = \frac{\frac{mg L}{2} + Fh}{L} \] \[ N_r = mg – N_f = \frac{\frac{mg L}{2} – Fh}{L} \]

If Nr becomes zero or negative, the rear wheel lifts.

We see that as horizontal braking force F increases, the weight is transferred from Nr to Nf by Fh amount. Therefore, it is much easier to brake with front brakes than with back brakes when driving on a bike!

When there is no braking, the front wheel acts with half bike weight force to the ground. When there is braking at the front wheel, the weight shifts to the front resulting in greater maximum static friction at the front wheel, of up to the full weight of the bike Fg.

Now let us consider braking force F. Force of braking pads Fbp produces kinetic friction Fk. If Fk is smaller than maximum static friction at contact, the static friction Fst is the horizontal braking force F.

\[ F = F_{static friction} \] \[ \textbf{Up to } \] \[ \mu_s N_F \]

If Fk exceeds maximum static friction, then front wheel starts skidding and the horizontal braking force F drops to equal kinetic friction at contact.

\[ F = F_{dynamic friction} = \mu_{d} N_f \]

To learn more about rotational mechanics of adjacent wheel-shaped/cylindrical objects please read Systems of Two Shafts: Rotational Mechanics – SciTechFocus

Gear Switching

On most modern bikes there are gears present so you can switch from low gear to high gears when driving.

You start with low gears numbered 1, 2 ,3 etc. Low gears have larger radiuses and so their torque will be greater at the point of force application:

\[ \textbf{Torque } T = F r \]

You could pedal with less effort on lower gear. However, the speed limit is small. Since the radius is large, faster wheel rotations will correspond to faster chain movement along the rim of a lower gear. To maintain high speed on a lower gear you will need to pedal very fast.

The problem with maintaining speed is less so when riding on a higher gear. Higher gears will have smaller radiuses so their rotations will correspond to smaller speeds and slower pedaling. However, to apply same acceleration you must apply greater force through smaller radius.

Your pedaling causes middle gear to rotate with angular speed. Angular speed omega (how fast you pedal) converts to tangential velocity v of the chain through the following relation:

\[ \textbf{(1)} v_{tanchain} = \omega_{cg} r_{cg} \]

Where rcg is the radius of the middle gear and wcg is the angular speed of the middle gear. Tangential velocity vtanchain is transferred from the middle gear to the back gear through chain movement. Therefore, the angular velocity at the back gear wbg can be found as follows:

\[ \textbf{(2)} w_{bg} = \frac{v_{tanchain}}{r_{bg}} = \frac{w_{cg} r_{cg}}{r_{bg}} \]

Angular velocity at the back gear wbg translates into tangential velocity of the back wheel v tanback through the following relationship:

\[ \textbf{(3)} v_{tanback} = w_{bg} r_{backwheel} \]

On lower gears, the radius of the back gear is large (rbg > rcg). This causes angular velocity (2) at the back gear wbg to be smaller than the angular speed of pedalling wcg. Smaller wbg results in smaller tangential speed (3) of the back wheel.

With higher gears, the radius of the back gear is small (rbg <= rcg). So, angular speed of back gear is greater than the angular speed of pedalling, resulting in larger tangential speed (3) of the back wheel.

This shows that on lower gears you must pedal faster to maintain target velocity v. While on higher gears you can pedal slower to maintain the same velocity v.

Determining Circular Path of a Turning Bicycle

It is known that a bicycle travels through a circular path when turning left or right. However, determining the exact path is more difficult. The key is that you must find the point where lines from each wheel axes intersect:

Figure 13. Circular Path of a turning bicycle.

The point of intersection of two axes lines will be the cetner of a circular path. This makes sense as in case of a 90 degree angle, the radius of a circular path will be the distance between two wheels. In case of a 0 degree angle. The circular path will extend indefinitely.

Findings Summary

We have explored the following findings regarding dynamics of riding a bicycle:

Finding	Example
Key factors determining whether you fall off to the side: angle of leaning, speed of movement and radius of turning.	The torque from gravitational force is greater for larger lean angle. So, to counteract it, you must travel at higher speed and at smaller radius.
Braking with the back wheel begins with squeezing the braking pads on the wheel. Slowing rotational movement of the wheel will slow translational movement of the bike.	In case when center of gravity lies in the middle of the bike, both front wheel and back wheel get half of net gravitational force acting on them. This determines max static friction force for braking.
Braking with front wheel will result in higher braking force than that of a back wheel.	Braking with front wheel will tip the bike over to the front and apply additional pressure on the wheel.
The larger the radius of a gear chain the easier it is to accelerate. However, the harder it is to maintain high speeds.	Rotational force T depends on radius r and applied force F. The larger the radius the smaller F will result in same torque T.
Center of the circular path is in the intersection of two lines from wheel axes.	When the front wheel is tilted at 90 degrees relative to the back wheel, the circular path will have radius equal to distance between front and back wheels.

Conclusion

The physics of riding a bicycle is a complex interplay of forces, balance, and motion dynamics. Throughout this document, we have explored how gravitational, centrifugal, frictional, and torque forces contribute to the stability, maneuverability, and efficiency of a bicycle. From understanding the effects of leaning and turning to analyzing braking mechanics and gear switching, we have seen how these fundamental principles dictate a rider’s control over the bicycle.

Maintaining balance requires continuous adjustments to counteract gravitational and centrifugal forces, ensuring that torque and friction forces are properly balanced. The mechanics of braking highlight the differences between front and rear wheel braking, demonstrating the importance of weight distribution and normal force. Additionally, the physics behind gear ratios emphasizes how torque and speed are optimized for different riding conditions.

By applying these principles, cyclists can improve their riding techniques, engineers can design more efficient bicycles, and scientists can further explore the mechanics of two-wheeled motion. Whether for transportation, sport, or recreation, understanding the physics of cycling enhances both performance and safety, making it not just a practical skill but also a fascinating study of applied physics. If you are interested in more of this type of articles, read Physics of Running!