How can you measure the mechanical power required to pedal a bicycle by observing how it slows down when you stop pedalling?

- a = Δv/Δt
- F = ma
- Power = Fv

The forces acting on a bicycle can be estimated by observing the rate at which it slows down when you stop pedalling. This technique can, in principle, be used for any transportation vehicle.

Bicycle7.mp4 [2] (right-click and choose "Save Link As..." to download to your computer)

**Experiments in real life**

In order to model real-life situations with a mathematical model, we need to perform experiments that verify and supply data for the model. In this experiment, we will collect data to obtain the real forces acting on a bicycle as it free wheels without any power input. With these data, we will extract information on the rolling resistance and air drag which will help us predict the power required to maintain any speed, without further experiment.

**Forces acting on a bicycle**

The forces acting on a bicycle depend on the quality of the transmission, the interaction between the tires and the road, the angle of inclination of the road, and the speed of the bicycle and the wind. By analyzing how a bicycle slows down, we can estimate the forces that are intrinsic to the bicycle, i.e. the rolling resistance and air drag (which is a known function of speed), and also factors particular to the experiment, i.e. angle of the road and wind speed.

These forces can be modelled as it follows:

Air Drag, D = (½C_{D}ρA)(v±u)^{2}

Here C_{D }is the dimensionless drag coefficient of rider and bicycle, ρ is air density, A the frontal area of rider and bicycle, and v and u are the bicycle and wind velocities respectively, relative to the road. The ± in this equation depends on whether you are pedalling with (-) or against (+) the wind. We assume here that u < v, otherwise D becomes negative.

Rolling friction, F_{r} = µ_{r}N

Here, µ_{r} is the dimensionless coefficient of rolling resistance and N is the normal force acting by the road on the bicycle.

Weight of the bicycle + rider, W = mg

N = mgcos(**Θ**)

If we sum all of the forces to get the net force **parallel to the road** acting on the bicycle, we get:

F_{net} = (µ_{r}±sin(**Θ**))mg + (½pAC_{D})(v±u)^{2} (1)

The ± in the first term of the equation depends on your direction. If you are going uphill, the weight component parallel to the road is acting on the same direction as the force of rolling friction (against your motion). Therefore, the components add. Similarly, if you are going downhill, they act in opposite directions, and subtract.

**How to calculate the forces acting on a bicycle you ride**

In order to measure these forces, the only instruments you need are: a speedometer that gives you time and speed recordings, a video camera, and a scale.

The experiment consists on pedaling the bicycle up to a certain speed (30 km/h in our case) and then let the bicycle free-wheel down to the lowest speed at which you can still balance it. While you do this, record the speed and time recordings given by the speedometer.

Later on, enter the readings into a spreadsheet, connecting each time with each speed (as time increases, the speed should decrease). Then measure and record the mass of the bicycle and rider.

Once you layout the data table, you can obtain the values of acceleration by dividing the difference between the following and previous speed recordings by the difference of their two time values (i.e the difference in speed by the difference in time). When you do this, you won't be able to get the first and last values of acceleration, as you don't have the previous and following speed and time recordings, respectively. For example, the acceleration at the 13th time depends on the 12th and 14th times and velocities:

a_{13}=(v_{14}-v_{12})/(t_{14}-t_{12}) (2)

Finally, multiply each of these acceleration values by the mass recorded (of the bicycle + rider) in order to get the net force acting on the bicycle at that given point. For example:

F_{13}=ma_{13 }(3)

Once we have a list of net forces acting on the bicycle at certain times, we can approximate the values of C_{D}, u, µ_{r}, and Θ. In order to do this, we can assign guessed initial values to these constants, and use the net force equation (1) to calculate the forces that were obtained from the experiment. The value of A is the approximate frontal area of the rider + bicycle (0.7 m^{2} in our case), ρ is the density of air in this case (1.225 kg/m^{3}) and g is the acceleration due to gravity (9.8 m/s^{2}). You should also create a scatterplot of force vs. speed for both the measured and calculated values. Use points for the data (with error bars estimated from the scatter of the points) and a line the theory.

To obtain values for C_{D}, u, µ_{r}, and Θ that best represent reality, you can either use trial and error (not bad for an rough estimate) or do a chi-square analysis. If you do not know how to do this, just refer to our Chi-square analysis video (http://www.youtube.com/watch?v=dLBNMty5nC0 [3]).

Once you have minimized the chi-square sum using the solver add-in in Excel, check the values of C_{D}, u, µ_{r}, and **Θ** and make sure they are realistic.

For our experiment, we played with the values mentioned and obtained:

C_{D}: 1.25 ± 0.09

u: -1.1 ± 0.1 m/s

µ_{r}: 0.05 ± 0.004

Θ: -0.06 ± 0.06^{o}

**Using results in future predictions**

Since we now have the constants in the net force equation (1), we can estimate what would the force to move a bicycle at at any speed (or road inclination, wind conditions).

To obtain the power required to provide the force, we multiply by the instantaneous speed: P = F_{net}v:

This is a very steeply rising curve; the v^{2} factor in the air drag term becomes v^{3} in the power equation. The increase in power required to move at 8 m/s compared to 7 m/s is much greater than the increase from 2 m/s to 3 m/s. In our experiment the maximum mechanical power was about 300 W, which ocurred at 8 m/s (29 km/h). This we only had to maintain for a few seconds, but a professional cyclist can put out about 400 W of mechanical power for a sustained period, which translates to 8.6 m/s (31km/h) on our bicycle. The world record for one hour's pedal on a bicycle enclosed by an aerodynamic fairing is more like 25 m/s (91 km/h); that for an unenclosed bicycle is a mere 16 m/s (56 km/h)1., but still almost double what would be achievable on our bicycle by a rider in street clothes.

Notes:

- This method assumes the drag from the free-wheel mechanism is the same as that from the transmission when pedalling. There is no particular reason why this should be true, but in a well-maintained bicycle, both effects are likely much smaller than the tire-road interaction.

- We have assumed all the mass of the wheels resides in the rim and tires, and therefore have not estimated the rotational energy of the wheel separately. (i.e. if I = mr^{2 }and ω = v/r, then E = 1/2 Iω^{2} = 1/2 mv^{2}).

- 1. International Human Powered Vehicle Association, www.IHPVA.org [4], http://en.wikipedia.org/wiki/Cycling_records [5]

© Physics and Astronomy Outreach Program at the University of British Columbia (Christian Villar, Chris Waltham 2011-12-07)