## Alternative Homework Assignment: Bicycle |

Kevin is riding a twelve speed touring bicycle in a road race. The race is very long, and Kevin would like to ride his bicycle as efficiently as possible. In order to do this, he is constantly shifting gears in order to maintain a maximum speed with minimal effort. Let us examine why this works, by considering how the force Kevin produces with his legs is translated into an accelerating force for the bicycle.

Before we begin this examination, it will be helpful to introduce some bicycle terms as Kevin, (an avid racer) would use them. The three terms that will be most useful are gear, sprocket, and cog. While in some applications, these terms may be interchangeable, or at least very similar in meaning, they have very distinct meanings with regards to a bicycle. A sprocket is attached to the pedal cranks and is the front object on the chain loop. Most twelve speed and ten speed bicycles have one large sprocket and one slightly smaller one. These are sometimes called the "front gears," but to avoid confusion, we will call them sprockets. Cogs are sometimes called the "rear gears." They are attached to the rear wheel of the bicycle and have the appearance of small sprockets. A gear is a combination of one sprocket and one cog. As a final note, cogs come in many varieties. The two most common, are "touring" and "racing." Touring cogs are standard on most bicycles bought in stores, but racers often prefer the smaller racing cogs.

Part 1:

a) To see how Kevin's leg force is translated into acceleration, we must first consider some more basic features of the bicycle. Consider the bicycle and rider as a single system. Estimate where the center of mass of this system is and make a simple sketch that indicates this.

b) Now consider the bicycle and rider at rest. What external forces are acting on this system? Draw a free body diagram of the bicycle/rider system to illustrate this.

c) Add a force to the free body diagram that accelerates the bicycle and rider along a horizontal road. Indicate on the sketch of the bicycle and rider where this force could have acted. What is the name of this force?

d) Use a new sketch of the bicycle and rider to help explain qualitatively how the force that Kevin's legs exert result in a force to accelerate the bicycle.

Part 2:

a) The system that you have just described can be called the bicycle's power train. It consist of three machines that are in series with each other. The first machine is a lever. Specifically, it is the pedal crank. A typical pedal crank is about 20 cm. long. If the Kevin pushes straight down on the pedal with one fifth of his body weight, what is the maximum torque produced on the sprocket? Write an expression for this torque as a function of the angle between the pedal crank and vertical up, for a full pedal stroke.

b) The second machine is one of the bicycle's gears. The rider chooses a gear by using levers to direct the chain to connect a specific sprocket to a specific cog. Typical sprockets have diameters of 23 cm and 18 cm. Touring cogs on a twelve speed have diameters that range from 13 cm to 5 cm, and racing cogs can be as small as 3 cm in diameter. When the torque acting on the sprocket is maximum, what is magnitude of the force the larger sprocket can exerts on the chain? What is the average force?

c) Repeat (2b) for the smaller sprocket.

d) Using the average forces calculated above, what are the maximum and minimum torques that can be delivered to the rear wheel using touring cogs?

Part 3:

a) A typical touring twelve speed has wheels that are 27" (69 cm) in diameter. If it is not possible for the tire to slip on the road, then what are the forces exerted by the tire due to the maximum and minimum torques that were calculated in part 2.

b) The coefficient of static friction between rubber and asphalt is approximately .8. Make a reasonable estimate of Kevin's weight, and the weight of his bicycle. Then estimate what percentage of this weight rests on the rear wheel. (A sketch may be helpful). Use this to calculate the largest force for which it is true that the bicycle's wheel doesn't slip.

c) What is the bicycle's maximum acceleration?

Part 4:

a) The bicycle's speed is determined by how fast its wheels turn. If Kevin can turn the pedal cranks at a rate of two revolutions per second, what is the maximum rate at which the rear wheel will turn?

b) How fast does the bicycle travel if its rear wheel is turning this fast without slipping?

c) Does one gear give the best acceleration, and the highest top speed?

d) What gear might Kevin want to use on a hill?

e) Why does Kevin's bicycle have multiple gears, and what general strategy of gear selection should he use to win a long mountain road race?

These problems written and collected by K. Vick, E. Redish, and P. Cooney.
These problems may be freely used in classrooms. They may be copied and
cited in published work if *the Activity-Based Physics* (ABP) *Alternative
Homework Assignments* (AHAs)* Problem site* is mentioned and the
URL given.

Work supported in part by NSF grant DUE-9455561 |

To contribute problems to this site, send them to redish@quark.umd.edu.

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Comments and questions may be directed to
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Last modified June 25, 2002