GLU: 3.1

How Far Can Your Bicycle Go?

Introduction / Motivation

Most of us have used bikes to get to school or to a friend’s house. As our riding terrain changes, we shift gears to accompany uphill, downhill, and straightaways. Bicycles depend on human power. Our legs produce peak power at a certain RPM (number of revolutions per minute), much like a car engine. Cars have low gears for starting and going up hills and high gears for going fast. Bicycles have lots of gears to accommodate different speeds and terrain.

Historically, bicycles didn’t have gears and the pedals were attached directly to the wheel like a kid’s tricycle or a unicycle. In order to go faster, the wheel was made larger, and eventually became known as the high-wheeler or penny farthing with a wheel up to 5 ft tall! Once mechanics were able to create a chain drive with gearing, the wheels were made a more manageable size and the gears took care of the speed. These bicycles are called safety bikes. The gears are created by having a front chainring attached to a rear cog with a chain.

When we shift gears the resistance of the pedals to our feet and legs changes, making pedalling either easier or more difficult. But how do gears help us go up and down hills? When is the best time to switch gears? How can math help us ride this bike? What is the optimal gear setting for maximum efficiency?

Vocabulary / Definitions

Word / Definition
Gear Ratios / The number of chainring teeth divided by the number of teeth on the rear cog.
Chainrings / The front gears of a bike attached to the pedals by the cranks.
Cog / A toothed wheel or gear that is part of a chain drive attached to the rear wheel
(also called the rear sprocket).
Crank / The arm which connects the pedal to the bottom bracket axle (the part of the frame around which the pedal cranks revolve).
Rollout / The distance the bike travels with one complete revolution of the wheel.
Development / The distance the bicycle travels for one revolution of the pedals.

For this activity, we will use the international standard of measuring and calculating the gear development using the following formula:

wheel circumference * chainring teeth/rear cog teeth= development

Remember:

circumference = diameter * pi

How fast can your bicycle go?

In this activity, we will experiment with a modern bicycle to find out how the range of gears helps us ride slowly up hills and quickly down the other side.

Measure the rollout of the bicycle wheel.

Mark on the floor where the starting point is, roll the wheel until it has travelled one complete revolution and measure back to the starting point on the ground.

Record that number here:

Rollout:

How does this relate to the diameter of the wheel?

Count the teeth of each of the chainrings (the gears at the pedals) and record them here:

Chainrings: big middle small

Count the teeth of each of the cogs (the gears on the wheel) and record them here:

Cogs: big (1) (2) (3) (4) (5) (6) (7) small (8)

How many gears are possible on this bike? Show your calculations.

Do you think someone would use every single one of these gears? If you don’t think so, why do you think there would be so many gears available on this bike?

What combination do you think is the lowest gear (which chainring with which cog) and why would you consider this the lowest gear?

How far do you think the bike travels per pedal turn in the lowest gear?

Measure the development of the lowest gear.

Put the bike in its lowest gear (small chainring and biggest cog (1)). Mark the starting point on the floor and pedal the bike one full turn of the cranks.

Measure the distance back to the starting point on the floor and record that number here:

Lowest gear development:

How does this compare with your guess?

What combination do you think is the highest gear (which chainring with which cog) and why would you consider this the highest gear?

How far do you think the bike travels per pedal turn in the highest gear?

Measure the development of the highest gear.

Put the bike in its highest gear (big chainring and smallest cog (8)). Mark the starting point on the floor and pedal the bike one full turn of the cranks. Measure the distance back to the starting point on the floor and record that number here:

Highest gear development:

How does this compare with your guess?

Gear Chart

Create a chart with all of the gear combinations filled in with the development.

The formula is the following: number of chainring teeth / number of cog teeth * wheel circumference = development.

Chainring teeth # will be listed across the top and cog teeth # will be listed down the side.

Big / Middle / Small
Cog 1 (Big):
Cog 2:
Cog 3:
Cog 4:
Cog 5:
Cog 6:
Cog 7:
Cog 8 :

Make a graph of your development measurements and the calculations from the gear chart:

Hint: Development will be on the y-axis and # of teeth of the cogs will be on the x-axis.

Hint: Use a different colour (or symbol) for each chainring, and also for your measurement.