Bicycle Science

© Brooke Clarke 2013
Bicycle Science
Climbing 15% Grade


Bicycles (Wiki) are based on many scientific principles and understanding these can help in making choices and maybe in coming up with a new and better design for some specific application.  I'll be looking at a Raleigh Mountain Tour bike because it's the one that was at hand, not selected for any reason other than free to me.

I live at about 1,000 feet elevation and my driveway and the street out in front have a considerable slope.  So I'm looking into a bicycle that I could ride to and from my house, rather than getting one that I would have to drive somewhere in order to ride.  It's my recollection that when the gear is too high and when going up hill you can stand on the pedal and nothing happens, i.e. you need low gearing in order to climb a hill.  Another factor is that if the speed is too slow you can no longer balance.  Would gyroscopic stabalization help?


This is a man's (it has a horizontal bar between the steering head and the top of the seat post) mountain (Wiki: Types, mountain) bicycle .  It uses derailer gearing (Wiki).
There are three chain wheels (front sprockets) with teeth counts of: 49, 44 and 30.  The rear cassette has 6 sprockets with teeth counts of: 34, 28, 24, 20, 15 and 14 teeth.
The rear tire is a 26 x 2.10" and the front is a 26 x 1.95".

One of the earliest bicycles was the Penny-farthing (Wiki).  It had a large diameter front wheel so that with one turn of the pedels the bike woud go forward a reasonable distance, i.e. at a normal pedeling cadence (Wiki) the gearing (Wiki) was approiate for a human powered vehicle (Wiki).  Later chain drive was used to provide gearing so that smaller wheels could be used.  Gear inches (Wiki) is a way of equating modern gearing to the diameter of a driven wheel.

The crankset (Wiki) consists of the pedel arms and chain wheel.

Climbing 15% Grade

When pedeling if you press down on the pedal and stand up, i.e. your full weight is on the downgoing pedel then there's some amount of power you can generate.  If you have very strong leg muscles and you sit and press down with one leg and at the same time lift with the other leg (this requires using pedel clips (Wiki)) then you can generate much more power.

Classical Mechanics (Wiki) can be used to analyze various aspects of bicycle operation.  For example to determine the gearing required to move a bicycle up a given slope a free body diagram (Wiki) and simple mathematics can be used.

For the free body diagram below showing the bicycle climbing a 15% grade (Wiki) we will assume all the riders weight is pressing down on the forward pedal and that the weight is split evenly between the front and rear.  W= 200 pounds (91 kg) for this example.

Free Body diagram bicycle
                climbing 15% grade

W:     Weight of rider
RRT:  force of Road on Rear tire Tangent to road
RFT:  force of Road on Front tire Tangent to road
RRN: force of Road on Rear tire Normal to road
RFN: force of Road on Front tire Normal to road

Note:  In a free body diagram of a static situation all the forces must sum to zero, that's to say there's no net force that would cause movement.

First assume the crank is welded to the frame.  Then:
RRN=W/2* COS(8.531) & RFN=W/2* COS(8.531), or RRN=RFN=W/2*COS(5.31) = 99.6 pounds.
RRT = RFT = W/2 * SIN(8.531) = 14.83 pounds.

Now assume the crank is not solid and that the gearing is such that there's no motion of the bicycle, that's to say the torque on the crank ends up generating just enough force to keep the bicycle from moving. Then:
The torque about the rear axle is 13" (half the wheel dia) * 14.83 = 192.79 in lbs. = 16.06 ft lbs
The torque about the crank pivot point is 6-5/8 * 200 = 1325 in lbs = 110.41 ft lbs.
Note the crank torque is much higher than needed to balance the rear wheel torque so with 1:1 gearing the bike will go forward.

It turns out that my driveway has a slope of 10.1 degrees which is a grade of 17.5 % (i.e. 17.5 feet rise for 100 feet horizontal).
AccuRemote Digital Angle

Now let's look at riding uphill.  The Wiki page for bicycle performance says that a comfterable walk for a 150 lb person takes 30 Watts, or for a 200 lb person maybe 40 Watts.
A Watt is a Newton Meter per Second or a Joule per second(Wiki).
Assume the bicycle is going up a 15% grade at 3 MPH that's the same as 3 Mi/Hr * 5280 Ft/Mi * 12inch/Ft * 1meter/39.37inch = 4828 meters/hr or 1.34 meters/sec along the road.
But only the vertical distance counts for calculating power and that's SIN(8.531) * 1.34 m/s = 0.2 meters/sec
The vertical force is 91 kg.
Joules = m * g * ht = 91 * 9.8 * 0.2 = 178 J for each second or 178 Watts.
It will actually be higher because of the weight of the bicycle which is about 16 kg, so the power is closer to 209 Watts.
The Wiki bicycle performance page says an amaetur cyclist can produce 3W per kg of body weight, or 273 Watts.
So, 273W / 209W * 3 MPH = 3.9 MPH top speed.  This can only be done using the lowest possible gear and at about 50 turns of the crank per minute (i.e. half normal cadence, i.e. a lot of straining.

In the lowest gear front: 30 teeth per turn and 50 turns per minute then there are 1500 teeth per minute of chain movement.
If a motor was connected to the chain with a sprocket that had 14 teeth then the sprocket would be turning at 1500/14 = 107 RPM.
Electric scooter 500 Watt motors operate on 24 VDC but they turn about 2600 to 3000 RPM, way too fast for a direct connection to the chain.
So a jack shaft is needed to provide a gear down by 24 to 30 to one so that the motor is turning a peak PRM (i.e. peak Watts).  Any lower gearing and the motor will not develop 500W, but instead a much lower power, this is why hub motors do not help climb hills.

There are gear head scooter motors (Electric Scooter Parts) that have a shaft speed of about 500 RPM, but would still need a jack shaft gear reduction.

Problem with derailer gearing

There is no simple way to know how to get into the next higher or lower gear from whatever your current gear is set to.
It may be as simple as shifting the rear gear up or down, but it also may require changing both the front and rear gear settings.
The following table came from an Excel spreadsheet that has been sorted by inches per turn so that the gearing is in order.
The first three speeds (30/34, 30/28, 30/24) are all with the front 30 tooth sprocket and changing the rear sprocket, but to get to the forth gearing (44/34) both the front and rear sprockets need to be changed.  The speed range is between 3.6 MPH and 28.4 MPH for cadence of either 50 or 100 turns per minute of the cranks.


Turns per min Front
in/turn 100 TPM
50 TPM

30 34 76 7.2 3.6

30 28 92 8.7 4.3

30 24 107 10.1 5.1

44 34 111 10.5 5.2

49 34 123 11.7 5.8

30 20 128 12.2 6.1

44 28 135 12.7 6.4

49 28 150 14.2 7.1

44 24 157 14.9 7.4

30 15 171 16.2 8.1

49 24 175 16.6 8.3

30 14 183 17.4 8.7

44 20 188 17.8 8.9

49 20 210 19.9 9.9

44 15 251 23.8 11.9

44 14 269 25.5 12.7

49 15 280 26.5 13.2

49 14 300 28.4 14.2

The top speed you can achieve depends on how powerful you are and to some extent the design of the bicycle.

There are gearsets that are internal to the rear wheel tha use internal gear wheels, not anything to do with the chain and these can have equal ratio increase/decrease for each gear setting.

For example the rear gears have the following ratios:

Ratio to next lower gear

As you can see the ratios are not anywhere near a constant ratio like you can get with the internal hub gearing.
For example the Rohloff Speedhub (Wiki) has 14 speeds where the ratio between all of them is 1.135.
The Shimano Alfine 700 (Wiki) has 11 speeds, but not equally spaced: first to second is about 29% the the others are about 13 to 14% each.

Many more details to follow.


Fig 1 Left overall
Fig 2 Right overall
Fig 5 Front chain wheels (sprockets)
Fig 6 Rear sprockets


Tilting Three Wheeler Patents - 3-wheel powered vehicles are legally motorcycles and have lower legal costs (insurance & license)
3 Wheel Recumbent Electric Bikes - this is one way to climb a mountain and maintain stability when going very slowly
Motorized Bikes - and Scooters - may have even lower legal costs than motorcycles if they meet some requirements
Cars - including 3-wheelers
Gyroscopes -


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  page created 23 Oct 2013.