Bicycle Science

I realized that I had barophobia (fear of gravity) about two thirds of the way up Dutchtown Zion when I saw my hear rate at 10 beats over what should have been my max. Gravity is great for keeping stuff stuck to the ground but sucks when you fight against it by climbing up a hill on a bike. Although a hill is just a geological formation we tend to personify them by describing them as annoying or rude and usually add some profanity as they get steeper and longer. This only adds to the phobia.

The best way to treat barophobia is to gain a better understanding of the forces that make it hard to climb hills so the fear can be rationalized away. This means understanding the science behind bicycling.

This pass summer I bought a new bike specifically for the hillier rides that I do. One of the first questions I had to answer is do I want a triple or a compact. If you search the blogosphere you will see passionate opinions on both sides which tend to sound like political arguments with no facts to back them up. To cut through all the noise I ended up turning to physics, math and a good book “Bicycling Science” by David Gordon Wilson to get some real answers.

There has been a lot of research on the science behind the bicycle and the equation below can answer most of the questions about the forces that act on a bike. With this equation and a little knowledge of physiology you can make some good decisions what type of gearing you want for your bike.

The equation:

Pr = Cfriction x V x P + Cair x (V + Vwind)^2 x V+ 9.81 x P x Slope% x V

Where Pr is the power generated by the rider
Where V is the speed of the bike
P is the weight of the rider and bike
Slope% is the steepness of the slope in percent
Cfriction is the coefficient of friction of rolling resistance
Cair is the coefficient of drag


For those who have arithmophobia (fear of math) let me break this equation down in to some simple explanations so you can you can understand how to use it to make some decisions about setting up the gearing on your bike. If you don’t care about the actual math you can stop reading and head to the Bicycle Science section of my web site which has some calculators that you can play with to get a feel of physics behind bicycling.

The equation has 3 parts:

The first part (Cfriction x V x P) calculates the power needed to overcome the rolling resistance of the bike. This includes the friction of the gears, bearing, tires, and other parts of the drive train that slow you down. The rolling resistance of a decent quality road bike is very small especially once you are rolling and up to speed. Usually only 3% to 5% of power generated by the rider goes to overcoming the rolling resistance so this part of the equation can usually be ignored.

The second part (Cair x (V + Vwind)^2 x V) calculates the power needed to overcome air resistance. The drag coefficient (Cair) is usually between 0.25(racing tuck) to 0.4(sitting up). The important thing to note here is that the power needed to overcome air resistance goes up by the square of the speed. So assuming a drag coefficient of 0.4, cruising along at 16 mph takes 146 watts going 2 miles an hour faster takes 208 watts, a 62 watt increase. Another interesting thing about air resistance is that weight doesn’t matter. Whether you are 100 lbs or 200 lbs it still takes the same amount of effort (146 watts) to cruise at 16 mph.

The third part of the equations (9.81 x P x Slope% x V) calculates the amount of power needed to climb up a hill. When you are climbing what is happening is that you are fighting the force of gravity. The two main factors here are the weight of you and the bike, and how quickly you are gaining altitude. The rate that you are gaining altitude depends on how steep the slope is and your speed. Not all 350 ft climbs feel the same. Climbing all the way to the top of Federal Twist you climb a little over 350 ft in a little less than a mile and the last half has a 15% slope. It’s a tough climb. On the other hand the climb up Rockaway Rd is also around 350 ft but it is 3 miles long and the slope stays mostly in the 3% to 4% range with no really steep spots. It’s actually an enjoyable climb.

A rider can only produce and maintain a specific about of power so as the slope gets steeper the rider speed decreases which is what the equation is telling you but you need to go beyond the math to understand why it feels so bad to climb a tough hill like Federal Twist. This is where you need to understand a little of the physiology behind riding.

This is a simplification but, the human body has basically two types of muscles, slow twitch and fast twitch. Slow twitch muscles are the ones that are used to generate slow continuous power so these are the muscles that you use as you cruise along that flat stretch of road. Slow twitch muscles are very efficient so as long as you each and drink correctly these muscles can generate power all day long.

Fast twitch muscles have the same strength as slow twitch muscles but as the name implies can apply their strength much faster. These are the muscles that kick in when you sprint away from the pack or climb a tough hill. The quickness comes at a price because Fast twitch muscles fatigue quickly so they can only be used in limited bursts. This is the problem will climbing steep hills.

As you start to climb you usually need to apply more power to the pedals. You only have a limited amount of power so as the slope gets steeper you need to slow your speed. If you stay in the same gear the force you need to put to the pedals increases and you switch from using slow twitch muscles to using fast twitch muscles. You can mitigate this by switching to a lower gear to keep the force on the pedals to a comfortable level. On steep hills you may run out of gears and have to stomp your way up the hill. In this case you will quickly use up your fast twitch muscles and eventually run out of power. Most of the time you will make it to the top before this happens but it will use up some energy and the next hill will feel harder.

This is where the science comes in. What you want to do is to find the gearing that will let you maintain a comfortable level of force on the pedals without having to use up your fast twitch muscles. This may sound hard to determine but can easily be measured by your cadence. When you are climbing the slower your cadence the more fast twitch muscles you are using. Everybody has a preferred cadence some a little faster some a little slower but the rule of thumb is that you want to try to maintain a minimum cadence of around 60 rpms as you climb.

Knowing this you can now use the information to calculate what gearing to need to get yourself up a hill of a certain grade. Let me take you through the calculation to show you how this work.

The first thing you need to know is how much power you can generate. To get a rough estimate of this find a few miles a flat road on a windless day and ride along this road as fast as you comfortable can. This shouldn’t be an all out sprint but you should be pedaling level that you think you could maintain for a half hour or so. Your average speed over these few miles gives you an idea of how much continuous power you can generate. To calculate you power just plugin the speed, V, into the equation (Cair x (V + Vwind)^2 x V). For example let’s say your average speed was 20 mph and you were riding pretty low on your handlebars giving you a drag coefficient of 0.3 your generated power would then be 214.4 watts.

Now that you know your max continuous power you can then calculate how fast you can climb a hill of a certain slope. To do this you use the equation 9.81 x P x Slope% x V and plugin V (the speed you are going), P (the weight of the rider and bike) and Slope% (the steepness of the slope in percent) to calculate the power required for the climb. Since your weight and the slope of the hill are unchangeable what you will have to do is to adjust the speed so that the power calculated is less than or equal to the power that you can generate. For example let’s say you and the bike weigh 180 pounds and the slope you are trying to climb is 10%. To climb that slope at 8 mph would require 285 watts of power which is more than you can generate but if you lower your speed to 6mph it only requires 214 watts of power which is doable.

Knowing the speed you can climb a hill still doesn’t tell you what gears you need but if you know what cadence you want to maintain you can determine the best combination of gears. To do this you need to understand what gear inches are.

Gear inches is the number of inches your bike moves for each rotation of the pedals. For example if you in a 52 tooth ring in the front and an 11 tooth chain ring on your cassette, then your gear inches are 128 which means you back wheel travels 128 inches for each rotation of the pedals. This translates in to 4.7 rotations of a 27 inch wheel.

Calculating gear inches is pretty simple. All you do is divide the number of teeth on the front chain ring by the number of teeth on the rear cassette then multiple by the wheel size. So the formula is.

Gear inches = front chain ring/ rear cassette * rear wheel diameter

You then can use gear inches to calculate your cadence for a given speed and gear ratio using the following formula.

Cadence = speed/(gear inches*Π*60*0.0000157828)

For example if you are going up a hill and can maintain 6 mph, for a 39/28 (front ring/rear cassette) gear ratio your cadence (rotations per minute) will be 54 rpm which should be a hard but doable pace. However if you are going up a steeper hill and can only maintain 4mph the cadence for a 39/28 gear ratio the cadence will be 36 rpm which is going to require a lot more strength. At this cadence you may not have the strength to push the pedals and if you do you won’t be able to keep riding for long. What you need in this case is a lower gear ratio. If you are going 4mph with a 30/32 ratio then your cadence will be 53 which you probably can handle.

If you are still following along you now have everything you need to determine the best set of gear for the type of terrain you ride. First you determine how much power you can generate. Then use your power number to determine the speed you will be able to climb the type if hills that you are going to ride. Then use the speed to determine the gear ratio you need to maintain the cadence you want. This will give you the lowest gear you will need. To make this easy for people who are not good with a calculator or spread sheet I have added a Bicycle Science page to my web site that you can use to do all the calculations I described here.

The math that I have shown here can be used to give you a good idea of the best gears to get you up the hills but there is no substitute for training and experience. Although the correct gearing will help you get up the hill if you don’t do some hill training or understand the how to maintain a comfort pace as you climb you will still feel like crap when you reach the top.