# Bicycle Transmission Physics

Bikes are pretty simple machines. Completely enjoyable without special mathematical treatment. But if you’re an obsessive engineering-type geek like me, you’ll feel better when you have a little math to go with all that exercise.

This is a real simplified run through power transmission in a bicycle as I understand it. The numbers check out as do the units of measure. Somebody call bullshit if you see it.

### Definitions

First, some definitions. These are just to clarify the physics meaning of these words.

Power – The rate at which work is done. If you bike on a flat road at constant speed, then your power output was constant for the whole trip.

Work – Power integrated over time. Power multiplied by time if power was constant

Force – A force acts on a body to produce an acceleration.

Torque – A measure of a force’s tendency to produce rotation. Think about a wrench. If you used a wrench that was 1 foot long and applied 25 lbs of force on it, then the torque on the nut is 25 ft*lbs. Double the distance, double the torque.

### Assumptions

Let’s assume that a random dude is driving a bike down the road. The dude weighs 200 lbs and he is driving an old-school ten speed. Let’s also assume that the losses of the bicycle mechanics are small and ignorable (called negligible in the business).

We’ll ignore the wind. Suffice to say that wind resistance increases the power required to move the bike. Quantifying this value is difficult because it depends on a lot of factors. We’ll also ignore rolling resistance. Just know that narrower, higher PSI tires give less rolling resistance.

### Pedals and Crank

Force is applied to pedals. If random dude is standing on the pedal and the pedal is parallel to the flat road, then he’s exerting 200 lbs of force downward on the pedal. Really, you cannot apply your full load into the pedals at all times and the average force will be less, but we’re simplifying. Also, if the pedal is not parallel, then only a trig ratio of the force on the pedal is producing rotation.

The crank is fastened to the bike chainring. The cranks job is to transform the force into a torque. The torque is then applied to the chainring which pulls the chain (no duh).

The torque on the crank is 111.5 ft-lbs if the crank is 170mm or 6.7 inches

200lbs * (6.67 inches/12 inches/ft) = 111.5 ft*lbs

### Chain

To calculate the force and torque delivered by the chain, you have to know these three things.

1. That power is equal everywhere in the system and it is calculated by rotational velocity multiplied by torque.
2. By using the same chain on the front and back gear, we know that the ratio of front and rear gear rotational velocity is the same as the ratio of teeth on the gears.
3. The ratio of torque is inversely proportional to the gear ratio. It has to be to make the power equal in the front and back even though the speed of the front and back gears is different.