Watt is Power?

The Physics of Cycling!

Watt is Power? Power Tap Bicycle Crank, drive side
Power Tap Chain Rings and Crank Courtesy of Saris Media Center
This entry is part of 3 in the series Physics for Cyclists

Watt is Power? — What is a Watt?

Us cyclists are concerned about many numbers. Heart rate, cadence, speed, weight, power, and those are just the most obvious (and not requiring needles). The standard unit of power measurement is the watt and the more watts we can produce and sustain the better!

Watts are not the only unit we use to measure power. The horespower (746 watts) is also a unit used to measure power in. We are also familiar with watts in the world of electricity where we often use watts to select lightbulbs and of course once a month we see a measure of kilowatt-hours, not a measure of power but of total energy.

So at least on an informal basis we know what a watt is. However, this article’s point is to make you intimately familiar with watts!

What is Power? — Joules is Work and Watt is Power

So what is power? Again we know the more watts or horsepower something generates the brighter things are or the faster some action happens.

Watt is Power? Sir Isaac Newton -- the father of modern Physics
Sir Isaac Newton a pioneer of physics
Think about the following scene. You have your boat up at the lake and you return from a fishing expedition. You ease your boat into the lift’s cradle unload the fish, the drinks, and the gear. Now you start to turn the big wheel to raise the boat exactly 1 meter above the water level. Yeah! You did work! In fact, we can calculate and attach a number to that work, but we’ll just say it took you five minutes to do the work.

The next day the UPS truck shows up and delivers an electric boat lift winch. You hook it up to the lift, you lower your boat, and go fishing again. On your return you use the lift to raise your boat exactly one meter above the water level and this time it only takes 30 seconds to accomplish the task. Question, was more work done by you spinning the big wheel or by the motor (for the sake of this discussion ignore you working your arms and all the rest of your body’s movements)?

Well, the answer is the work accomplished in both cases is identical! Some of you may protest that the motor worked harder because it did the job more quickly. I will say congratulations, you understand what power is, you just have it confused with work!

Power is the time rate at which we do work. Lifting the same weight the same distance is always going to be the same amount of work no matter if you do it slowly or you employ a motor to do it quickly. The difference is the motor can generate more power and hence accomplishes the task more quickly.

How much power in each case? Without working in numbers the exact power is undetermined, but we do know the motor was 10x more powerful than you.

This is probably not a big suprise to most cyclists who pay attention to their power numbers (even the SWAGuestimated numbers from Strava or other app). You ride a route and have a rocking night and so you try it again the next day and come back slower, we all notice the power numbers on the slower night are lower.

Watt is Power? — Starting to Get Real

Work is also something we have an intuitive feeling about but most likely not an exact one. We will limit our discussion here to moving things around, such as you and your bicycle. Also, to make it easy we will consider my ride up Rib Mountain and then ONLY the vertical componet. Yeah, we wave away many complications but it really will give us an understanding of what is going on. One last thing, we will be working metric units.

Work is simply the force applied to an object multiplied by the mass of the object times the distance you moved the object [update June 24, 2016 — add in distance factor]. Lift a one kilogram object one meter into the air and you accomplish 9.8 joules of work.

Watt is Power -- Formula for work calculation 1 Kg lifted 1 meter into the air
Calculating the work done by lifting 1 Kg of mass 1 meter into the air

In my case, my mass is 91.6 Kg and I climbed 221 meters. So the work done is:

What is Power? Calculation of my work climbing up Rib Mountain
Calculating the work I did climbing Rib Mountain

The 9.8 m/sec2 is a constant and is the acceleration due to gravity.

Now that we have the work done we simply divide that result by the seconds the effort took and officially that number is 1,106 seconds (18’26”) and that result is 179 watts over the whole effort. This is obviously an average.

Interestingly enough Strava reports an average power of 27 watts, I am guessing Strava assumes flat ground when doing its power calculations (and in that case it also has to assume some value for air/wind resistance as well as rolling and other resistive forces). I knew right away the Strava number was incorrect as I reported earlier the ride was not very difficult but it certainly was not an easy 27 watt effort. I suspect had I not stopped at the park entrance to check to see if I had to pay a fee I would have done it in about 18′ flat giving a power of 184 watts. Not bad, which yields a power to weight ratio of 2.01 watts/Kg. Its early in the season.

What is Power? — Power On the Flats and Downhills

Performing the same exercise on the flats is trickier. One of Newton’s laws states that an object in motion stays in motion unless acted on by a force. Essentially it says there is no force required to keep your speed and direction the same. However, we live in the real world and we have to apply power to keep the speed up to overcome air resistance and friction.

Because it is harder to directly measure air resistance and friction in a real world setting power meters determine power indirectly. Power meters are usually some form of strain gauge and determine the force you are applying to the pedals and that combined with speed allows the meter to compute power.

Watt is Power? — No Such

Watt is Power -- Gottfried Wilhelm von Leibniz
Leibniz (pictured immediately above) and Newton pioneered the math behind power
Also when you look at your power meter or a power reading on Strava you are looking at some very fancy mathematics (well, that should be obvious now). Both power meters and Strava make it look like you have a power number at a specific point in time. Well, not really. You can not have a specific power at a point in time as power is work divided by time the work is done. How much work can you do in zero seconds or the flip side is any work in zero seconds is ludicrous power!

You are right to complain I’m being pedantic. We solve this problem by simply calling the power over a very tiny portion of time as the “instantaneous power”. This is where the mathematicians start to get all excited about limits and derivatives.

Watt is Power? — The Two Giants

What is this, no Einstein photos? Unless we are talking about photovoltaics powering a hidden motor or perhaps keeping our devices charged while riding, Einsteinian physics just doesn’t apply. Unless you are riding your bike near the speed of light the physics we need to consider is the stuff of Newton!

So there you have it you now know watt a watt is!

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