## Friday, January 27, 2012

### Power up a Tower: Part I

After reading my friend Stan's excellent blog for some time, I've decided to do a bit of number crunching, too. My end goal is to be able to predict performance and benchmark other athletes depending on the height of a climb. I won’t be able to do it all in one post, so I expect this will be one of many.

Like Stan, I believe an important measurement is power. To get a good understanding of power, here is a brief explanation on the difference between work and its close relative, power.

Work, is defined as force (e.g. weight) multiplied by distance. The standard SI unit of measurement of work is called a joule (J). It takes 9.8 joules of work to lift 1 kilogram (kg) of mass one meter (m) straight up in earth’s gravity field [written as 9.8 J = (1 kg x m2)/s2]. The 9.8 factor comes from earth’s gravity field acting on a mass (i.e force = mass x acceleration). Basically, a mass will drop (accelerate) at a rate of 9.8 meters per second each second (9.8 m/s2).

Power is closely related to work. It is defined as the amount of work done per a given time period. The standard SI unit of measurement of power is called a watt. For example, it would take 9.8 watts of power to lift 1 kilogram (kg) of mass one meter (m) straight up in exactly one second [written as 9.8 watts = (1 kg x m2)/s3]. Likewise it would take 4.9 watts of power to lift up the same 1 kg mass if the work is spread out across two seconds. In essence, you’ve done the same amount of work at half the power output, but it has taken twice as long to do so.
Power and work are very import in stair climbing because you are primarily acting against gravity when you climb up a stair case. This is unlike other sports like running or cycling that primarily act against friction and drag (unless you are climbing uphill).

Here is an illustrative example. If a 100 kg climber (about 220 lbs) races up a 100 meter stair case (roughly 30 stories) in 300 seconds (5 minutes) then that climber has done:
100 kg x 9.8 m/s2 x 100m = 98,000 joules of work
98,000 joules / 300s = 326.67 watts (on average during those 5 minutes)

So how would your climbing ability compare with this person? It is hard to tell just by looking at a person’s power output. After all, a lighter climber (say someone who only weighs 110 lbs (about 50 kg) only needs to do 49,000 joules of work to climb this stair case. Similarly, the lighter climber would only need to exert a paltry 164 watts to climb up this staircase!
A good measure of relative performance would be to compare the power to mass ratio of these two climbers. In the case above, each climber has an equal 3.27 watts/kg power to mass ratio.

Interestingly enough, to calculate this ratio you don’t even need to know how much someone weighs! Since mass is used to calculate power, you effectively eliminate mass from the equation since you divide by mass when you calculate the power to mass ratio. This simplifies things greatly:
Power to mass ratio (watts/kg) = [Height of climb (meters) x 9.8 m/s2] / Climbing time (seconds)

This ratio gives a good performance indicator when climbing up given height. But how do you compare performance between climbs of different heights? The answer is somewhat tricky. After all, athletes can only maintain a difficult pace (i.e. a high power output) for a relatively short amount of time. The length of the race really determines what pace you are able to maintain. Here is an illustrative example: Runners can’t sprint an entire marathon. Conversely, runners can’t competitively jog a 100 meter dash.

To better make this point, here are the power outputs and power to mass ratios of the last two races I participated in. I also included data for Sproule Love and a couple other fellow Tower Masters so you can see some relative data points. Weights are approximate except for my weight and Sproule’s. I specifically asked him how much he weighs after our race up One Penn Plaza :-)
*note* I think One Penn Plaza climb is actually about 10 meters taller than my estimate above. Does anybody have a tape measure I could borrow?

As you can see, since the Trump Parc is only 100 meters tall, the climbers were able to put out around 20% more power than in One Penn Plaza which about twice as tall.

Also note that although I had a larger power output than Sproule, he climbed much faster because he weighs a good 25 lbs less than I do. He has a superior Power to Mass ratio.

Finally, can you tell who use to be an elite marathon runner? As you can see from the two data points above, Steve has a very good power to mass ratio on tall climbs. I have a feeling he’ll kick my butt at the ESBRU, which is nearly 100 meters taller than One Penn Plaza (^_^).

So now that I’ve written about work, power, and the power to mass ratio, I hope to show how power output will vary depending on the height of a climb. It will probably be a while before my next post since it will take a bit of time to collect the data I’m going to need. Until then, I’m going to be busy climbing stairs. I hope to improve my power to mass ratio in time for the ESBRU!

#### 1 comment:

1. Thanks for putting these power equations up, Alex. I've been training with HR and would like to move to training with power. These will allow me to easily do the calculations I need.

I hope to see you at Strat this weekend! -Bob Toews