Friday, February 25, 2011

Power-to-weight ratio and speed; or, I suck at math

On Wednesday, a group of SF2G'ers rode into work via Calaveras Rd.  Brooks showed up with a shiny new PowerTap, and as we were climbing up Calaveras, we were chatting about power:weight ratio.  We were both working on PR times on the climb, so not much glucose was available for higher-level thought.  Which is to say, we were not able to resolve the question posed.  Which is fine, because, hey, who really wants to think a lot when there's such beautiful scenery around and lactate build-up in your legs to distract you?

But while waiting at the coffee machine in the break room at work today, it dawned on me - Duh, the math for this should be really simple; we're just talking about ratios, here.  Then again, I suck at math.

Anyway, the basic question was: If you remove X amount of weight from the bike/rider combination, what does that translate to in terms of additional speed (unspoken assumption was at a given constant power output).

So, let me kind of talk through it, and I'll let my readers judge my reasoning and math skills...  Note I'm going to ignore most units of measurement here, because I'm pretty sure it doesn't matter.

For the purposes of this discussion, let's assume that the rider is going to produce a constant power output.  Power is angular velocity x torque, so if you both increase cadence 10% and reduce torque 10%, you still have the same power output.  Right?

So, let's say that a rider loses 5% of their body weight.  For a given speed, that would seem to dictate a 5% reduction in power output.  But we stipulated earlier that our rider is going to maintain a constant power output.  Seems to me that we then get to apply that 5% to either torque or angular velocity; the former would manifest as pushing a bigger gear, the latter simply spinning at a higher cadence.

Because it's easier for me to wrap my head around, I'll go with higher cadence.

So let's say our intrepid rider is turning 34x23 @ 90RPM up a climb.  That's 10.4mph.  5% reduction in body weight would allow 5% increase in cadence to keep the same power output; 94RPM, 10.9mph.  10% reduction, 99RPM, 11.3mph.

If you didn't want to spin that fast, you could shift to a higher gear (increase your torque) and reduce your cadence (angular velocity) to maintain the same power output.

So, just to be clear, it seems to me that e.g. a 10% reduction in body weight would allow for a 10% increase in speed at a specific power output.  Or, just as importantly, it would allow a 10% reduction in power at a specific speed, which might be a big deal if previous efforts were at or slightly above LT power.

I'm concerned only about climbing here, and so am purposely ignoring the effects of wind drag, etc. which aren't major factors when climbing.

Anyway, like I said, I suck at math.  Is there a flaw in my logic here, or do I have it right?