Sunday, December 11, 2005

Math Question

This morning I posted this to the Fixed Gear List but I figured I'll put forth the question into the blogosphere as well:

Here's a math question for you. In my copy of "The Dancing Chain" by Berto, Shepherd and Henry (1st edition) on page 117 in a quote from a Cyclo catalog, a cyclist identified as C.A.P. recounts climbing a 1 in 4 hill in a 41 inch gear. Later on this page there is the comment:

"C.A.P.'s claim of riding up a 1 in 4 (25 percent) slope was a little optimistic. With a cyclist's entire weight on the pedal, the maximum rideable gear is seven times the slope denominator. On his low gear of 41, C.A.P. could, with maximum effort, have ridden up a slope of 1 in 6. For touring, an ideal gear is only twice the denominator. Thus a gear of 41 is comfortable to pedal up a slope of about 1 in 20 (5 percent.)"

Now this maximum rideable gear being seven times the slope denominator seems bogus. If I'm reading this right with a 70 inch gear I should only be able to climb a 1 in 10 (10% grade). Yet I've gone up 15% and even 18% climbs with a 70 inch gear. I'm not looking for a bunch of anecdotal "I've climbed hill X" stories but it does seem to me there is some kind of mathematical maximum which I think would have to factor in crank length, rider weight, gravity, the coefficient of friction (at some steep point tires will slip on the road) and maybe some other stuff. Any engineers with time to kill and fresh batteries in their 41GX want to take a shot at this?

BTW in general "The Dancing Chain" is a really cool book, even if it does focus on shifting and coasting and other things that clutter up the riding experience.

Kent Peterson
Issaquah WA USA
http://kentsbike.blogspot.com/

P.S. SHAMELESS PLUG: "Shiftless Bum" T-shirts with the No-Derailleur logo are available online at http://www.cafepress.com/BikeThere

5 comments:

ryan said...

Maybe it has something to do with putting more than your maximum weight on the pedal (pulling up on the rear pedal, on the handle bars, etc, all allow you to put more weight into the pedal stroke).

Cellarrat said...

I think there are to many factors...

Crankarm 165mm-185mm
I like low 60 gear inches...
Fixed or free?
Size of tire
What you are riding on..

I have surpised myself a few times on climbs with how steep they can be. It also seems ez on a fixed then I try the same climb with a freewheel and a lower gear I can't make it...

Who knows? I am just gonna go ride and not think to much about it =)

Moishe said...

I thought about this a lot and started doing some math, but I think Ryan's right -- you can put much more than your weight on the pedal by pulling on the bars. Momentum is also key for short hills.

I think this is one of those problems where it looks like everything should cancel out into one unitless number, but it's not that simple.

btw, congrats on your new job!

Al Maviva said...

Hey for anybody who wants to do the math, check out the Human Powered Vehicle Gear Ratio Calculator at http://www.soulbikes.com/gears/

Al
- All 58 inches of me.

Anonymous said...

The inconsistency between analytical and practical experience is almost certainly due to a static analysis by Berto but the problem is actually a dynamic (and thus more complex) one.