[An earlier version of this article appeared in the March 1995
issue of *ExtraOrdinary Reflections*, the newsletter of Bombay
Bicycle Club. This article will display best on a graphical
browser with table support. -Ed.]

The Article Appendix: The Math JavaScript Calculators

Scott Ellington

February 14, 1995

As you may have noticed during some of the Sunday Bombay rides, southern Wisconsin is far from flat. Remember Vermont Church Road? Climbing hills is always work, but if it seems really miserable, chances are your lowest gear just isn't low enough. According to the US Geological Survey, the average grades for at least 100 vertical feet for some of our favorite hills are:

Denzer Road, northbound | 8.7 percent |

Observatory Road, northbound | 9.5 percent |

Enchanted Valley Road, northbound | 11.1 percent (Ugh!) |

Mounds Park Road, southbound | 10.0 percent |

Even the first hill up Old Sauk Road westbound averages 10 percent for 50 vertical feet. So, for southern Wisconsin 10 percent is a reasonable value for maximum grade, unless you do Enchanted Valley often. Elsewhere in the Driftless Area, there are probably grades of close to 15 percent. The climb up to Wyalusing State Park comes to mind.

A few words about the mechanics of bicycle gearing: The chain goes around two sprockets, the chainring by the pedals, and the freewheel on the back wheel. A low gear requires a small chainring and a large freewheel sprocket. Most bicycles have either two (double) or three (triple) chainrings. The smaller chainring on a double can't be all that small, so a triple is necessary to get really low gears. The gear ratio is the number of teeth on the chainring divided by the number of teeth on the freewheel. This ratio is often multiplied by the wheel diameter, usually 26 or 27 inches, to give the "Gear Inches".

The speed at which a cyclist can climb a steep grade depends on the amount of power the cyclist is capable of producing long enough to reach the top, and at a tolerable level of exertion. The appropriate gearing allows the cyclist to maintain a reasonable cadence at this speed. Very strong cyclists can climb at a good cadence in a fairly high gear, but the rest of us need much lower gears. Even among active cyclists, there is a tremendous range of power capability, so there needs to be an equally large range in bottom gears. A reasonably fit recreational rider can sustain a power of about 200 Watts, while a competitive racer generates about 400 Watts. Most casual riders are probably comfortable at about 100 Watts. (A Watt is a unit of power; there are 746 of them in one horsepower.)

The easiest way to get an idea of how much power your body is capable of providing is to measure the speed you can sustain on a flat road, with no wind, down on the drop bars. Since the larger hills in southern Wisconsin take at least 15 minutes to climb, use the speed you can sustain for at least that long. This is what I'll call your "flat speed". Don't be discouraged by the numbers that follow: If you really work as hard on the flats for 15 minutes as you do climbing a steep hill, you'll probably go a lot faster than you expect.

Now, I'm going to make the assumption that the recreational cyclist wants to be able to climb major hills without resorting to anaerobic exercise, which means not exceeding the power limit given above. Competitive cyclists, of course, can produce considerably more power for short periods, but they aren't really the folks who need advice on gearing. For those of us who don't like to suffer that much, the aerobic mode assumption seems reasonable.

I use 60 RPM as the minimum acceptable cadence for hill climbing. A higher cadence might be more efficient, but it leads to truly ridiculous gear ratios, as you'll see. A lower cadence, which may require getting off the saddle, is really tough on knees. If your knees will take it, you may be able to get away with a higher bottom gear, at least until your knees get a little older.

Now here's the most important point: If the lowest gear isn't low
enough, it is **impossible** to maintain a reasonable cadence on a steep
grade. For any combination of weight, grade, cadence, and gear ratio,
the laws of physics dictate how much power is required. If the gear
ratio doesn't match the available power, the cadence must be lower.

The table below shows the gear ratios needed to climb 10 percent grades at a cadence of at least 60 RPM, with no extra touring gear on the bike.

Rider | Flat Speed | Power | Gear Inches | Chainring/ Freewheel |
---|---|---|---|---|

Casual | 17.5 MPH | 100 Watts | 16 | 20/34 |

Recreational | 22.0 MPH | 200 Watts | 31 | 28/24 |

Competitive | 27.7 MPH | 400 Watts | 63 | 42/18 |

Now you can see why the competitive types can climb hills with
those tiny freewheels! But for those of us who don't do time trials
at 28 MPH, much lower gearing is appropriate. While it may not be
easy to get a bike set up with a 16 inch gear, it can be done. And
remember, the 100 W rider can average 17.5 MPH on the flats. If you
add 35 pounds of touring gear, even a 200 W recreational cyclist needs
something like a 28/30 combination (25 inches), a ratio of LESS that
1:1. And there **are** grades in southern Wisconsin steeper than
10 percent.

The chart below lets you determine the required gearing for various conditions. I've assumed a total weight of 175 pounds without touring gear, but the cyclist's body weight doesn't make much difference since heavier riders are generally stronger.

For example, suppose you find that, working as hard as you ever want to work on your bicycle, your flat speed is 19 MPH, and you want to be able to comfortably climb a 10 percent grade without touring gear. The chart shows you need a low gear of 20 inches, which corresponds to a 24 tooth chainring and a 32 tooth freewheel. That requires triple chainrings, because a 24 won't fit on a double.

How strong do you have to be to get away without a triple? Well, about the lowest possible gear with a double is something like 36/32, or 30 inches. That corresponds to a flat speed of at least 21.6 MPH. So, unless you're pretty strong, or like to suffer a lot, you really NEED a triple chainring. A typical double-chainring road bike, say with a 40/28 bottom gear, is really only suitable for someone whose flat speed is at least 23 MPH. Needless to say, there a lot of people with that kind of gearing who aren't anywhere near that strong. If the bottom gear isn't low enough, the only way to climb a long hill is at a painfully slow cadence. Or to walk.

In summary, if the speed you can maintain on the flats is less than about 17 MPH, get triple chainrings with the lowest ratio you can get, like 20/34. Unless you can maintain at least 22 MPH, you still need a triple. If you can really manage 28 MPH, go for the corncob freewheel. Individual preference plays a role, of course. Even with the optimum gearing, getting to the top of the hill is still work.

Of course, there's an easier way to determine if your gearing is right: If people with bigger freewheels than yours are passing you going uphill, your gears aren't low enough.

For climbing steep grades, the required gear ratio is:

where

G = 1.64 * 10^4 * P

C * GR * W

G = Gear ratio, inches P = Power, Watts C = Cadence, RPM GR = Grade, Percent W = Total weight, pounds

Wind resistance and other friction are neglected, a pretty good approximation when climbing steep grades, where most of the energy goes into simply overcoming gravity.

The approximate power is:

P = 0.0188 * V^3

where V is the flat speed in MPH. All friction except air resistance is neglected, but this is a reasonable approximation for speeds above about 15 MPH, with the rider in the crouched position on the drop bars. At lower speeds the rolling resistance of the tires becomes significant. Combining the two equations gives the required gear in terms of flat speed:

G = 308 * V ^ 3

C * GR * W

Finally, for 27 inch wheels:

Gear inches = 27 * Chainring

Freewheel

Reference: Frank Rowland Whitt and David Gordon Wilson,
*Bicycling Science*, MIT Press.

Enter the weight of your rig-- bike, gear, and self-- the cadence you like to keep, your power output, and a grade-- and click on the "crank" button to compute the required gear ratio. I've provided some reasonable defaults; just type over them with your own values.

This next calculator takes your wheel size, chainring size, and freewheel size and computes the gear ratio.