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The Physics of Bicycle Wheel Size

Why Do We Care?

We care because there is considerable misinformation out there with regard to the advantages and disadvantages of different bicycle wheel sizes, and edification is a noble pursuit. ūüôā I hesitated to use the word physics in the title here because I thought it might turn some people off. However, physics is¬†ultimately what we’ll be talking about so put on your thinking caps (i.e., helmets) and buckle up my friends.

When we talk about wheel size we’re really talking about mountain bike wheel size since virtually all road bikes have the same size wheels. If you’re not familiar with the different size wheels available for mountain bikes, you might want to start with¬†our introduction to Mountain Bike Wheel Size.

The Primary Misconception

There is one very common misconception that I’m trying to overcome here, and that is the misconception that larger mountain bike wheels come with a huge penalty because they are heavier (though that is true) and because the additional weight is concentrated at the outer edge of the wheel (also true). It is undeniable that larger diameter wheels & tires weigh more¬†than smaller diameter wheels and tires. However, it is not true that the additional weight results in a significant penalty all, or even much, of the time.

Forces and Friction

This discussion is not intended to be submitted to a peer-reviewed physics journal, nor even to an introductory physics textbook for that matter, so I will take some liberties with terms in order to keep this discussion approachable. For example, I may not differentiate between weight and mass even though they are not technically the same thing. Likewise, rather than always using the term centripetal force, I may use angular momentum because I believe it is more widely understood, even though they are not precisely the same thing.

That said, when it comes to analyzing the physics of a bicycle wheel, there are really only a few forces and a couple sources of friction. The forces are:

  • the force exerted on the wheel by the rider¬†pedaling
  • the centripetal force of a¬†wheel¬†in motion
  • the force of gravity

and the primary sources of friction are:

  • friction of the hub rotating around¬†the axle
  • friction between the tire and the ground

Are there other sources of friction in a bicycle? Of course! There is friction at every interface of the¬†drive train, within the chain itself, wind resistance, rubbing brakes (if this applies to you, then you should really stop reading this and go fix your friggin’ brakes), bottom bracket, headset, and more. However, I argue that all of these sources of friction are virtually identical when comparing different mountain bike wheel sizes. Technically, is the friction in the hub greater for a larger wheel because the rotating mass is greater? Sure, but you won’t be able¬†to convince me that the difference can be perceived by the rider. However, you say that the larger diameter wheel has a larger contact patch with the ground and therefore greater friction there, right? Arguably, yes. However, the misconception that we’re discussing here is that the alleged penalty is due to the¬†weight of the larger wheel and tire rather than the increased contact patch. In fact, many people argue that the increased contact patch is one of the primary advantages of larger diameter wheels and tires.¬†Therefore, for all practical purposes, the sources of friction can be ignored with regard to a discussion of the impact of wheel and tire weight. So, that leaves us with the forces.

We previously identified two of the forces at work as “the force exerted on the wheel by the rider¬†pedaling” and “the centripetal force of¬†a¬†wheel¬†in motion.” Let’s briefly examine¬†the relationship between these forces. Let’s say we just got on the bike to start our ride.¬†At moment zero¬†the bike is at rest, so the¬†centripetal force of the wheel is zero. As we exert the other force (by pedaling), that increasing force is transferred to the wheels. Okay, now we’re up to speed and just cruising along at constant velocity. At constant velocity (think about a very brief moment of coasting), the¬†force exerted on the wheel by the rider¬†pedaling is zero and only the centripetal force is at work. The point here is that we can consider these forces in isolation.

This is How I Roll (i.e., Rotating Mass)

Okay, now we’re getting into the physics but we’ll keep the math relatively simple, if not nonexistent. We¬†could consider the circumference of our wheel as made of an infinite number of points, much like calculus breaks area problems into small segments. However, for the purposes of simplicity, let’s reduce the number of parts¬†of our wheel to just eight arc segments, each spanning 45 degrees. We can analyze the forces on the wheel by analyzing the forces at the center of each segment. If we superimpose a compass onto our wheel, the centers of each of the eight segments correspond to the four primary compass directions, abbreviated as N, E, S¬†&¬†W¬†along with their midpoints, NE,¬†SE, SW &¬†NW. If we picture our wheel/compass rotating clockwise, then our bicycle is moving from left to right.

To keep things simple for now, let’s assume that our wheel is rolling with constant velocity on level ground. With these assumptions made, we can draw a vector (arrow) at each of our compass points to represent centripetal¬†force (or angular momentum if you prefer). Each vector is the same length but they all point in different directions, tangential to the wheel of course. At North, the vector points to the right (or the direction of travel of the bike). At South, the vector points to the left, or backward. At East, the vector points straight down. And at West, the vector points straight up. At each of the other four points, the vectors are angled at 45 degrees either up or down.

What about the other force, gravity?! I’m glad you asked. It may seem¬†that we have thus far only accounted for one of the two forces we previously identified.¬†However, let’s make another¬†assumption. Let’s assume that in increasing the diameter of our wheel we increased the weight by exactly half a pound. Since half a pound is eight ounces and we have eight wheel segments, we’ve increased the weight of each wheel segment by exactly one ounce.

The segment that is half way up the trailing side of the wheel (West) is being raised vertically, so the entire ounce has to be lifted. However, other segments have angular vectors and the entire ounce isn’t being raised vertically. Furthermore, for each segment being raised (against gravity), there is segment on the opposite side of the wheel that is being lowered (gravity assist, baby!). In other words, the gravitational work of lifting segments is countered by the gravitational assist of lowering the other segments.

Conclusion:  At constant velocity on level ground, there is zero penalty due to the increased weight of our larger wheel. Cool.

What About When You (Have To) Accelerate?

Fair question. When¬†we swing a leg over the bike at the trailhead, the bike is (hopefully) not moving. So, yes,¬†we have to accelerate in order to get up to the constant velocity discussed above. To visualize the forces at work, let’s assume that in order to accelerate we have to¬†make the trailing half of the wheel go faster than the leading half of the wheel. While that may be an oversimplification, it works for our model. So the vectors going up (trailing side of wheel) have to be even greater than the vectors going down (leading side of wheel). At that point the vectors no longer cancel each other out. Therefore, accelerating with¬†a heavier wheel does indeed require more work.

Quantitative Conclusion: There’s no getting around it. When you have to accelerate, a heavier wheel does require more work.

Okay, I’m going to take a slight detour here (on level terrain, haha) and turn¬†from the¬†quantitative to the¬†qualitative. Bear with me. While our analysis indicates that accelerating with¬†a heavier wheel does require more work, let’s ask ourselves some¬†qualitative questions about mountain biking.¬†How much of the time are we actually accelerating? Or accelerating significantly? While you could make a theoretical argument that on rough terrain we‚Äôre constantly either accelerating or decelerating, or a 50/50 split, I would argue many of those small incremental accelerations are arguably not particularly significant. On a loop ride on smooth terrain and no breaks, we might very well accelerate at the beginning, ride a relatively constant velocity, and decelerate at the end. In cross country riding (not just racing, but riding), I would argue that moments of significant acceleration comprise the vast minority of mountain bike rides.

This is where my extremely smart best buddy Rich, along with many of you I expect, chimes in and says “Dude! Apples and orangutans! We started out talking about rotating mass and ended up with insignificant moments of acceleration!”

With all due respect, it’s not¬†apples and orangutans at all. We’ve demonstrated¬†that at constant velocity there is no penalty for a heavier wheel.¬†Therefore, it is only moments of acceleration that matter. Since mountain biking can involve rides with only minimal moments of acceleration or significant moments of acceleration (i.e., technical terrain), it‚Äôs reasonable to turn to qualitative measures at this point.

Qualitative Conclusion: If you think you’re constantly accelerating, then a larger diameter wheel may indeed result in a significant penalty. However, if you don’t think¬†you’re spending a significant portion of time accelerating, then a larger diameter wheel should¬†not result in a significant penalty.

But Hills Are Like Accelerating, Right?

Wrong. Accelerating is like accelerating. Channeling Douglas Adams for a moment, hills are just like, well, tilted flats. Honestly, I think this may be the hardest part for many people to absorb. I’ve thought about why that might be and I think the explanation could be as simple as the fact that accelerating and climbing¬†are the two times when we have to put in significant effort on a bike. That’s when we have to do the most work. That’s when we feel it in our legs. However,¬†the forces at work during¬†accelerating and climbing are¬†not the same.

So, let’s ride uphill. First, we’re back to constant velocity just to keep things simple. Okay, now use your mental prowess to visualize¬†an angled line under the wheel/compass sloping from lower left to upper right. But please don’t make it too steep, as I’m on a singlespeed.

To account for the fact that the wheel is now going uphill, let’s add¬†antigravity vectors to our mental diagram. They are anti-gravity vectors because they have to overcome the force of gravity in order to make the wheel go uphill. So we add vectors pointing up at the center of every segment. Therefore,¬†the segments on the leading half of the wheel still benefit from downward centripetal force (or angular momentum, whatever), but that benefit is reduced by gravity.¬†Meanwhile, the segments on the trailing half of the wheel already had vectors pointing upward so the antigravity vector gets added to those because the whole damn wheel has to go up, up, up! However, because we’re climbing at a constant velocity, the angular vectors cancel each other out again and we’re just left with the anti-gravity vectors.

However,¬†the whole damn bike (and rider) has to go up, up, up! So, if we‚Äôre climbing at a constant velocity then the increased weight of the wheel isn‚Äôt any different than weight that is anywhere else on the bike or the rider.¬†Constant velocity is key here. When the wheel is rotating at a constant speed, the vectors on opposite sides of the wheel cancel each other out and the only thing left is gravity. So it doesn‚Äôt matter where the weight is.¬†Raising 8¬†ounces of weight requires the same effort regardless of where the weight is. So at a constant velocity, 8¬†ounces on the wheel isn‚Äôt any different than 8¬†ounces anywhere else. Once that wheel is spinning at a constant velocity, no acceleration is necessary (by definition). So it‚Äôs not that the weight doesn‚Äôt matter but that the location of the weight doesn‚Äôt matter at that point.¬†Don’t conflate climbing and¬†accelerating. They are not the same.

Conclusion:  At constant velocity, even when climbing, there is zero rotational penalty due to the increased weight of our larger wheel. Very cool.

Revisiting Friction

We’ve come a long way in a relatively short period of time, and we’ve come to some conclusions that clearly go against the conception¬†that the additional weight of a larger diameter wheel results in a significant penalty all, or even much, of the time. So, let’s revisit one¬†of our assumptions about friction that I think people have the hardest time swallowing. That assumption is that the friction between¬†the tire and ground gan be ignored. Some people might make the argument that riding uphill is like accelerating because the friction between the tire and the ground is constantly slowing down the wheel so, in order to maintain our theoretical constant velocity, in a real world with friction we’re actually constantly accelerating.

It’s true that we ride in a real world with friction, and I agree that the primary friction we encounter is friction with the ground. Hell, another word for that friction is traction and we all appreciate¬†good traction. However, remember that the whole point of this discussion is to address the¬†conception¬†that the additional weight of a larger diameter wheel results in a significant penalty¬†compared to a smaller diameter wheel. So, first, our whole discussion is rooted in the weight difference rather than any frictional difference because the conception is that the weight makes a difference. Second, even if we were to take friction into account,¬†I think most of us can agree that on level ground the increased friction of a larger diameter tire over a smaller diameter tire is negligible. If that’s true on level ground, then it’s also true on a hill. The frictional difference between the two tire¬†sizes doesn’t change just because we’ve tilted the ground. The coefficient of friction of¬†each wheel against the ground is exactly the same as it is on level ground.¬†Therefore, for the purposes of comparing one wheel size with another, our assumptions stand up.

Summary of Conclusions

As we did in the acceleration section above, our conclusions can be categorized into quantitative and qualitative.

Quantitative

There are two¬†quantitative¬†conclusions that we’ve reached.

  1. At constant velocity, even when climbing, there is zero rotational penalty due to the increased weight of our larger wheel. When climbing at constant velocity, it’s not that the weight doesn’t matter but that the location of the weight doesn’t matter.
  2. However, during moments of acceleration, there is indeed a penalty due to the increased weight of our larger wheel.

Qualitative

So it all comes down to¬†acceleration. The more you accelerate, the bigger the penalty.¬†If you think you’re constantly accelerating, then a larger diameter wheel may indeed result in a significant penalty. However, if you don’t think¬†you’re spending a significant portion of time accelerating, then a larger diameter wheel should¬†not result in a significant penalty.

Overall Conclusion

At the risk of being verbose (I know, it’s too late ;-), let’s participate in one more analogy, or perhaps it’s just a visualization exercise. Let’s picture ourselves on a bike climbing up a really long but really consistent grade. Picture¬†pavement if that helps with the consistency. I expect most experienced cyclists would agree that the most efficient way to climb that grade is to get spun up to a nice efficient cadence and then maintain it. I would argue that is how we actually ride most of the time, at a relatively constant velocity. Therefore, it is my opinion that the penalty due to the¬†increased weight of a¬†larger wheel is negligible on many, if not most, rides.

Bonus Argument in Favor of Larger Wheels

A bonus argument? Cool! The simple version of the bonus argument in favor of larger wheels is that a tire with a larger¬†circumference will cover more ground per rotation. Therefore, if two riders are pedaling along at the same RPMs, the one with larger diameter wheels and tires will be going faster. In order to mostly, but not entirely, eliminate the impact of gearing, let’s consider two singlespeed mountain bikes. My first singlespeed (SS1) has 32:18 gearing (32-tooth sprocket in front and 18-tooth cog in back), resulting in a 1.78 gear ratio. My second¬†singlespeed (SS2) has 33:20 gearing, or a 1.65 gear ratio. So, for every rotation of the cranks on SS1, the rear wheel does 1.78 revolutions and for every¬†rotation of the cranks on SS2, the rear wheel does 1.65 revolutions.

If the wheels were the same size, SS1 would be¬†faster due to the higher gear ratio. However, the wheels aren’t the same size. SS1 has 26″ wheels and SS2 is a 29er. Generally speaking, the diameter of bicycle wheels and tires (e.g., in this case 26″ and 29″) refers to the outer diameter of the tire. While this isn’t exact, it’s a reasonable approximation for us here. (Since SS1 and SS2 are actually my real bikes, I can tell you that the actual measure of the outer diameter of the tires is more like 25.5″ and 28.5″ respectively. But I digress. Let’s stick with outer diameters of 26″ and 29″ for now.) Circumference equals pi times¬†diameter, so every revolution of the 26″ wheel covers about 81.7″ of ground. Similarly, every revolution of the 29″ wheel covers about 91.1″ of ground. Therefore, for each rotation¬†of the cranks:

  • SS1 covers 1.78 revolutions x 81.7 inches per revolution¬†= 145″
  • SS2 covers 1.65 revolutions x 91.1 inches per revolution¬†= 150″

So, with two riders pedaling side-by-side at the exact same cadence, SS2 goes 5″ further for every rotation of the cranks. After just a dozen rotations of the cranks, SS2 is five feet ahead of SS1. To slow SS2 down to match SS1 would require us to change the gear ratio on SS2 to about 1.6 (145″ /¬†91.1 inches per revolution), or 32:20 instead of 33:20.

Okay, at this point this latter argument admittedly gets more qualitative than quantitative. I assert that most riders when switching from a 26″ singlespeed to a 29″¬†singlespeed wouldn’t make that large a change in their gear ratio and, therefore, would gain at least some speed advantage from the larger wheels. I would argue that the reason it doesn’t feel necessary to make this large of a change in gearing when increasing wheel size is that, once spun up to speed, the larger wheels don’t require additional effort that is proportional to their size. In fact, you may recognize that this is really just¬†another¬†way of stating the primary argument above, that no additional effort is required to turn a larger (i.e., heavier) wheel at a constant velocity. ūüôā

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