Momentum and gyroscopic difference between wheel sizes

Question

A common folk wisdom in mountain biking is that smaller wheels (like 26") are "quicker" or "more flickable" and "better for courses that require lots of speeding up and slowing down", whereas larger wheels (like 29") tend to "carry more speed" through track sections due to their "greater momentum". This is usually attributed to the "smaller rotation weight" of the smaller wheels and presumably the resulting angular momentum.

Angular momentum is proportional to radius. But the speed that a wheel turns at a given bike speed is also proportional to radius. So the amount of momentum stored in a wheel should be exactly the same, regardless of wheel size. And the relative change in wheel angular momentum required to accelerate or decelerate to bike by a certain amount should also be exactly the same. Similarly, the amount of angular momentum required to change the tilt of the bike should be the same.

Is my understanding of the physics correct, and therefore the folk wisdom is either wrong or based on other factors than wheel momentum (such as differences in bike geometry, weight/linear momentum, or rolling resistance)?

Or is my understanding of the physics incorrect, and there really is a "angular momentum" difference between wheel sizes?

In theory, bike wheels are not discs, but since the same tire casings and same rim extrusions are often used for both 26 and 29 inch versions of rims and tires, let's assume the circumferential density of the tire and rim stay the same for different wheel sizes.

Answer 1

This is usually attributed to the "smaller rotation weight" of the smaller wheels and presumably the resulting angular momentum.

This is mostly untrue (assuming constant mass).

If you have 1.0 kg mass at the perimeter of the wheel (counting mass of spokes only partially according to a certain percentage), and are going at 8 m/s, then you have 0.5*1.0*8^2 = 32 Joules of rotational energy on the wheel. This figure is the same on every possible wheel size.

A smaller wheel, if of the same weight, does indeed have for example smaller moment of inertia, but it's rotating faster so that cancels it out in the energy figure. In angular momentum figure, it doesn't completely cancel out so gyroscopic effects could be very slightly different, but that's irrelevant as it has been proved that bicycles with counter-mass rotating in opposite direction attached to the wheels (to cancel out angular momentum and gyroscopic effects) are perfectly rideable. Bicycle riding does not depend on angular momentum, so that's not what matters. Rotational energy is what matters.

What could reduce rotational energy in the wheel on smaller wheels however is the smaller mass of the rim and tire. However, the effect is small, 559mm and 622mm wheels have only slight difference in mass, and you could achieve the small mass change by selecting a lighterweight rim and a lighterweight tire, plus a lighterweight tube of course. The claim that 559mm wheels are somehow more different than 622mm wheels than what you can achieve by small rim, tire and tube mass changes is untrue therefore: you could achieve the same mass by swapping the rim, tire and tube, and the mass is the only figure that counts.

Hub does not contribute to angular momentum or energy at all because it's in the center and spokes are usually of similar weights and their contribution percentage to angular momentum and rotational energy is about the same.

Angular momentum is proportional to radius.

Yes, true, but this is also true: when considering bicycle riding, angular momentum is irrelevant.

What is relevant for acceleration for example is rotational energy and that only depends on mass at the perimeter, not radius.

But the speed that a wheel turns at a given bike speed is also proportional to radius. So the amount of momentum stored in a wheel should be exactly the same, regardless of wheel size.

You are forgetting that angular momentum is L = Iw = mvr where I = mr^2. Thus, when using rotation rate, angular momentum is proportional to radius squared, when considering linear movement rate, angular momentum is only proportional to radius. Thus, angular momentum does increase with wheel size. However, angular momentum is irrelevant and doesn't affect bike handling, as have been proved with the experimental bikes where angular momentum is cancelled by counter-rotation.

And the relative change in wheel angular momentum required to accelerate or decelerate to bike by a certain amount should also be exactly the same.

You're considering the wrong figure. If you accelerate, you are interested in energy not angular momentum. Energy is not dependent on wheel size if mass is the same. However a smaller wheel could have lower mass, but to achieve that effect, you have to have a larger difference than just 559mm vs 622mm. Consider for example Brompton vs 622mm. But you won't enjoy the Brompton. It will transfer all obstacles as larger bumps to you, the rider. A 622mm bike on the other hand is well enjoyable.

Or is my understanding of the physics incorrect, and there really is a "angular momentum" difference between wheel sizes?

There is difference in angular momentum. There is no difference in angular energy (assuming identical mass). However, angular momentum is irrelevant, but angular energy is relevant.

In theory, bike wheels are not discs, but since the same tire casings and same rim extrusions are often used for both 26 and 29 inch versions of rims and tires, let's assume the circumferential density of the tire and rim stay the same for different wheel sizes.

If circumferential density is the same, then smaller wheel does indeed have smaller mass. However the effect is not large. But if this effect is present, then indeed a smaller wheel does have both smaller angular momentum and also smaller angular energy. But not by much.

Answer 2

Rotation weight and angular momentum have very little, if anything, to do with it.

29" wheels have a smaller 'strike angle' when they hit an obstacle. Therefore they roll over bumps easier. The larger diameter also means when the wheel drops into a hole, it drops a smaller distance. While it is barely measurable in real world conditions, over thousands of undulations per kilometer, the difference is less effort for the same speed, or more speed for the same effort. Switching to my old 26er from my 29er, the front wheel often 'drops' unexpectedly into holes that my 29er would have rolled right over (though only for the first few kilometers until I get dialed back into the bike).

29ers also have a larger tire volume for same tire width, which means you can run lower pressures for same effective load capacity. Lower pressure MTBing is used to increase traction and small bump compliance. A modern 29er now has much wider tires than old 26er (2.4" vs 2.1"), so direct comparisons are hard.

26" wheels are often talked about as being more agile, but again it's not about the weight or angular momentum. The axle is lower and wheel stiffer (see later comment though). A lower axle leads to a lower center of gravity, and stiffer wheel leads to greater responsiveness. Forks are shorter, all being equal, making fork flex less noticeable.

Frame geometry plays a huge role in this. The real comparison is early 29ers vs 26ers, not a modern 29er. In the early days the were made by taking a 26er frame, making the seat tubes and stays a bit longer, increasing the fork length, and calling it a 29er. These bikes went better in a straight line, and rolled off rough stuff much better than any 26er before them, but they cornered liked a beached whale. The wheels were relatively flexible and poorly supported by low spoke angles.

A decade on…

• We have boosted axle widths to make spoke angles more realistic
• Forks now carry much bigger diameter stanchions, making them probably stiffer than 26" forks of a decade ago
• The frame geometry has completely changed

Overall the difference between 29er and 26er, while still there (if they still built them), would be hard to detect.