Abstract
A moving bicycle is said to stay up because of the gyroscopic effect of its wheels. This paper asks whether that explanation is the right order of magnitude——the answer is that it is not. No new mathematical theorem and no new law is claimed.
First, the gravitational torque on an $80$ kg bicycle-and-rider leaning $5$ degrees is $\mathbf{68.3765}$ N$\cdot$m (Section 2).
Second, this is the core of the paper. The front wheel’s gyroscopic torque is only $\mathbf{1.2750}$ N$\cdot$m, a gap of $\mathbf{53.6286}$ (Section 2).
Third, quadrupling the speed brings it only to $5.1000$ N$\cdot$m (Section 2).
Fourth, the two become equal at $\mathbf{268.14}$ m/s, that is $\mathbf{965}$ km/h (Section 2).
Fifth, what remains is steering: a $5$ degree lean is balanced by a turn of radius $29.1385$ m (Section 3).
Sixth, the separator is whether a mechanism is individually necessary, not whether it acts at all (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——the gyroscopic effect, caster trail, and the inverted pendulum are all standard. We do not build bicycle dynamics——the four-degree-of-freedom linearised model and its eigenvalue analysis (weave, capsize) are not entered. They are named and no more. We prove no self-stability——the speed range over which a bicycle is self-stable is not computed. What is computed here is the size of torques. We perform no experiment——no measurement on a real machine is made. The numbers are on parameters stated explicitly. Control is not treated——the rider’s reflexes and learning are not the subject. Tyres are not treated——contact patch deformation, slip angle, and toe are not the subject. Relation to earlier papers: Paper 357 showed maximum power and maximum efficiency to be different points——there there were two objectives; here there are two explanations. Paper 356 showed quantities of the same dimension to be different quantities——here two torques of the same dimension are compared in size. Paper 300 showed whether two things share a root is decidable——by that test, gyroscopic restoration and steering restoration have distinct roots. Paper 353 showed a claim turning false when its condition is dropped——confusing “it acts” with “it suffices” is the same error. What is added is giving both torques as numbers under one set of conditions, showing the ratio to be $53.6286$, solving backwards for the $268.14$ m/s at which they are equal, giving the time available from the inverted-pendulum time constant, and putting the separator on individual necessity.
1. Notation and the Question
Setting: Treat the front wheel as a thin ring of mass $m_w=1.5$ kg and radius $r_w=0.34$ m.
$$I_w=m_wr_w^2=0.1734\ \mathrm{kg\,m^2},\qquad T_{\text{gyro}}=I_w\omega\Omega,\qquad T_{\text{grav}}=Mgh\sin\theta$$Take bicycle and rider together as $M=80$ kg with centre of mass at $h=1.0$ m.
Observation 1.1 Two explanations are commonly heard. The gyroscopic effect of the wheels, and restoration by caster (trail)——both are real effects.
Observation 1.2 This paper denies neither. What is asked is only the size——because “it acts” and “it suffices” are different things.
Observation 1.3 Treating the wheel as a ring is an upper estimate. A real wheel has spokes and a smaller moment of inertia——so the comparison is made with the gyroscopic side favoured.
Question: Is the gyroscopic effect enough to arrest a fall?
2. Two Orders Short
Calculation 2.1: Both torques under one set of conditions (computed in this paper).
| Quantity | Value |
|---|---|
| Front-wheel moment of inertia $I_w$ | $0.1734$ kg m$^2$ |
| Wheel angular speed $\omega$ at $5.0$ m/s | $14.7059$ rad/s |
| Steering rate $\Omega$ | $0.5$ rad/s |
| Gyroscopic torque $T_{\text{gyro}}$ | $\mathbf{1.2750}$ N m |
| Gravitational torque at $5$ degrees $T_{\text{grav}}$ | $\mathbf{68.3765}$ N m |
| Ratio | $\mathbf{53.6286}$ |
Observation 2.2 This is the core of the paper. A factor of $53.6286$ is not “slightly short”——it is nearly two orders of magnitude. The gyroscopic effect cannot be the main actor.
Observation 2.3 Speed does not close the gap. At $20$ m/s ($72$ km/h) the gyroscopic torque is $5.1000$ N$\cdot$m——still far from $68.3765$ N$\cdot$m.
Observation 2.4 The equalising speed can be solved for. $T_{\text{gyro}}=T_{\text{grav}}$ at $v=\mathbf{268.14}$ m/s——that is $\mathbf{965}$ km/h. It is no longer a bicycle.
Observation 2.5 Every estimate was taken in the gyroscope’s favour. Wheel as a ring, a standard rather than a light wheel, and a steering rate that is not small——and still this is the gap.
Observation 2.6 Care is needed here. The claim is not that the gyroscopic effect is absent——$1.2750$ N$\cdot$m is real. The claim is that by itself it is not an explanation.
3. What Remains Is Steering
Calculation 3.1: Times from the inverted pendulum, and the balancing turn (computed in this paper).
| Quantity | Value |
|---|---|
| Time constant $\sqrt{h/g}$ | $\mathbf{0.3193}$ s |
| Time to fall from $1$ degree to $10$ degrees | $\mathbf{0.9558}$ s |
| Turn radius balancing a $5$ degree lean at $5$ m/s | $\mathbf{29.1385}$ m |
Observation 3.2 There is about a second. $0.9558$ s to fall from $1$ degree to $10$——long by the standard of human reaction time. So the rider gets there in time.
Observation 3.3 The way to recover is to bring the contact patch back under the centre of mass. Steer into the fall——a $5$ degree lean is exactly balanced by a turn of radius $29.1385$ m.
Observation 3.4 This steering torque is the same order as the gravitational one. It comes straight out of the balance $\tan\theta=v^2/(Rg)$——it does not pass through a small intermediate quantity as the gyroscopic route does.
Observation 3.5 Distinct roots by the criterion of Paper 300. Gyroscopic restoration comes from conservation of angular momentum, steering restoration from the geometry of balance——two mechanisms explaining one fact, with different roots.
4. “Acts” Is Not “Necessary”
Observation 4.1 The literature shows neither gyroscope nor trail to be individually necessary. A design cancelling both has been made self-stable——this paper does not reproduce that experiment. It is cited and no more.
Observation 4.2 What this paper can show independently is a size. The ratio $53.6286$——it states “not enough by itself” in orders of magnitude. It is an agreement reached from a different road than the literature’s.
Observation 4.3 The two claims have different strengths. “Two orders too small” is carried by the computation here——“individually unnecessary” is carried by the literature. They are not mixed.
Observation 4.4 To be stated honestly——this paper does not fully answer “then what holds it up”. What is shown is that steering remains as the only candidate of the right order——self-stability itself is a four-degree-of-freedom eigenvalue problem and does not reduce to a single mechanism.
5. The Separator: Individually Necessary?
Observation 5.1 The separator——not whether a mechanism acts, but whether it suffices alone, and further whether it is necessary alone.
| Question | Gyroscopic effect | Steering | Carried by |
|---|---|---|---|
| Does it act | yes ($1.2750$ N m) | yes | this paper |
| Does it suffice alone | no ($1/53.6286$) | right order | this paper |
| Is it necessary alone | no | no | the literature |
Observation 5.2 The three rows are different questions. A “yes” in the first does not settle the second——everyday explanations collapse all three into one.
Observation 5.3 The rightmost column is the discipline of this paper. Write, row by row, what carries which claim——different provenance means different strength.
Observation 5.4 The separator can be moved. Make the wheel extremely heavy and the gyroscopic torque grows——but bringing the ratio $53.6286$ to $1$ needs $53.6286$ times the moment of inertia, a front wheel of some $80$ kg. At that point the vehicle is not a bicycle.
Observation 5.5 To be stated honestly——$68.3765$ N$\cdot$m and the ratio $53.6286$ are values on parameters stated explicitly. Another bicycle gives other numbers——what does not move is the structure: gyroscopic torque scales with speed and gravitational torque does not depend on it.
Observation 5.6 One last thing——“is the effect real” and “does the effect explain it” are different questions. The gyroscopic effect is real, and it is not enough——stop at confirming that something is real and you are satisfied without ever having looked at the order of magnitude.
6. The Census and the Close
- $I_w=0.1734$ kg m$^2$ (Calculation 2.1): an upper estimate, as a ring.
- $T_{\text{gyro}}=1.2750$ N$\cdot$m (Calculation 2.1): $5$ m/s, steering $0.5$ rad/s.
- $T_{\text{grav}}=68.3765$ N$\cdot$m (Calculation 2.1): $80$ kg, $5$ degree lean.
- Ratio $53.6286$ (Observation 2.2): nearly two orders.
- Only $5.1000$ N$\cdot$m at $20$ m/s (Observation 2.3): speed does not close it.
- Equal at $268.14$ m/s (Observation 2.4): $965$ km/h.
- Time constant $0.3193$ s (Calculation 3.1): inverted pendulum, $h=1$ m.
- $0.9558$ s from $1$ to $10$ degrees (Calculation 3.1): long enough for a person.
- A $5$ degree lean balances at radius $29.1385$ m (Calculation 3.1): steering is the right order.
- Three different questions (Observation 5.2): acts / suffices / necessary.
Closing: A moving bicycle is said to stay up because of the gyroscopic effect of its wheels——and the gyroscopic effect is certainly real. But it is two orders too small——treating the front wheel as a thin ring (which overestimates its moment of inertia), riding at $5$ m/s and steering at $0.5$ rad/s gives a gyroscopic torque of $\mathbf{1.2750}$ N$\cdot$m. The gravitational torque on an $80$ kg bicycle-and-rider with its centre of mass at $1.0$ m, leaning $5$ degrees, is $\mathbf{68.3765}$ N$\cdot$m. The ratio is $\mathbf{53.6286}$ — not “slightly short” but nearly two orders of magnitude. Speed does not close the gap——at $20$ m/s ($72$ km/h) the gyroscopic torque reaches only $5.1000$ N$\cdot$m, and the two become equal at $v=\mathbf{268.14}$ m/s, that is $\mathbf{965}$ km/h. It is no longer a bicycle. What remains is steering——as an inverted pendulum the time constant is $0.3193$ s, and falling from $1$ degree to $10$ takes $0.9558$ s, long by the standard of human reaction time. So the rider gets there in time. Recovery means bringing the contact patch back under the centre of mass — steering into the fall — and a $5$ degree lean is exactly balanced by a turn of radius $29.1385$ m. That restoration comes straight out of $\tan\theta=v^2/(Rg)$, so it is the same order as the gravitational torque. But the strengths of the claims must not be mixed——“two orders too small” is carried by the computation here; “individually unnecessary” is carried by the literature. A design cancelling both gyroscope and trail has been made self-stable, but this paper does not reproduce that experiment. What this paper says independently is the ratio $53.6286$, reaching the same conclusion by a different road. One thing separates them——not whether a mechanism acts, but whether it suffices alone, and further whether it is necessary alone. The three are different questions, and a “yes” to the first does not settle the second. Bringing the ratio to $1$ would need $53.6286$ times the moment of inertia — a front wheel of some $80$ kg — and at that point the vehicle is not a bicycle. To be said honestly——this paper does not fully answer “then what holds it up”. What is shown is that steering remains as the only candidate of the right order. Self-stability itself is a four-degree-of-freedom eigenvalue problem and does not reduce to a single mechanism. One last thing——“is the effect real” and “does the effect explain it” are different questions. The gyroscopic effect is real, and it is not enough. Stop at confirming that something is real and you are satisfied without ever having looked at the order of magnitude.
References
- Kooijman, J. D. G., Meijaard, J. P., Papadopoulos, J. M., Ruina, A., Schwab, A. L. (2011). A Bicycle Can Be Self-Stable Without Gyroscopic or Caster Effects. Science.
- Meijaard, J. P., Papadopoulos, J. M., Ruina, A., Schwab, A. L. (2007). Linearized dynamics equations for the balance and steer of a bicycle. Proceedings of the Royal Society A.
- Jones, D. E. H. (1970). The stability of the bicycle. Physics Today.
- Whipple, F. J. W. (1899). The stability of the motion of a bicycle. Quarterly Journal of Pure and Applied Mathematics.
- Åström, K. J., Klein, R. E., Lennartsson, A. (2005). Bicycle dynamics and control. IEEE Control Systems Magazine.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). Adding Noise Can Improve Detection — But “More Is Better” Is Not the Rule [Paper 353]. Zenodo. 10.5281/zenodo.22696835
- Yamagishi, Y. (2026). The Current Flows and Does No Work [Paper 356]. Zenodo. 10.5281/zenodo.22697124
- Yamagishi, Y. (2026). Maximum Power and Maximum Efficiency Are Not the Same Point [Paper 357]. Zenodo. 10.5281/zenodo.22697180
Reference
Yamagishi, Y. (2026). The Gyroscope Is Two Orders Too Small to Explain Why a Bicycle Does Not Fall [Paper 359]. YOI: https://www.yamagishiyuuki.com/yoi/359
DOI: 10.5281/zenodo.22697293
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 359. This paper is a preprint and has not been peer reviewed.