Yo-yo

Freestyle

A real yo-yo is 56 mm across on a 900 mm string — 6% of the frame, with a gap three pixels wide — so the wide view draws it 2.6× larger than life to make the gap, the coil and the response pad visible at all. Only the picture is enlarged; the physics uses the published dimensions throughout. Tick Close camera to see it at true scale.

Pick a round, press Start the music, then hold Space and let go to throw.

Elements

Every element name below comes from a source this build opened: a Guinness World Records title, the National Yo-Yo League's own worked clicker examples, or the Wikipedia article. Eight of the durations are Guinness rate records divided out, so this yo-yo performs them at world-record pace. The 2A elements are here on purpose: IYYF scores nothing for an out-of-division element but still deducts for missing one, and this app enforces that rather than hiding it.

Bench

Three sweeps the engine computes live. Nothing here is a picture of a result — each one is the engine running, so moving a slider moves the physics.

What the engine found

Two claims are commonly made about how a yo-yo works, and this page treats both as hypotheses. The first is confirmed, with a caveat that turns out to matter more than the claim. The second is confirmed, but for a different reason than the one offered — and there is exactly one mechanism that gets round it, which the usual account does not anticipate and which the engine found.

1. The descent — confirmed, as a limit

The claim. "A body unwinding on a string should descend at g/(1 + I/(m·r²)) where r is the axle radius — for a thin axle, a small fraction of g."

Confirmed, and it is also published. MIT 8.01SC Example 21.3, "Torque, Rotation and Translation: Yo-Yo", derives it for an object it calls a yo-yo and settles the ambiguity in so many words: the constraint is ay = b·αz "where b is the axle radius of the Yo-Yo". This engine reproduces that number to 1 part in 10⁹ when given a string of zero thickness, which is the source's own assumption. For the Duncan Imperial's thin fixed axle it comes out at 0.171 m/s², which is g/57.4; for the size C bearing race of a YoYoFactory Abyss it is 1.024 m/s², or g/9.58.

The caveat is bigger than the claim. The formula is the constant-radius limit, and a real yo-yo's winding radius is not constant: string wound on the axle builds a coil, and that coil is thickest at the start of the throw. Differentiating the exact energy quadrature gives

ẍ = ( g − J·r·(dr/dw)·ω² ) / (1 + J),   J = I/(m r²)

and the coil model makes r·(dr/dw) a constant, so the correction is the same size at every point of the throw and is scaled only by the spin. Its sign is always negative, so the yo-yo is everywhere slower than the textbook number for its instantaneous radius. On an Imperial the coil goes from 11.49 mm down to 2.50 mm — a factor of 4.6 — and the consequence is visible: the yo-yo reaches its peak descent speed 56.7% of the way down and arrives at the bottom slower than it was in the middle. On the Abyss the coil only changes by a factor of 1.54, the peak moves to 91.3%, and the effect is nearly invisible. How much a yo-yo's descent departs from the textbook curve is a design choice, set by the ratio of coil radius to axle radius, and nothing published says so.

What is not settled by anything published. No source this build could open states which radius the string winds on for a ball-bearing yo-yo. The bearing's inner race is clamped to the axle and cannot turn under the string, so the string must ride the outer race — 6.35 mm for a size C, not the 3 mm or so of the axle inside it. That inference is this build's, not a source's, and it changes the descent acceleration by a factor of four.

2. The return — confirmed, and the reason is not friction being small

The claim. "The return is a binding phenomenon, not a spin phenomenon. A perfectly slipping axle should sleep indefinitely and never come back, however fast it spins."

Confirmed. The engine was swept over sixty spins from 0.1 rad/s to 10¹² rad/s with the string fully out and the loop free. It never rose, at any of them. Change one flag — the catch — from the same state, with the same parameters, the same integrator and the same step, and it climbs home.

But the reason offered is not the reason the model gives. The natural account, and Wikipedia's, is that the friction is too small: "there may not be enough frictional force to overcome the weight of the yo-yo, which is necessary to begin winding up the string," and a tug makes "the static friction force rise above the gravitation force". That reads as a threshold you could cross by making the axle rougher. In this model you cannot. Under the rolling branch the torque on the body is exactly T·r, so the contact has to carry the whole of the string tension — not most of it, all of it. A loop around a shaft has both ends of its wrapped arc on the same strand, so they share the load, and the capstan difference it can transmit is T·tanh(μφ/2), which is strictly less than T for every finite friction coefficient and every finite wrap. A bare loop is structurally incapable of what rolling asks, and raising μ does not fix it, and raising the spin does not fix it. At the calibrated friction a single wrap carries 49.7% of the tension against the 100% rolling needs.

What does fix it is one wrap of actual string. Wound string is not a frictional coupling at all: a wrap cannot slide around an axle without winding or unwinding, so any wound length at all makes the constraint kinematic, at any friction coefficient. That is why the throw works and the sleeper does not, and it is the same thing the published account is describing when it says the bind works by the string being "doubled over inside the string gap". A bind manufactures a dead end, and a dead end is the one thing a capstan needs in order to carry a full tension. Getting there by friction alone would take 6.97 turns of wrap to come within a thousandth of rolling, which is not a loop any more.

A correction to this page's own sweep, kept because it is instructive. The first version of that sweep asserted the capacity is strictly below 1 everywhere, and the sweep falsified it — not physically but arithmetically. Math.tanh rounds to exactly 1.0 in double precision once its argument passes about 19.06, so above μφ ≈ 38 rad the shortfall is smaller than a double can hold. At the calibrated friction that is about 35 turns. The mathematics is unchanged; what the sweep discovered is that beyond a few dozen turns of wrap the distinction between "a very tight capstan" and "a knot" is below the precision of the arithmetic, which is a fair description of the physics too.

3. The mechanism the usual account does not anticipate

Everything in §2 is about couplings proportional to tension, and rolling needs all of the tension, so they always fall short. There is one coupling that is not proportional to tension: a ball bearing's viscous drag, which is proportional to spin. The bearing drags its own outer race; the string holds that race still only while the drag torque stays below the torque needed to start lifting the yo-yo, r·m·g. Above

ω* = m·g·r / c_visc

the race turns, the string winds on, and the yo-yo climbs with nothing having caught. The engine does this, and bisecting on the spin at which its behaviour changes lands on the predicted number. So the strict form of the claim — never, however fast it spins — is false in this model.

It is also unreachable. With the bearing drag calibrated to a documented 150-second sleep, ω* is 7,962 rad/s, which is 76,029 rpm: a rim speed of 223 m/s, and 12.6 times the 631 rad/s (6,023 rpm) this model gets from the hardest throw it will accept. Nobody has ever spun a yo-yo there. The honest statement is therefore narrower than either the usual claim or the correction: a slipping yo-yo does not return at any spin a person can produce, and the barrier is a shortfall in what the contact can transmit rather than a shortfall in spin — but a coupling that scales with spin rather than with tension does have a threshold, and it is about thirteen times out of reach.

4. Gravity gives exactly enough to come back, and not a joule more

The least spin at the bottom that can still carry the yo-yo home follows from energy alone, with no integration: ωmin = √(2·m·g·L/(I + m·raxle²)). US 6,354,905 states the qualitative fact — "If the yo-yo's speed drops too low, the yo-yo will not be able to climb back up the string" — without a number; for the Abyss on a 0.90 m string the number is 213.8 rad/s, or 2,042 rpm.

Now drop the yo-yo instead of throwing it. With every dissipation switched off, it arrives at the bottom spinning at 213.799135 rad/s — the climb floor to eight significant figures, which is no coincidence: the potential energy that took it down is exactly the potential energy needed to bring it back. So a real yo-yo released from rest can never return, because every real loss — bearing drag, air, and above all the impulse of the bind itself — has to be paid out of an account that starts at exactly zero. The bind is an impulsive engagement and multiplies the spin by I/(I + m·r²), which on a size C race is a 10.4% cut on its own. The engine puts the least release speed that can still get home at 1.08 m/s for the Abyss, 0.85 m/s for the LOOP Classic and 0.60 m/s for the Imperial. Every yo-yo instruction ever written says to throw it; this is why, and how hard.

5. A tied string, and why it has to be thin

A knotted yo-yo reverses at the bottom rather than sleeping, and the reversal is impulsive: the string switches sides of the axle and the tension impulse changes both velocities at once. Conserving I·ω − s·m·r·ẋ across it gives a closed form for what survives,

ω_after/ω_before = (I − m r²) / (I + m r²)

which costs energy as the square. On the Duncan Imperial's 2.5 mm axle that is a factor of 0.9652, so 93.2% of the energy survives the turnaround. Knot the same string to a size C bearing race instead and the factor is 0.7912 — only 62.6% survives. A tied yo-yo has to have a thin axle, and the arithmetic says how thin. Nothing published makes this connection; it falls straight out of the impulse.

6. What did not work, and what is missing

• No measurement of a real yo-yo exists that this build could open. Not one. Every number it uses is a manufacturer's design spec, a record time, a textbook idealisation, or this build's own calibration. There is no published moment of inertia for any yo-yo, no friction coefficient, no spin-down curve.
• The moment of inertia here is too low, and the sign of the error is known. It is computed by integrating the published silhouette at a uniform effective density, because no mass distribution is published. A real metal yo-yo is machined hollow with an undercut rim, which puts more mass further out. So the real I is higher than the 2.314 × 10⁻⁵ kg·m² used here, which means the real sleep is longer and the real tilt slower. The radius of gyration comes out at 0.665 R, below a uniform disc's 0.707 R, where a rim-weighted yo-yo should be above it.
• The mapping from a simulated performance to a 0-10 Freestyle Evaluation score is invented. The architecture, the weights, the category names and IYYF's own descriptions of them are transcribed; which statistic each category reads is this build's, and it is not what a judge does.
• The bearing drag coefficient is a fit, not a measurement. It was chosen so a ball-bearing sleeper lasts 150 seconds, the middle of Wikipedia's "about 1-4 minutes". Every number that depends on it — including ω* in §3 — moves with it. A bearing four times freer puts ω* four times higher, and still out of reach.
• "Maxwell's wheel" has no physics article on English Wikipedia — the title redirects to Color triangle, a colour-vision topic — and none in German or Czech either. Two encyclopedias assert that a yo-yo and a Maxwell pendulum are the same physics; no dataset this build could open demonstrates it. The intended validation against measured Maxwell-wheel data did not happen.
• There is no Guinness record for the longest yo-yo sleep. Across all 29 yo-yo records — obtained by crawling 49,562 record URLs rather than guessing slugs, because that site returns HTTP 200 with a 404 body — there is no spin-duration record of any kind. The 30 min 28.3 s figure that circulates is a community record repeated by two encyclopedias, not a Guinness one, and this page does not present it as one.
• The trick economy rests on one record. The drag of a single string contact in a mount was fitted so that a maximal throw supports exactly the Guinness record of 36 gyroscopic flops on one throw. That is a single data point from a single performance by a single person, and everything about how much a trick costs in the game above rides on it.
• The string length is invented. IYYF publishes no string rule beyond "string for the exclusive use of yo-yoing". The 0.90 m used here is a guess, and the bench lets you move it because the answer moves.

7. How the instruments were checked

An instrument that never fires is not evidence, so the engine is broken on purpose, one edit at a time, and at least one instrument has to reject the result. 61 deliberate mutants were run against five instruments — the engine harness, the outward-normal proof, a Monte-Carlo mass-property oracle, a closed-form energy oracle that never integrates the way the engine does, and a second World Yo-Yo Contest judge with every rule typed again from the transcriptions. All 61 were caught, none escaped.

The interesting part is the first pass, where six escaped, and every one of the six located a hole in the tests rather than a harmless edit:

• Reversing the triangle winding escaped, because outward normals and outward faces are two different claims. Every normal still pointed out and the yo-yo simply vanished behind back-face culling. There is now a per-triangle winding test with its own reversed control.
• Negating the string tube's normals at source escaped — and it turned out not to be a mutation at all. The same expression feeds the vertex position and the vertex normal, so negating it puts the vertex on the far side of the spine with the correct outward normal for its new place: the identical tube, every ring rotated half a turn. Proved by comparing the two meshes vertex by vertex under a half-turn index shift, 24 of 24 identical to 10⁻⁹. The mutant now negates only the normal.
• Moving the string-rub radius from the rim to the axle escaped, because nothing tested the calibration target itself. A maximal throw must now support exactly the Guinness record of 36 gyroscopic flops, and a three-string mount must cost strictly more sleep than a one-string mount.
• Flipping the sign of the Coriolis term escaped, because nothing tested the pendulum. The sleeping yo-yo's period must now be 2π√(L/g), and with gravity switched off and the string paying out, d²ψ̇ must be conserved — which is exactly the term a wrong Coriolis sign breaks.
• Reaching the hand without stopping escaped, because nothing checked what happens after the yo-yo is home. It must now stop dead, satisfy the string relation exactly, and not move again.
• Mistyping the random-number generator's constant escaped, because two runs of the same generator agree even when the generator is wrong — and the judging oracle was importing the engine's copy. There are now frozen known-answer vectors, and the oracle types the generator out a second time.

Six real defects in the engine were found the same way, by an instrument rather than by inspection: a fixed-axle yo-yo charged Coulomb drag while its string was still wound and descended 49% slower than its own textbook formula; the parameter builder silently omitted the threshold that decides whether a coil counts as wound, so every direct test of it compared against undefined; the WildCard round reported a four-category Freestyle Evaluation breakdown for a round that scores none; Trick Diversity counted tricks the player dropped; the tension solver's single refinement pass left the trajectory faintly dependent on the outer step size; and arriving in the hand kept integrating for the rest of the step, breaking the string relation by seven micrometres after every return. A seventh came from the browser and from nothing else: a rolling yo-yo whose tension had fallen to zero only went slack if it was descending, so a yo-yo swung over the top of the hand and bound there stayed glued to a constraint the string was not supplying, and died instead of coming home.

8. Corrections, and whose claim each one was

• Correcting a common description of the scoring. It is often said that the World Yo-Yo Contest judges a "clicker technical-execution score plus performance-evaluation categories". That describes the WYYC14 system — Technical Execution 60 plus Technical Evaluation 20 plus Performance Evaluation 20, four sub-categories in each. It is superseded. WYYC2026 is two criteria, Technical Execution 60 and Freestyle Evaluation 40, with eight categories in a single block. Variation and Rareness are gone, Music Use split in two, Cleanliness became Control. The 2014 document is still on IYYF's own site, which is how the old system keeps being repeated.
• Correcting a circulating record. A Guinness longest-sleep record is often quoted. There isn't one.
• Correcting the usual account of the mechanism, not its conclusion. It frames the return as binding rather than spin, which this model agrees with; it does not say that the barrier is the whole tension rather than a large friction, nor that a spin-proportional coupling has a real threshold. Both are in §2 and §3.
• Correcting a published source. US Patent 59,745 (1866) is for a bandelore, not a "whirligig", and the second inventor is Charles Hettrick, not Hettrich. Its own first sentence: "Our invention is an improved construction of the toy, commonly called a bandelore, and consists in forming the same of two disks of metal, coupled together at their centers by means of a clutch and rivet." The word clutch is in the 1866 patent.
• Correcting folklore. The story that yo-yos were used as weapons in the Philippines has no historical evidence; Wikipedia records it as a marketing invention of Duncan's own demonstrators. And the rule that a yo-yo string should reach your navel is folklore: no governing body publishes a string-length rule at all.
• Correcting an assumption about the sport. Competitive yo-yo has essentially no equipment specification. "There are no special restrictions on the maker, type, parts or modifications used for the freestyle yo-yo." No diameter, no weight, no gap, no bearing, no string. The only genuinely stable numbers across IYYF, the US National Yo-Yo League and their older editions are the deduction ladder (−1 / −3 / −5 / disqualification) and the 1A-5A division definitions.

9. Sources this build could not open

Named because they bear on the result, not to pad a list. Nothing on this page rests on any of them.

• Minkin, L. & Sikes, D., "Yo-Yo Jerk Dynamics in the Vicinity of the Lowest Point", The Physics Teacher 58, 656-658 (2020), DOI 10.1119/10.0002735. By its title this is the closest published work to §5 and to the bind. The publisher returned HTTP 403. Not read.
• Krupa, E. & Tanska-Krupa, W., "Newton's second law and the physics of the yo-yo", Physics Education 32(3), 185-187 (1997), DOI 10.1088/0031-9120/32/3/019. Paywalled; only the abstract is public. Not read.
• The 1965 genericness holding. Donald F. Duncan, Inc. v. Royal Tops Manufacturing Co., 343 F.2d 655 (7th Cir., 4 May 1965) — the docket, the date, the court and the opening line were recovered from CourtListener, but the body of the opinion returned an empty response, and two mirrors returned 403. The citation is documented; the court's actual wording is not, so this page does not paraphrase the holding beyond naming the case.
• Guinness World Records' own search is JavaScript-only and returns HTTP 200 with a 404 body for a guessed slug, so guessing silently fabricates results. The 29 records used here came from its sitemaps.
• arXiv returned zero results for every yo-yo query — and also for a ti:pendulum control, then began rate-limiting. That is evidence about a rate limiter, not about the literature. No conclusion is drawn about what arXiv holds.
• No lab manual with measured Maxwell-wheel data was opened, so the intended external validation of the descent formula against somebody else's measurements did not happen.

The rules, as published

Competitive yo-yo turned out to be the rare case where the assumption holds: a real governing body exists, and it publishes a current, complete, openly readable rulebook with no paywall and no login. The International Yo-Yo Federation has run the World Yo-Yo Contest since 2013; the WYYC2026 edition is the one this app implements, and the contest was held in Himeji, Japan on 13-16 August 2026.

The one dimension IYYF does publish is the floor: "the organizer may mark a box (6 m by 4 m) on the stage" — and stepping outside it carries no penalty at all, only the risk of being out of the judges' view.

The trap. IYYF's own site still hosts the 2014 document, labelled in its sidebar "Freestyle Rule 2015". Most secondary writing describes that system. Building from it would ship a nine-year-old scoring architecture. See §8 above.

The score

Final = Technical Execution (60 max) + Freestyle Evaluation (40 max) − Major Deductions

Technical Execution is counted on two hand tally counters, one positive and one negative: "Judge only considers success, difficulty, risk and variation of each trick performed." It is the only component that is normalised between judges — and IYYF never says how it normalises. The National Yo-Yo League does publish a method, with a worked example, and this app uses it: each judge's highest-scored player is rescaled to 60 and everyone else pro-rata against that judge's own top score, then averaged across the panel. That is the United States' method, not the world body's, and this page does not claim otherwise.

Freestyle Evaluation is eight categories, each 0-10, totalling 80 and then halved to make up the 40-point share. Prelim and Semi-Final score only four of the eight. IYYF does not say what happens to the arithmetic in those rounds; four tens is already forty, so this app leaves them unhalved and flags it.

What competitive yo-yo does not specify

Where the three published systems disagree

IYYF, the US National Yo-Yo League and Japan's national body publish materially different systems, and IYYF's own mission statement still lists "unify rules and judging system" as a goal. Two examples with hard numbers. The cross-style difficulty coefficients, which answer "how much harder is 3A than 1A": IYYF gives 2A 1.28, 3A 1.35, 4A 1.19, 5A 1.27; the US gives 2A 1.40, 3A 1.50, 4A 1.30, 5A 1.60 — and they do not even agree on the ordering, IYYF ranking 3A hardest and the US ranking 5A. And the round weighting: IYYF holds Technical Execution at 60% in every scored round, while the US uses 70/30 in Prelim and Semi-Final and 60/40 only in the Final. There is no single published answer. The one genuinely stable number set across all of them is the deduction ladder.

Provenance

The five tags are DOCUMENTED, MEASURED, DERIVED, CALIBRATED and RECONSTRUCTED. Every number this app uses carries one of them, plus a sixth marker for entries documented of something adjacent. Folding those into DOCUMENTED would flatter the tally, so they are counted separately. The page harness recounts all of this from the shipped files and fails the build if the figures below have drifted.

91 entries: 35 documented · 9 documented but qualified · 19 measured · 11 derived · 3 calibrated · 14 reconstructed. 38.5% strictly documented, 9.9% qualified, 81.3% sourced or derived, 15.4% reconstructed.

What qualified means here, case by case: standard gravity is a BIPM constant rather than a yo-yo fact; every trick duration taken from a Guinness rate record is the pace of the fastest human alive rather than a typical one; the clicker banding and the two repetition rules come from the United States national rulebook rather than the world body's, because the world body publishes no per-element values; and the sleep-time bands are a tertiary encyclopedia's summary rather than a measurement.

About, and what this is not

This is an independent reimplementation, not anyone's product. No manufacturer, federation or rights holder is involved in or endorses it, and no code, art or asset from any of them is used. It is written from published documents, and the documents are named throughout.

What it replicates

• The International Yo-Yo Federation's WYYC2026 freestyle scoring: the two criteria and their 60/40 weights, the eight Freestyle Evaluation categories with IYYF's own descriptions of each, the four scored in the earlier rounds, the 0-10 band table, the Major Deduction ladder, the four round lengths, the six division definitions and the four gate conditions for "Competitive Yo-Yoing" — which are enforced by the engine and not merely printed.
• The National Yo-Yo League's clicker banding, its two repetition rules, and its normalisation method with its worked example, all marked as United States rules rather than world ones.
• Three real yo-yos at their makers' own published dimensions: the YoYoFactory Abyss and LOOP Classic, and the Duncan Imperial — for which Duncan publishes a weight, a response type and a material, and no dimension at all, so its geometry is reconstructed and labelled.
• Element names drawn only from sources this build opened: Guinness World Records titles, the NYYL rulebook and the Wikipedia article. Eight durations are Guinness rate records divided out.

What is this build's own decision

• All of the physics. Nobody publishes a yo-yo dynamics model. The clutch, the coil-radius model, the impulse treatment, the gyroscopic tilt and the trick-cost economy are this build's, derived from stated assumptions and tested against independent oracles rather than against data, because there is no data.
• The string: 0.90 m long and 0.95 mm thick, both invented, because no rule or spec publishes either.
• Three calibrated coefficients, each fitted to exactly one documented behaviour and each printed next to the target it was fitted to: the bearing drag to a 150-second sleep, the fixed-axle friction to a 15-second one, and the drag of a single string contact to the Guinness record of 36 gyroscopic flops. A fourth, the bearing's rolling resistance, is not fitted — it is held at a typical bearing value and counted as reconstructed, because calling it calibrated would credit it with a fit it never had.
• The mapping from a simulated performance to a Freestyle Evaluation score, and the rival field the normalisation runs against.
• A single-player game where the real thing is a stage performance in front of a panel.

The original, credited

The yo-yo is an ancient toy, documented since 440 BC and appearing on Ancient Greek vase paintings; the Greek word for it is unknown. It was called a bandelore in England at the end of the eighteenth century, and that is the word used in US Patent 59,745, granted to James L. Haven and Charles Hettrick on 20 November 1866 — the earliest patent this build read, and already describing "a clutch and rivet". Pedro Flores, a Filipino immigrant, opened the Yo-yo Manufacturing Company in Santa Barbara in 1928, and it is Flores's design that replaced the knot with a looped slip-string — the single modification this whole app is about. Donald F. Duncan bought the company around 1929. In 1965 the Seventh Circuit held in Donald F. Duncan, Inc. v. Royal Tops Manufacturing Co., 343 F.2d 655, that "yo-yo" had become common speech. Tom Kuhn patented the take-apart "No Jive 3-in-1" in 1979 and introduced the first successful ball-bearing yo-yo, the SB-2, in 1990. Bandai's Hyper Yo-Yo, from 1997, sold 27 million units in two years. Steve Brown developed counterweight (5A) play in 1999. The word itself probably comes from the Ilocano yóyo.

Controls

Space hold and release to throw · B bind · T tug · ← → swing · Q E correct the lean · 1-9 the first nine elements · R reset. Everything is also a button.

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