The 56-Year Argument About a Hopping Hoop

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Thank you to BetterHelp for sponsoring this video! To get 10% off your first month of therapy, go to betterhelp.com/standupmaths to sign up today.
If you are need of urgent mental health support please check for crisis help lines available in your country.
USA: www.mentalhealth.gov/get-help...
UK: www.nhs.uk/mental-health/advi...
Plus if you are in Europe you can search Mental Health Europe: www.mhe-sme.org/
Check out Ben's Hopping Hoop Geogebra file:
www.geogebra.org/m/ddsfjbvy
Thanks to Lisa Mather for making the hopping hoops for me. And an extra shout-out to Joel Tatarek-Gintowt for modelling and 3D printing the ball bearing casement.
Here are all the papers mentioned:
A Mathematicians Miscellany (1953) by John Littlewood
archive.org/details/mathemati...
The Hopping Hoop (1997) by Tadashi F. Tokieda
doi.org/10.1080/00029890.1997...
Hopping Hoops Don't Hop (1999) by James P. Butler
doi.org/10.1080/00029890.1999...
The Hopping Hoop Revisited (1999) by Timothy Pritchett
doi.org/10.1080/00029890.1999...
The rolling motion of an eccentrically loaded wheel (2000) by W.F.D. Theron
doi.org/10.1119/1.1302324
The amazing variety of motions of a loaded hoop (2007) by W.F.D. Theron and M.F. Maritz
www.sciencedirect.com/science...
The dynamics of an eccentrically loaded hoop (2010) by Andrew Taylor and Mary Fehrs
aapt.scitation.org/doi/10.111...
CORRECTIONS
- Yes, I know, 08:33 worst cycloid ever. No need to tell me.
- At 12:23 I say "no friction" when I am talking about the hoop not skidding, and yes, that should be "infinite friction".
- Let me know if you spot anything else!
Huge thanks to my Patreon supports. They keep me rolling along. / standupmaths
Filming and editing by Alex Genn-Bash
Props by Lisa Mather and Joel Tatarek-Gintowt
Written and performed by Matt Parker
GeoGebra and extra material/research by Ben Sparks
Hoop catching by Nicole Jacobus
Dog wrangling by Lucie Green
Hoop chasing by Skylab the Dog
Music by Howard Carter
Design by Simon Wright and Adam Robinson
MATT PARKER: Stand-up Mathematician
Website: standupmaths.com/
US book: www.penguinrandomhouse.com/bo...
UK book: mathsgear.co.uk/collections/b...

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26 фев 2023

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Комментарии : 1,8 тыс.

@hamishmcfarlane9931 Год назад

for a stand-up mathematician, he sure sits down a lot

@doogierusty Год назад

He IS standing up:)

@treesap2 Год назад

But he could just be a stand-up guy who is also a mathematician.

@jackdog06 Год назад

The number of times you stand up and the number of times you so down are always the same each other

@treesap2 Год назад

@@jackdog06 not true. Sometimes you lay down or fall down and have to stand up.

@Faroshkas Год назад

@@jackdog06 not true. If you haven't stood up yet, it will be different.

@MikuJess Год назад

My favorite part is that the only paper that addresses the "is it intuitive" part of the question proves it by existence. "We've written a paper refuting another paper about this problem, we're WAY beyond intuitive."

@gazeboist4535 Год назад

Truly, one of the most convincing proofs I have ever seen.

@JoQeZzZ Год назад

And ironically their proof was my immediate intuition. While it rolls, it's connected to the ground, while it hops it's not: so at the boundary between the two behaviours it needs to disconnect. Now it won't move straight up because that's not how gravity works and it won't move straight down because that's not how ground works. Therefore it has to slip, which is not how "rough" usually works. That's a contradiction so it don't hop.

@bertram-raven Год назад

@@JoQeZzZ But it did hop. Your supposition ignores the "massless" part of the original thesis. To me this proves empirical research is a flawed concept as it requires interpretation using a common frame of reference between the known and unknown. This is impossible. In other words, the hopping behaviour of the hoop is intuitive precisely because it is not intuitive.

@JoQeZzZ Год назад

@@bertram-raven it hopped when he threw it, it didn't hop when he dropped it from an unstable equilibrium, which was the question posed. Also, any physical object isn't infinitely rough, and it having to slip means it won't because this perfect mathmatical object can't You can't solve mathmatical puzzles by empirical research, it's a math puzzle.

@aukora129 5 месяцев назад

@@bertram-raven It hopping when he did it means nothing, the problem posed requires a supposed massless hoop with infinite friction on its contact point. Whether or not it is massless, there is infinite friction by definition of the question, which means it cannot slip and therefore cannot hop.

@GoodishEats Год назад

15:40 "a little joke there for all the joke fans" is such a fantastic line

@vigilantcosmicpenguin8721 Год назад

It was a good joke, if you will.

@PsiVolt Год назад

absolutely stealing that one

@yoavsigler4457 Год назад

I love that whenever Matt makes a silly joke he unapologetically pauses to acknowledge it.

@bertram-raven Год назад

In my mind a silent drum is playing "Ba doom, boom" every time he does.

@jamiewarren2307 Год назад

Matt, I must say that you missed a golden opportunity to say, "Out with the old and in with the μ"

@davidanoble Год назад

I haven't even watched the video yet but I can tell this would have been a top-notch Matt Parker joke 😂

@whiskeytuesday 2 месяца назад

What's old is μ again

@Theexplorographer Год назад

On today's Stand Up Maths, Matt explains why we balance the tires on our cars.

@yarati4584 Год назад

Now I want a car that hops.

@Theexplorographer Год назад

@@yarati4584 Just search RU-vid...lots out there.

@alextopfer1068 Год назад

@@yarati4584 just unbalance the tyres enough and give it enough angular velocity (not legal advice, might void warranty).

@AcidDotCom Год назад

@@yarati4584 Mythbusters made a car where the axle was not in the middle of the wheel...not very comfy ;)

@ericchambers9023 Год назад

Just visit Ohio. With no annual inspections required, there's some pretty bad vehicles on the road. I've seen pickup trucks with a rear tire so out of balance, the tire left the ground every rotation........Just another day in Ohio.

@Chris-cb9tx Год назад

I'm glad there are people out there willing to go the distance and do the science on these important subjects.

@PauloJRFonte Год назад

Maybe not important, but it warmed my physicist's soul.

@SilverLining1 Год назад

Not important? Ever heard the phrase "Reinventing the wheel"? This is a very important question as it pertains to unbalanced wheels. Since no wheel is perfect, literally every wheel meets this criteria, up to imperfections in the circumference. We're talking about one of the most primitive types of motion here, so even if nonstandard rolling only occurs in extreme cases, some engineer is bound to run into these scenarios in the future.

@Yora21 Год назад

I often wonder how much time and effort goes into making these papers? Are these things that get spontaneously created in one afternoon by three people hanging out?

@PauloJRFonte Год назад

@@SilverLining1 I thought you were being ironical.

@PauloJRFonte Год назад

@@Yora21 In my field a paper can represent several man-months of work. All easy things were done long ago.

@osmium6832 Год назад

I think it's funny how the generated subtitles for the episode categorically refused to type the word "yeet" and instead say "eated" or "heating". Edit: Oh wait! At 22:20 the subtitles *will* say Yeet as long as it's a proper noun as part of the phrase "Yeet Theory". It still won't type lowercase yeet if it's used as a verb.

@AkiSan0 Год назад

i think the "if it is yeeted fast enough, it will hop" is a pretty distinct way of phrasing the solution to this problem and should be written down on stone tablets, for future generations to decipher.

@emceeboogieboots1608 Год назад

Unfortunately the kids will now be going with "angular momentoring" it seems, and yeet will fall into disuse as a word that never lived up to its full potential 😕 Like wamblecropt or ultracrepidarian

@Wtfinc Год назад

yeah it bugged me when he said at the end "we dont know" but we do its just not white and black like people like,its all grey. i love gray. people are so blinded by binary options. so no, hopping hoops dont hop unless yeeted and they will skid with no friction.

@unclebrat Год назад

Perhaps it can be constructed as "Parker's Conjecture."

@Jkirek_ Год назад

The absolute best part of this is that all parties involved at some point spent time rolling weighted hoops around

@hens0w Год назад

There are lots of scientist that don't science (lots as is too many, I don't want to guess the proportion) so its very possible that one of the authors never tried the physical experiment and just did the math(s) and thought that counted (Cliff Stoll's numberphile podcast appearance shows him giving the correct view of these people, I got the felling he thinks its more common than I do)

@anonym3017 Год назад

The people that disagreed clearly didn’t. Otherwise they would have immediately seen that they are wrong. Just unbalance a wheel on your car and take it on the freeway and it becomes evident.

@akaraven66 Год назад

I'm sorry the best part of this video was the dog.

@karlharvymarx2650 Год назад

@@anonym3017 Maybe, but IMO the problem is stated so poorly a variety of interpretations may be possible leading to experimentally testing different interpretations leading to different results that might contradict one another despite being valid as the experimenters restated the question. For example, as I interpret the question, an unbalanced car wheel is a different problem because the imbalance isn't what moves the car, the engine does. The car and wheel have a huge mass compared to the imbalanced mass. The wheel is attached to a spring and damper system. You are likely thinking about what happens when the wheel is moving much faster than it would if driven by gravity pulling the mass from the top position on the hoop to the bottom. And Etc. At the moment I can't quite put my finger on what strikes me as stupid about the question from the book but something about it bothers me in ways different than, say, asking the student to ignore air resistance as is common in physics problems. Maybe it is that mass, and things that go with it like inertia and weight, and friction of the hoop are assigned impossible values and are essential to know approximately what would happen. Or maybe I just got up on the wrong side of the bed this morning.

@anonym3017 Год назад

@@karlharvymarx2650 might be the near weightless hoop as that makes everything quite a lot harder to imagine. Also I'm pretty sure that both are the same phenomenon just with a very different manifestation due to a vast difference in driving forces. After all the thing that's causing it in both cases is the center of mass not being on the axis of rotation.

@valdemar91 Год назад

I cant believe that after all these years, Matt still manage to trick me into learning about maths, while I wait for the dad humor jut show up. Well played, sir. Well played.

@philvogelfilms Год назад

This is exactly the kind of wholesome dog-chasing-hoop content I come to RU-vid for

@jillslack9013 Год назад

"If you yeet something it shall hop" Will forever be known as the Matt Parker Law.

@brune02 Год назад

I love how the auto-subtitles can't quite get the word "yeet"

@tfkroks16 Год назад

I need this on a t-shirt. As well as "I contradict references"

@kumoyuki Год назад

The word "yeet" is inherently funny. Seriously, trying saying it out loud. The Parker Law of Hoops needs no further wordsmithing. Proof: by inspection ;)

@Corwin256 Год назад

What if it works? Can something be named after Parker if it works?

@user-zc1vi8tx7b Год назад

@@brune02 Eat them

@NeatNit Год назад

Important: I tried Better Help the first time Matt was sponsored by them. They *do not* have therapists worldwide, only in USA.

@standupmaths Год назад

I’ll ask and report back!

@mayube9292 Год назад

*some* of their therapists are certified in other countries, but afaik they only verify that they're certified in the US, so if you're not in the US you just have to keep asking therapists if they're licensed to practice in your country until you find one that is (provided there is one that is)

@beardiemom Год назад

@@standupmaths Also, *please* look into the controversies surrounding BetterHelp and them matching up patients with therapists that were not educated on their mental health struggles as well as underpaying therapists and loading them up with too many patients. BetterHelp exploits therapists and puts vulnerable people seeking safe therapy at risk. They are not a service worth supporting.

@patfre Год назад

They’ve also just been caught selling your data to companies so yeah don’t trust them

@JacksonBockus Год назад

They’re also a scam.

@nikitaelizarov7444 Год назад

The dog is having fun despite having little to no conception of maths. That's the way to go.

@jfbeam Год назад

Yes, but does the dog hop when rolled?

@klausstock8020 Год назад

Mathematician: goes to great length to figure out the trajectory of abody with rotational symmetry and an aerofoil cross section when travelling through the air, spends month in research. Dog: does that the same in a few split seconds, and also calculates a parabolic intercept course to catch the frisbee mid-air in real time. I guess dogs are pretty good at math.

@k0pstl939 Год назад

We love skylab!

@hhjones9393 Год назад

I really like that Littlewood covered it all 70 years ago. That last line was a doozy! He knew, he was sandbagging to give the next generation something to do.

@blaise3004 Год назад

What a fascinating niche subject, the phase diagrams were absolutely lovely!

@standupmaths Год назад

I’m going through a real phase diagram phase.

@PhilBoswell Год назад

@@standupmaths I'd like to see some where they varied the mass ratio: the weight only being 3× that of the wheel seems unreasonably light to me, I'd want to see what happens with heavier weights.

@joelstuder8543 Год назад

@@PhilBoswell Yes, it would be interesting to see a phase diagram for when the mass ratio and coefficient of friction are both very high values (which most closely models the original thought experiment)

@Reth_Hard Год назад

My answer: It depends. I give permission to anyone who want to use this to write a scientific paper...

@simonmultiverse6349 Год назад

Roll it downhill. The gravity vector will stay the same .... aaaaand..... the upsy-downsy mass-acceleration of the weight will get stronger and stronger and, eventually, it will get very jumpy indeed! :)

@thechumpsbeendumped.7797 Год назад

I’ve lived this very situation. I was in a Zorb with my 8 year old niece and as we picked up speed we completely left the ground multiple times rolling down a smooth hill. It was a hell of a ride and at the time I put it down to me being 4-5 times the mass of my niece and on further trips down the hill I avoided partnering with someone of such a different size and the rides were far smoother with no airtime at all.

@HeavyMetalMouse Год назад

I love how this comment casually drops the word 'Zorb', as though daring me to go look up what such a thing is. :)

@danielrowson3379 Год назад

@@HeavyMetalMouse I am shocked that someone doesn’t know what a zorb or ‘zorbing’ is. But basically it’s a human sized hamster ball. You get inside and roll down a hill. Very fun.

@vaclavtrpisovsky Год назад

Be careful, hopping while zorbing may lead to leaving the track. Lives have been lost to bouncing zorbs.

@thechumpsbeendumped.7797 Год назад

@@vaclavtrpisovsky I did warn the organisers of this problem at the time. Fortunately, this was a purpose-built facility with retaining banks, fences, troughs/grooves in the hill to guide the direction of the Zorb and a level run-out area at the end to stop them, so plummeting to our deaths (like in the famous ski slope vid) was impossible. What I was concerned about was the risk of whiplash due to the repeated impact with the ground on every revolution.

@joejoejoejoejoejoe4391 Год назад

Every science experiment needs an 8 year old in peril. My nieces are 11.

@alexny1173 Год назад

Really love this format. A kind of video literature review or journal club. Everything explained really clearly. Great job!

@Blake-jl8lh Год назад

I discovered this late night when I was in my undergrad. I had a jar of peanut butter I'd snack on and it had fallen on its side. The peanut butter had settled to one side and I rolled it aggressively across my desk and I was delighted to see it jump

@fleurbird Год назад

I was hooping you would make a new video

@bl4cksp1d3r Год назад

Ironically, I was hopping he would do so

@bbbb98765 Год назад

Hoopefully that's the last pun

@Elitekross Год назад

For the joke fans

@jayfredrickson8632 Год назад

He had a question and decided to roll with it

@GalileoCap Год назад

Does increasing the mass ratio (so that the mass represents more of the total mass) cause the hoop to hop at lower initial speeds? Because then, as it reaches 100% it should approximate the original idea of the hoop being weightless.

@Matyanson Год назад

Yes, exactly what I extecte to be in the conclusion. Behaviour as the ratio approaches the limit

@JBLewis Год назад

This is much clearer formation of the question I had!

@rantingrodent416 Год назад

I had this same thought, but maybe the math makes this uninteresting because the acceleration due to gravity is independent of mass?

@louisvictor3473 Год назад

@@rantingrodent416 Acceleration, but not the forces. If you're on the moon, a feather and a bowling ball dropped on you from the same heght wll hit you exactly at the same time, but only one of them wll kill you.

@tristanridley1601 Год назад

I believe even as the ratio approaches 100% it will always require some initial angular momentum. I'd like to see those phase diagrams though.

@tequitofoot Год назад

I love those phase diagrams-- what a satisfying and clever way to display that info. Kudos to the authors. Great video!

@easye9186 Год назад

Glad you are encouraging mental health support. Love your videos and books

@renerpho Год назад

Littlewood's "A Mathematician's Miscellany" is a little known treasure trove. The mathematical problems are brilliant (as your example shows), and as are the anecdotes and jokes he tells. "I once challenged Hardy to find a misprint on a certain page of a joint paper: he failed. It was in his own name." "A good, though non-mathematical, example is the child writing with its left hand 'because God the Father does'. (He has to; the Son is sitting on the other one.)"

@eekee6034 Год назад

That misprint quip is great! XD

@renerpho Год назад

@@eekee6034 Yes, he drops quite a few of those. The title of the book itself is a joke (referencing Hardy's 1940 book "A Mathematician's Apology").

@proloycodes Год назад

@@renerpho i knew it had to be a reference

@vigilantcosmicpenguin8721 Год назад

That's what math is really about.

@secondengineer9814 Год назад

This should be added to the list of fun intro to Physics experiments. Imagine the molasses filled cylinder, the hopping hoop cylinder, and a standard cylinder, all being demonstrated. Oodles of fun

@muninrob Год назад

Here in Alaska, it's an "oil filled cylinder" using heavy gear oil - honey & molasses cylinders hop irregularly instead of rolling slower than expected

@iPsychlops Год назад

I just wanna say as a therapist who works in a university counseling center, I really appreciate that you took the time to point out that students might get free mental health care at their school.

@wafkt Год назад

Indeed. I took advantage of therapy from my school as an undergraduate student, and now as a professor at that same school, I strongly encourage my students to seek help from our counselling services and not (as too many do) suffer in silence when life gets tough. I share my personal experience with my students in an effort to dispel any stigma associated with meeting with a therapist. The wonderful therapists at my school have helped several of my students who were on the brink of dropping out because being a student had become so challenging and the stress of it all made it difficult for them to see that there was hope.

@Verrisin Год назад

I agree. People who worry about hoops hopping need help with overthinking.

@iPsychlops Год назад

@@Verrisin lol probably, but there are lots of other reasons.

@iPsychlops Год назад

@@wafkt thank you for helping to reduce the stigma. I love when students say a professor suggested they come here.

@kmn1794 Год назад

@@Verrisin I deadpan disagree. People who worry about other peoples overthinking about hopping hoops need help overthinking their own underthinking.

@DeathbyPixels Год назад

This kind of thing is like real life's equivalent of wizards arguing over the particulars of a certain spell.

@NoriMori1992 10 месяцев назад

D&D rule arguments are the best arguments! 😁

@D4M14N1989 Год назад

“If you yeet something it shall hop” - words to live by.

@storyspren Год назад

Skimming sounds like "it hops, but by so little you can't properly see it"

@IceBug1337 Год назад

Engineers wonder why this 56 year argument wasnt a 56 minute argument.

@marianaldenhoevel7240 Год назад

That's because only math can ever be entirely sure. Even if the real world, engineering, physics and, god help us, common sense all violently disagree. And no matter how useless the answers are: Math will have the final word.

@ModernEphemera Год назад

Would take quite an engineer to build a massless hoop

@robingrimm3443 Год назад

@@ModernEphemerayeah it honestly feels like the proofs so far are ‘we can’t recreate it irl’ but they also haven’t created the conditions in the first place

@Cryous Год назад

@@robingrimm3443 a relatively intuitive way to think about this is that a massless hoop is merely a geometric constraint. I think a lot of people assume that angular momentum is conserved when the normal force between the hoop and ground reaches zero, thus the hoop should rotate about the point mass into the air. However, a point mass with it’s rotational axis through itself has no inertia and thus cannot have angular momentum. This, if it occurs, results in the skimming effect previously described where the inertialess hoop rotates about the point mass as the point mass follows the parabolic path. It’s pretty cool and I think relatively intuitive

@JoQeZzZ Год назад

@@marianaldenhoevel7240what do you think physics is if not applied math? Physicists know their math and love a "massless hoop with infinite friction"

@THVEssays Год назад

"If you yeet something, it shall hop" I actually laughed out loud at that.

@irgyn Год назад

"Language!" got me. Also, as a material scientist, I appreciate you bringing phase diagrams unto the people.

@sachiperez Год назад

It would have also been interesting to see what happens when the weights are on a larger circumference than the traction part of the ring. Think of a monorail type set up where the outside of the wheel carries the weight but dips below the track.

@Mnaughten601 Год назад

I think that is a good thing to test, however one of the papers specifically says the wheel can’t go below the floor(or something like that).

@drdca8263 Год назад

@@Mnaughten601 That was just one of the assumptions they were using though? OP is suggesting a variation on the setup, which would therefore have/consist-of different assumptions .

@Mnaughten601 Год назад

@@drdca8263 I agree with that, I was just implying that it would be a different problem allowing that assumption. But nonetheless an interesting experiment.

@johanullen Год назад

I agree. My first thought was that the parable is so close to the circloid that if the centre of mass is inside the circle, it won't hop. Only if the hoop has additional higher angular momentum or the centre of mass is so close to (or outside) the edge of the circle that the parable is ahead of the circle and the circloid will the hoop hop.

@drdca8263 Год назад

@@johanullen parable?

@spacedoubt15 Год назад

The hoop hops naturally (ie without skidding) not on the way down but on the way back up - when the energy transfers away from the ground and pulls the hoop up with it. You can see that's what's happening in most of your slow-mo examples too.

@kasmirperriman9360 Год назад

I wish i could subscribe more than once. I love everything you do matt! You make maths so fun (and your theme is epic)

@addymant Год назад

If you have health insurance in the US, the ACA requires many plans to offer mental health and substance abuse, potentially with no co-pays or deductibles. Look into that before paying an expensive middle-man mired in controversy.

@michaelbauers8800 Год назад

some employers also offer employee assistance programs.

@JonathanMandrake Год назад

I actually knew they could hop because as a kid, I had a ball with a heavy weight in one part to make it more interesting

@Fidtz Год назад

Impossiball. I loved that until I decided I wanted to modify the weight and destroyed it

@krautbrain Год назад

And that is why one of your balls hangs a little lower then the other.

@PhilBagels Год назад

Yes, a ball should do as well or better.

@CraigClarkson Год назад

This makes me think of a similar phenomenon in air instead of rolling on the ground, oh and a sphere instead of a hoop. The spitball pitch in baseball.

@heighRick Год назад

Matt, an excellent video. Thanks so much, helps a lot!

@jamescomstock7299 Год назад

Such a fun topic and review of the papers! Thanks, Matt!

@michaelsommers2356 Год назад

1953 was 56 years ago? Thanks for chopping a dozen years off my age. I feel younger already.

@peterfireflylund Год назад

1953-2010? (Look in the description)

@standupmaths Год назад

That last paper was 2009 bug humbug.

@satyris410 Год назад

I'll tell my dad, he'll be delighted

@michaeldunkerton3805 Год назад

You can tell people that's your Parker age, anyway

@wazoheat Год назад

@@standupmaths If we argue about it in the comments then the argument is still ongoing, right?

@Jimorian Год назад

Adding a pupper chasing the yeeted hoop is no cap a poggers strat to catch Steve.

@wiseSYW Год назад

4head kekw

@SlenderSmurf Год назад

This would be enjoyed by those who enjoy jokes 5-10 years ago

@kieran8266 Год назад

Hm. Thumbs down!

@nbooth Год назад

Sadly, Google translate doesn't support this language.

@jrrarglblarg9241 Год назад

@@SlenderSmurf yeet

@animarain Год назад

Your videos make my heart smile!!

@artificercreator Год назад

Interesting stuff! thanks for the rad content!

@TonyHammitt Год назад

I get it now, in the massless hoop with infinite friction, you still need contact to use the infinite friction to get the hoop to hop. But as soon as it tries to start to hop, it loses the friction and it "skims" a bit until it re-contacts the floor.

@gormster Год назад

I think that’s what’s going on with the physical modelling papers, but I think the original argument of the 2001 paper is more like, at the point where the weight suddenly switches from rolling to ballistic trajectory, there’s a discontinuity of angular acceleration, which you don’t get in reality. There has to be some non-zero transition period, and at the point where the cycloid and the parabola meet, their second derivatives are not equal. I *think* that’s what they were saying, at least. I could be wrong, I’m just trying to glean this from the video.

@friedrich8322 Год назад

It seems like the problem is that a "zero mass hoop" and "zero friction" is not really physical, else how would the hamiltonian of that system look like? It makes more sense to look at physical hoops and to take limits afterwards. The free parameters seem to be the ratio of point mass to hoop-mass and friction. If you let both of these quantities tend to infinity (either independently or along some limited class of paths in R^2), one could look at the limiting behaviour of it hopping or not hopping. If it hops for all such limits, then I would say that it makes sense that the zero-mass infinite-friction hoop hops. otherwise the question does not seem well defined (and could be well defined by asking for what types of sequences of frictions and mass ratios there is a limiting hop-behavior and how it looks like). Do you know if there is a paper that examines this Matt? EDIT: also "proof: by inspection" is absolutely hilarious

@EebstertheGreat Год назад

It's not actually "zero friction" but rather zero rolling resistance. Friction is required to make the hoop roll. The zero rolling resistance assumes that the hoop always rolls without slipping and that deformation of the hoop is negligible. The latter is true for the toothed wheel design but not for the tire. The former is never true in this experiment, because if the mass ratio is large enough, that turns out to be physically impossible--you must get some slipping (or "skidding" or "skimming"), or else the wheel or ground must deform. In practice, if the wheel is stiff and the ground is strong, then you get some kind of slipping.

@kanucks9 Год назад

@@EebstertheGreat Aha! And if you use a rack and pinion instead of smooth surfaces, the 'skimming' becomes a jump, because of the angle of contact. Any real roughness should produce the same effect. The real unstable equilibrium in this experiment is the initial conditions preventing a hop 😆

@OhOkayThenLazySusan Год назад

Yes! The initial question is poorly defined. 🙏

@drdca8263 Год назад

@@EebstertheGreat I’m a bit confused as to how it could be ill-defined. It seems we should be able to set up differential equations or inequalities for all the different cases.. uh, So, relating the angular velocity to the horizontal velocity conditional on the bottom of the hoop being on the ground and conditional on there being a non-zero vertical component of the force between the hoop and the weight, Relating the velocity of the weight to the angular velocity of the hoop along with the velocity of the hoop, Acceleration of the weight determined by the force of gravity on the weight and the force from the hoop, This force being what it has to be in order to keep the other things true, ... err... I guess if the hoop is not currently touching the ground, the angular velocity of the hoop is no longer specified... Ok yeah I guess I can see now how you would have to do this by taking limits... Oh, hey, Do I recognize your username from the old XKCD forum?

@EebstertheGreat Год назад

@@drdca8263 In the simplest analysis (Littlewood's), the floor actually imparts a negative normal force on the wheel. A more careful analysis shows that his simplification requires incompatible conditions: Newton's laws, an impenetrable floor, and a no-slip condition with 0 normal force. The problem happens at the interface between the rolling stage, where the no-slip condition is imposed, and the hopping stage, when the normal force instantly becomes zero. And yeah, I used to post there sometimes.

@inkyskink Год назад

Omg I can't believe Matt actually said "Contradicts Reference Three" out loud in his video! I hope he can keep his account after something like that!

@rarapas Год назад

What a lovely fun video! Good work Matt!

@CodeWookiee Год назад

Great video, Matt! I'd love to see this done with a single round weight embedded in the edge of a circular disc of Aerogel. That would probably provide the closest experimental adaptation of the thought experiment that I think we could manage at this time. Add some marks round the edge, meet up with Smarter Every Day or Slow Mo Guys and film it with a Phantom, and we might see the skidding / skimming in action. :)

@labibbidabibbadum Год назад

Polystyrene would probably do the trick, and be a few orders of magnitude cheaper :)

@couldntcareless7884 Год назад

My intuitive understanding is that it depends on speed with which it’s launched. Because the weight will want to go in a parabola, but the hoop will try to keep it on a cycloid. If the parabola goes over the cycloid (the starting speed is big enough) it will hop, otherwise no. Unless the material is spring-y enough, then it may still hop, though for a different reason.

@YatriTrivedi Год назад

Great way to illustrate how published papers are part of a larger conversation and how they cite, support, and refute each others' claims.

@jamesahibbard Год назад

Thanks for sharing your needs for therapy as a student. Unfortunately, as someone from the east coast, I can't help but say that I'd probably need therapy if I had to live in Perth too.

@TopchetoEU Год назад

yet again you havent failed us with utrerly useless but fascinating maths. keep bringing us this amazing content matt!

@thechumpsbeendumped.7797 Год назад

Utterly

@leonsteffens7015 Год назад

@@thechumpsbeendumped.7797 butterly

@1224chrisng Год назад

if he failed us, then it'd be a Parker Video

@simonmultiverse6349 Год назад

Going downhill, now... it will accelerate and the upsy-downsy tendencies will eventually overwhelm any smooth rolling movement.

@acrossthevioletsky Год назад

Not useless, these maths may be useful for machines that use unbalanced wheels

@migfrarummet1907 Год назад

To get the mass centered on the rolling edge, you could put the extra weight outside the perimeter and roll it like a train wheel, unless I'm missing something.

@artswri Год назад

Great fun, really enjoyed, thanks!!

@graemev6799 Год назад

Good on you Matt, you are a good human!

@Smona Год назад

6:10 correction: yote

@jhonbus Год назад

The paper "Hopping Hoops Don't Hop" should clearly just have been called "Hopping Hoops Don't".

@terencetsang9518 Год назад

Hoppin't Hoops

@DanielHallmark Год назад

"They hopped along in exactly the same way hoops don't." - nod to Douglas Adams

@AlwinKristen Год назад

This made my day! :D Thanks!

@Hailfire08 Год назад

6:05 I agree; as it spins, it rotates around its center of mass, and, assuming it's spinning arbitrarily quickly and can't phase through the floor, in the limit of fast rotation it'll touch the ground at a single point, unless the center of mass is the center of the disk (i.e. it's balanced)

@ChristopherLien Год назад

I think your "diagram" with the cycloid and the parabola can help with a more intuitive explanation for why you need more "yeet" to get a hop. The parabola followed by a mass widens with horizontal velocity; which makes it more likely that it will cross the cycloid. I do think the coefficient of friction is important though; if the wheel has no mass, it can't counteract the horizontal velocity of the weight except through aerodynamic drag (which is probably negligible in this case), or through the friction with the table. If the friction is infinite, then the mass of the table is effectively pulling back on the weight until it reaches the point where it's horizontal velocity is zero, and its corresponding parabola can't be outside the cycloid. If that friction can be broken even for a moment, the momentum of the weight can break free from the mass of the table, and follow the parabola instead of the cycloid.

@dielaughing73 Год назад

Yes, a thought experiment relying on a no-slip motion of a massless body is somewhat meaningless since it can't be replicated in reality, nor does it follow the laws of motion in any intuitive manner. I'm satisfied with Matt's demonstration of Yeet Theory, as corroborated by the phase diagrams and the LED paper, as proof that hoops can hop under certain real-world conditions. I suppose a simulation using a decent physics engine could answer the original question with its assumptions intact.

@jacobtinkle9686 Год назад

"If you yeet something it shall hop" Beautiful wording, just magnificent Matt.

@StrengthOfADragon13 9 месяцев назад

So as I understand my intuition on the subject is that if you let it roll under gravity it won't ever hop. But if you have it rolling faster than the path the weight would take under gravity then it will hop. I love how your videos always make me think more than I would have expected

@tyruslawhorn Год назад

Of all the videos that could possibly send me into an existential crisis, this was the one to do it... It basically boils down to, "Does a hypothetical object placed in an impractical scenario behave intuitively?" On one hand I start thinking about the resources spent amongst everyone who has tried to provide an answer to something so arbitrary. Whether those resources be money, raw materials, someone's time... Those things all have value, some more than others. And all of them were just basically wasted... Conversely, hopping hoops and, for that matter, non-hopping hoops don't have any value whatsoever... There is no scenario in which the results will ever matter. To anyone... Ever... But then on the other hand, if I was still entertained the entire time, does that mean..? Keep up the great work, Matt

@smorcrux426 Год назад

19:06 that's the floor where my father's office is! And given that fine hall is pretty small and I'd assume there's only one hallway in the floor then I probably walked in the exact place where that hoop was rolled

@Darxide23 Год назад

It seems to work pretty much intuitive to me. Although I wasn't quite right as to why it hops, but I still figured it would if you gave it a push. My reckoning was that if you just let it go and let gravity take over, by the time the mass reaches 270 degrees from the top (or pointing backwards toward the origin point of the hoop) then there wouldn't be enough energy left to lift the hoop off the ground. I assumed that the angular momentum of the weight lifted the hoop upwards at that point which seems to be what happens if you watch the clips where Matt gives them a yeet (I want that word in a mathematical paper now). They seem to hop upward as the mass lifts upwards since giving them a yeet imparts more energy than the acceleration due to gravity and for a moment the mass is able to counteract the force of gravity. Thus a hop. That was how it went in my head and the clips of Matt rolling hoops in the hall seem to verify that to me, at least on a casual viewing. I didn't take into account any friction or slipping or skidding in my mental model, though.

@isaackvasager9957 Год назад

This is exactly what I see going on too. haha. It's so weird that these people spend all this time writing papers when the actual reason seems pretty obvious. It's why it only "hops" when you really "yeet" it. You need that intertia when the weighted point is coming back UP and around to get it to hop. It has nothing to do with the weighed part "falling" from horizontal as the math people seem to be talking about.

@lexinwonderland5741 Год назад

I really really hope you go overboard on the second channel with the phase diagrams and various factors in the model to be tweaked!!! especially if you can add the cycloid/parabola time plots back in as examples at some point

@GetMoGaming Год назад

LOVED the happy dog running in, lead trailing behind 🤣😆 (@3:33)

@peanutbuttereggdirt1 Год назад

The dog joining in with the hopping was the highlight.

@phildman132 Год назад

Those physics students definitely had fun on this one. I love these sorts of ideas!

@tofystedeth Год назад

Weirdly my favorite part of the video was the Geogebra animation because when the parabola was added it looked like one of giant bowlegged robots from the cover of Yoshimi Battles the Pink Robots stepping over the cycloid. I rewound that part several times while chuckling.

@bethmmq3840 Год назад

Thanks Matt for this interesting crossover between maths and physics. I'm just surprised you didn't mention another, much more popular "Little" mathematician: Little Richard, who hopped on that wagon very early on, in 1956, with his famous paper "Slippin' and Slidin'".

@PilotsWorld Год назад

This is the kind of video I can roll with.

@43201Spaceghost Год назад

Many papers like to change the parameters of the original thought experiment. It would certainly hop given the original parameters. If you rolled the wheel along a sensitive scale, you should see a weight difference indicating that the hoop is attempting to revolve around the point mass (or in reality around a barycenter). On a side note I really like the phase diagrams.

@tomfeng5645 Год назад

Copying a Tony in the comments since I don't see how to put it better, "in the massless hoop with infinite friction, you still need contact to use the infinite friction to get the hoop to hop. But as soon as it tries to start to hop, it loses the friction and it "skims" a bit until it re-contacts the floor." If you consider a "zero height" hop still a hop I guess. I suppose that means the real issue is, what exactly counts as a hop anyways?

@shanewilson3653 Год назад

In practice the point mass is not a point but a area and can exert a rotational energy due to its own internal leverage with enough energy to accelerate the remaining light disk at a rate that is faster then gravity accelerating it downward. This seem intuitive in the practical instance and would be expected to be a lot more intense as the rotational speed of heavier mass increases. Also as mentioned above the additional input from rotation around the barycenter.

@shoto5892 Год назад

given the original parameteers of the thought expierment, there is no possibility for it hopping since hopping would require a friction smaller than infinity, however any other action like skidding would require this as well, so there is no way for this to work, at least not with the known actions a hoop can do.

@vibaj16 Год назад

that doesn't necessarily mean it would hop

@DCSWCCkingpin1 Год назад

RU-vid needs to add a LOVE button for videos like this

@_wetmath_ Год назад

never thought I would be so excited to see a circular disk jump

@RomanNumural9 Год назад

It would be interesting to roll it along a scale and see how the down force changes. If the force decreases then you know it could hop if it was a massless hoop. The footage could be used to identify skidding.

@krautbrain Год назад

@Standupmaths. Why are the controversies about BetterHelp being shadowbanned on your channel? Is this your work? Hope not.

@YLLPal Год назад

I really hope that conjecture is original to you. I totally agree with it just from the centre of mass being outside of the centre of rotation... I'm typing this while watching, and seeing you read my thoughts live.

@johnathansaegal3156 Год назад

06:20 ... decades ago when I was in high school I worked at a tire shop (when it was legal for minors 16 & 17 years old to work with heavy equipment and vehicles), and one reason people had problems with their shocks and struts were unbalanced tires. The wear on the tires as well as the additional wear to the suspension was due to the unbalanced tires "hopping" as they drove down the freeway at high speed. Even when affixed to an axle, due to the velocity and angular momentum of the unbalanced tire, the suspension was not strong enough to overcome the "hopping" of the tire which led to uneven wear on the rubber tire. This phenomenon was well understood even by people who worked in the shop who were high school dropouts. It just made sense to those of us who dealt with such tires everyday. A properly balance tire (hoop) led to even wear because the tire was always in contact with the road surface. Introduce "hopping" and you get wear that is uneven and easy to spot for the tire professional as to which side was weighted more than the other. Until now, even after taking and acing physics in high school and college, this is the first time I ever heard of this topic as being an actual question in the mathematical community. From the experience of my few years of working tires and suspension I saw the results of this common aspect of unbalanced tires. It is why, when you have unbalanced tires, your steering wheel vibrates and the ride feels rough and uneven, especially at higher speeds. This was simply common sense to us in the shop.

@johnathansaegal3156 Год назад

I am sure there is a formula to calculate at what speed (velocity) the hop-factor begins and what speed it stays on the table/ground. I would also consider the force of gravity at the elevation the experiment is performed.

@ohiojosh78 Год назад

I just started the video but my initial thought is that the hoop would hop but in the opposite orientation from what you described. I feel as though the momentum of the weight coming up on the back side of the rotation would cause it to lift. We'll see.

@EvDelen Год назад

More footage of your dog chasing the hoop please!

@lwm-laughwithmemes2006 Год назад

Yes

@XavierXonora Год назад

It would be cool to roll/propel one of the physical models on a scale to see how much force the hoop puts on the ground at various points of the roll

@richardforster1239 Год назад

The 1991 Further Maths Mechanics A-Level exam (sorry, can't remember which exam board) had a very similar question to this as the final question on the paper. I don't think there was any discussion of hopping but it was all about the motion of a hoop with a point mass. It was worth the most points and nearly beat me. I remember giving myself "just 10 more minutes" to make more progress before I would stop and go back to check what I had done on the other questions. But I did make some progress and then when stuck again I gave myself "just 10 more minutes" yet again. I think I gave myself that ultimatum 3 times before finally finishing.

@jurjenbos228 Год назад

The math is cool, the slo-mo of matt crawling is priceless.

@liamogrady5868 Год назад

What happens if in magical theory land we provide the right initial conditions to put the hoop at a triple point in the phase diagram? I personally like the idea of the hoop just exploding (Edited for clarity)

@TestTestGo Год назад

I suppose that would mean that several forces on the wheel would be in perfect balance, so for example the acceleration from the falling weight at its most forceful position is perfectly balanced by the maximum friction of the wheel as it tries to slip. Meanwhile the gravity on the wheel is perfectly balanced by the upward force caused by the accelerating weight. That has the interesting side effect that if the wheel were rolling on a set of scales it would never hop but its measured weight would be zero at that instant. Magical theory land only though because if the weight is effectively zero why is there still friction?

@sorenlily2280 Год назад

It would be similar to a triple point in chemistry. A chemical well-balanced at it's triple point will chaotically shift between its different phases, but it doesn't make some magical new phase. The wheel would similarly shift easily between the different phases of the diagram if the conditions were just right, but it won't exhibit some magical new behavior in defiance of classical mechanics.

@koosb8162 Год назад

@@sorenlily2280 No explosions? :-(

@rufusdrumknottvgc Год назад

I just realised that these videos are basically literature reviews on a very specific topic presented in fun video format. You're a very useful reference for experts in other fields wishing to reach wider and more mainstream audiences with their science! Also as if this video wasn't fascinating enough, there's even a dog in it. You absolutely spoil us, Matt.

@FAB1150 Год назад

I have already watched the video, it's the n-th time I've seen the thumbnail and yet I still stop and look at the electrical outlet wondering what's wrong with it without even noticing the "hoop"

@dendrofilen Год назад

Hey Matt thanks for an amazingly nerdy video, I love it! However I’m a bit confused when the hoop should hop. At 11:30 you show that it should hop when P is at 90 degrees because the parabola is outside the cycloid. But they look tangent to me. At 180-270 degrees though the parabola is clearly outside. So shouldn’t it hop when the weight is on the way up? However the phase-diagrams with the experimental results at 17:30 shows it actually hops at 300-360 degrees. 🤔 Would appreciate if you could clarify this.

@Jehty21 Год назад

At 11:30 When P is at 90 degree the parabola is outside of the cycloid. Not at the point P, you have to follow the parabola just before it hits the y-axis. I am a bit confused as well, but the way I understand is that it doesn't hop at 90 degrees in real life. Instead as stated at 20:47 it hops at 3pi/2 to 2pi (270 to 360 degrees). That's also collaborated by the clip at 5:45 (I think). So it hops on it's way up. So the question is why doesn't it hop at 90 degree even though the parabola is outside of cycloid? I guess an explanation is given at 12:45 (basically friction). But then he says at 20:45 that the paper states that the paper at 12:45 is wrong 😕

@JosephDavies Год назад

It's disheartening to see that much of this research was going on while I was at Uni, where I was desperately asking people in the math department about the existence of projects of this very sort and getting nothing but blank stares and confused dismissals. This is exactly the sort of niche-but-fascinating interrogation of geometry and physics that I wanted to work on and that would have captivated me.

@vysje 9 месяцев назад

Very interesting analysis! It would have been nice if the phase diagrams would also include the hoops stiffness. I would think that for a non-infinite stiffness the hoop will be compressed when the weight rotates from the top to the side (parabola is below the cycloid), stored in potential energy. When the weight then moves further down and the parabola gets above the cycloid, the hoop is decompressed, and the potential energy is converted to hopping kinetic energy. This would shift the H-area in the diagrams to the left and down.

@chrisprou9216 Год назад

excellent montage!!!!!

@WeeWeeJumbo Год назад

the motion of these objects is unnatural and disturbing to watch, like a physics glitch in an open world video game

@DavidGuild Год назад

Reality has several physics glitches. Check out "laser cooling" where you shoot lasers at something and it gets colder.

@SirWilliamKidney Год назад

I would actually enjoy seeing Ben Sparks on the channel sometime, he's a really great maths communicator aside from being a geogebra wizard!

@mooncowtube Год назад

This all makes good sense, and I'm surprised the estimable Tadashi didn't spot it: if the parabola only comes outside the cycloid AT or AFTER the horizontal position, the force the weight will apply to the hoop will be ALONG or DOWNWARDS and so will require skidding (which a sufficiently rough hoop would not do). In order to get a hop, the parabola needs to head outside the cycloid BEFORE the horizontal position -- even just momentarily -- so that the force the weight applies to the hoop can lift it from the surface. Contributions from angular momentum in the hoop (which a weightless hoop would not have) or yeeting (good word) are ways to lift that divergence point above the horizontal. Very interesting topic -- thanks for the survey!

@ColinTimmins Год назад

What a cool way of joining all the bits for the large wheel. Very stylish. =]

@andrewmirror4611 Год назад

In my opinion, if the hoop doesn't roll it "hops enough", meaning there is enough of vertical up pull to affect the roll as opposed to a regular hoop rolling. And by that I mean a regular hoop after a full roll will trace 2пr, and if it was covered in wet rainbow paint, the hoop with a weight would probably not trace 2пr and the rainbow stain on the floor would be different

@brunizzl Год назад

I loved, when matt said "it's animating time" and the hoop animated all over the place.

@hurtee1807 Год назад

I think Littlewood is correct, but Matt and all the papers misunderstood what he meant with „when the radius vector becomes horizontal“. The assumption everybody made is that this happens at 90 degrees angular displacement. But it also happens at 270 degrees angular displacement, and I think this is what littlewood meant, and also what would be intuitive. At 90 degrees the vector of the parabola and the vector of the cycloid switch, but both still point downward. The point mass wants to go further on the horizontal plane than the cycloid path allows it to, but the infinite friction just cancels this vector and the hoop itself doesn’t hop or slide or slip or skim because it has no moment of inertia due to having zero mass. I believe that the first moment that the vector of the parabola is pointing further upwards than the vector of the cycloid is at 270 degrees angular displacement (if you account for the moment of inertia forced upon the point mass) and this is the first moment when the hoop is lifting of the ground and therefore hopping.If I read the phase diagrams correctly, then they, too, only find a hopping action after 270 degrees angular displacement under realistic circumstances. again I believe that the lower limit when the mass of the hoop tend towards zero and the friction tends towards infinity approaches 270 degrees.