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."
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.
@@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.
@@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.
@@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.
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)
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.
@@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.
@@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.
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.
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
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.
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.
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.
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?
@@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.
@@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)
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! :)
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.
*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)
@@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.
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 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 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.
The word "yeet" is inherently funny. Seriously, trying saying it out loud. The Parker Law of Hoops needs no further wordsmithing. Proof: by inspection ;)
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.
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.
@@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.
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.
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
Here in Alaska, it's an "oil filled cylinder" using heavy gear oil - honey & molasses cylinders hop irregularly instead of rolling slower than expected
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.)"
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.
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 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 .
@@drdca8263 I agree with that, I was just implying that it would be a different problem allowing that assumption. But nonetheless an interesting experiment.
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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. :)
Yeeting the system gives it linear inertia which needs to be maintained. The inertia of the weights is more than the hoop, so at certain points it's going to be easier to rotate the hoop around them than them around the hoop, and so a hop. Is my theory anyway.
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.
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.
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
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
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.
@@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 😆
@@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?
@@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.
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.
@@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
@@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
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
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.
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.
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.
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.
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 😕
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)
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
As a Mechanical Engineer, a discussion of the phase diagram of a rolling wheel pleases me. In reality, shaft balance is really important for machines operating at higher speeds (shafts, wheels, turbines ext). We tend to model the force on the system due to mass misalignment as F = sin(w*t)*d*m*w^2 and figure out at which speeds the magnitude of the force gets too high and just not yeet the shaft that fast. The w^2 is the kicker because it makes the forces get high quickly as the speed increases. Honestly, the hopping hoop is interesting in part because the speeds are low and hopping isn't guaranteed (Also, Coulomb friction is nonlinear and a pain to deal with).
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.
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?
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.
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.
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)
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?
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.
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.
Surely the rigidity of the hoop will play a role in the hop/no-hop outcome. The more flexible tyre might be storing some elastic potential energy and releasing it at the right time as a bigger hop?
I had been thinking of the lower-rigidity case as locally (slightly after the point of contact) acting like a lever in the event that the wheel deforms. Not sure if that makes any sense, but possibly that's a part of the mechanism that could force it upwards in the less-rigid case?
@@nikkiofthevalley true yeah I guess I can see what you mean. Just wanted to raise this point though that I think there will definitely be some effect the rigidity will produce and probably worthwhile adding to the physics model :D
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"
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.
I conceptualize this using orbital mechanics - for any given hoop there is an orbital speed of the point mass for which it is stable. Given normal gravitational acceleration in the first cycle this orbital speed can never be exceeded. But if you introduce additional energy into the system (yeeting the hoop) you introduce additional energy into the orbiting body which causes it to orbit at a higher “altitude” from the center of the hoop. If this altitude is greater than the hoop’s radius the imbalanced centrifigal forces on the hoop result in a pulling/lifting force that causes the hop (the point mass trying to ‘orbit’ a point other than the center of the hoop and pulling the rest of the hoop with it) I’m not sure how to describe this concept mathematically but it seems consistent with all the behavior described
Someone needs to roll these over a giant load-cell to measure the down-force accurately. I like the idea of "skimming" - where the vertical component of the resultant force on the ground is zero, but there remains a small horizontal component. I'm having a job visualising the extrapolation of that into hopping though... \o/
@Stand-up Maths Why are we assuming that if the wheel has no mass then the mass would follow the parabola? (9:00) We aren't discussing a mass-only object. The object is a mass+wheel system. It has mass (mass if the mass + mass of the wheel - some+zero). Ergo it has both inertia of the mass and rigidity of the wheel shape. Hence it should have friction regardless of the mass of the wheel alone. Hence it should be able to hop (skid) no problem regardless of the fact that the wheel alone has no mass.
The wheel hopped after about 270 degrees when the weight was going upwards - isn't that what you would expect? the wheel is pulled in the direction of the centrifugal force, in this case up - did I miss something?
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.
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.
For a mass constrained to a wheel, this all makes sense. But consider any unbalanced mass on constrained to any spinning surface. This creates oscillating outward forces as felt in vibrating motors (rumble packs etc). When this forces exceed those which bind it (whether the weight of the wheel under gravity, the material strength of a disk, or the strength of an axel) the mass will break free of its constraints (Hop under gravity, shatter a disk, snap an axel).
i actually used "as one easily sees" in one of my proofs in university... though to be fair, i dont remember if i got points for that exercise or not 😆
You can get away with it sometimes. For instance, if an equation has a few obvious trivial solutions, you could say "by inspection, f(x) = 0 and f(x) = -x are solutions" or something. As a step in a proof of course, not the whole thing. Some textbooks like to use that phrase a lot.
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'".
"The hoop lifts off the ground when the radius vector to the weight becomes horizontal." On a horizontal line bisecting a circle, there are two points of contact, and when the circle is in motion those points are the leading edge of the circle and the trailing edge. The hoop will lift off the ground (if it has sufficient speed to overcome the force of gravity) when the weight passes the horizontal plane at the trailing edge of the hoop.
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.
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!
I loved the giant hop the tyre did when Matt ran into frame after it and excitedly pointed. Very clear it was a hop and not a bounce. 9:30 So to counter the Mass of the wheel, to make it "weightless" you need to apply Force. That's physics F=MA. So mathematically if you cancel out the weight of the wheel by applying force, you can achieve the "weightless" wheel and it should hop.
Better help shares your private information with Facebook. From what I know they can't be trusted, please do some research into the sponsors you pick. Otherwise I loved your video
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."
Truly, one of the most convincing proofs I have ever seen.
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.
@@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.
@@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.
@@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.
for a stand-up mathematician, he sure sits down a lot
He IS standing up:)
But he could just be a stand-up guy who is also a mathematician.
The number of times you stand up and the number of times you so down are always the same each other
@@jackdog06 not true. Sometimes you lay down or fall down and have to stand up.
@@jackdog06 not true. If you haven't stood up yet, it will be different.
The absolute best part of this is that all parties involved at some point spent time rolling weighted hoops around
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)
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.
I'm sorry the best part of this video was the dog.
@@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.
@@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.
15:40 "a little joke there for all the joke fans" is such a fantastic line
It was a good joke, if you will.
absolutely stealing that one
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.
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
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.
Perhaps it can be constructed as "Parker's Conjecture."
I love that whenever Matt makes a silly joke he unapologetically pauses to acknowledge it.
In my mind a silent drum is playing "Ba doom, boom" every time he does.
On today's Stand Up Maths, Matt explains why we balance the tires on our cars.
Now I want a car that hops.
@@yarati4584 Just search RUclips...lots out there.
@@yarati4584 just unbalance the tyres enough and give it enough angular velocity (not legal advice, might void warranty).
@@yarati4584 Mythbusters made a car where the axle was not in the middle of the wheel...not very comfy ;)
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.
This is exactly the kind of wholesome dog-chasing-hoop content I come to RUclips for
Matt, I must say that you missed a golden opportunity to say, "Out with the old and in with the μ"
I haven't even watched the video yet but I can tell this would have been a top-notch Matt Parker joke 😂
What's old is μ again
I'm glad there are people out there willing to go the distance and do the science on these important subjects.
Maybe not important, but it warmed my physicist's soul.
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.
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?
@@SilverLining1 I thought you were being ironical.
@@Yora21 In my field a paper can represent several man-months of work. All easy things were done long ago.
What a fascinating niche subject, the phase diagrams were absolutely lovely!
I’m going through a real phase diagram phase.
@@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.
@@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)
My answer: It depends.
I give permission to anyone who want to use this to write a scientific paper...
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! :)
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.
Important: I tried Better Help the first time Matt was sponsored by them. They *do not* have therapists worldwide, only in USA.
I’ll ask and report back!
*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)
@@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.
They’ve also just been caught selling your data to companies so yeah don’t trust them
They’re also a scam.
Really love this format. A kind of video literature review or journal club. Everything explained really clearly. Great job!
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.
I love how this comment casually drops the word 'Zorb', as though daring me to go look up what such a thing is. :)
@@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.
Be careful, hopping while zorbing may lead to leaving the track. Lives have been lost to bouncing zorbs.
@@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.
Every science experiment needs an 8 year old in peril.
My nieces are 11.
I love those phase diagrams-- what a satisfying and clever way to display that info. Kudos to the authors. Great video!
"If you yeet something it shall hop" Will forever be known as the Matt Parker Law.
I love how the auto-subtitles can't quite get the word "yeet"
I need this on a t-shirt. As well as "I contradict references"
The word "yeet" is inherently funny. Seriously, trying saying it out loud. The Parker Law of Hoops needs no further wordsmithing. Proof: by inspection ;)
What if it works? Can something be named after Parker if it works?
@@brune02 Eat them
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.
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.
Yes, exactly what I extecte to be in the conclusion. Behaviour as the ratio approaches the limit
This is much clearer formation of the question I had!
I had this same thought, but maybe the math makes this uninteresting because the acceleration due to gravity is independent of mass?
@@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.
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.
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.
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
Here in Alaska, it's an "oil filled cylinder" using heavy gear oil - honey & molasses cylinders hop irregularly instead of rolling slower than expected
"If you yeet something, it shall hop" I actually laughed out loud at that.
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.)"
That misprint quip is great! XD
@@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").
@@renerpho i knew it had to be a reference
That's what math is really about.
This kind of thing is like real life's equivalent of wizards arguing over the particulars of a certain spell.
D&D rule arguments are the best arguments! 😁
"Language!" got me.
Also, as a material scientist, I appreciate you bringing phase diagrams unto the people.
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.
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.
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).
@@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 .
@@drdca8263 I agree with that, I was just implying that it would be a different problem allowing that assumption. But nonetheless an interesting experiment.
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.
@@johanullen parable?
The dog is having fun despite having little to no conception of maths. That's the way to go.
Yes, but does the dog hop when rolled?
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.
We love skylab!
“If you yeet something it shall hop” - words to live by.
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
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.
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.
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.
I agree. People who worry about hoops hopping need help with overthinking.
@@Verrisin lol probably, but there are lots of other reasons.
@@wafkt thank you for helping to reduce the stigma. I love when students say a professor suggested they come here.
@@Verrisin I deadpan disagree. People who worry about other peoples overthinking about hopping hoops need help overthinking their own underthinking.
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.
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.
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.
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.
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. :)
Polystyrene would probably do the trick, and be a few orders of magnitude cheaper :)
Yeeting the system gives it linear inertia which needs to be maintained.
The inertia of the weights is more than the hoop, so at certain points it's going to be easier to rotate the hoop around them than them around the hoop, and so a hop.
Is my theory anyway.
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
Impossiball. I loved that until I decided I wanted to modify the weight and destroyed it
And that is why one of your balls hangs a little lower then the other.
Yes, a ball should do as well or better.
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.
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.
yet again you havent failed us with utrerly useless but fascinating maths. keep bringing us this amazing content matt!
Utterly
@@thechumpsbeendumped.7797 butterly
if he failed us, then it'd be a Parker Video
Going downhill, now... it will accelerate and the upsy-downsy tendencies will eventually overwhelm any smooth rolling movement.
Not useless, these maths may be useful for machines that use unbalanced wheels
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
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
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.
@@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 😆
Yes! The initial question is poorly defined. 🙏
@@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?
@@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.
Engineers wonder why this 56 year argument wasnt a 56 minute argument.
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.
Would take quite an engineer to build a massless hoop
@@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
@@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
@@marianaldenhoevel7240what do you think physics is if not applied math?
Physicists know their math and love a "massless hoop with infinite friction"
I was hooping you would make a new video
Ironically, I was hopping he would do so
Hoopefully that's the last pun
For the joke fans
He had a question and decided to roll with it
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
"If you yeet something it shall hop" Beautiful wording, just magnificent Matt.
I loved, when matt said "it's animating time" and the hoop animated all over the place.
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.
some employers also offer employee assistance programs.
Interesting stuff! thanks for the rad content!
1953 was 56 years ago? Thanks for chopping a dozen years off my age. I feel younger already.
1953-2010? (Look in the description)
That last paper was 2009 bug humbug.
I'll tell my dad, he'll be delighted
You can tell people that's your Parker age, anyway
@@standupmaths If we argue about it in the comments then the argument is still ongoing, right?
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.
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.
Glad you are encouraging mental health support. Love your videos and books
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.
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 😕
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)
The dog joining in with the hopping was the highlight.
I wish i could subscribe more than once. I love everything you do matt! You make maths so fun (and your theme is epic)
The paper "Hopping Hoops Don't Hop" should clearly just have been called "Hopping Hoops Don't".
Hoppin't Hoops
"They hopped along in exactly the same way hoops don't." - nod to Douglas Adams
The math is cool, the slo-mo of matt crawling is priceless.
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
As a Mechanical Engineer, a discussion of the phase diagram of a rolling wheel pleases me.
In reality, shaft balance is really important for machines operating at higher speeds (shafts, wheels, turbines ext). We tend to model the force on the system due to mass misalignment as F = sin(w*t)*d*m*w^2 and figure out at which speeds the magnitude of the force gets too high and just not yeet the shaft that fast. The w^2 is the kicker because it makes the forces get high quickly as the speed increases. Honestly, the hopping hoop is interesting in part because the speeds are low and hopping isn't guaranteed (Also, Coulomb friction is nonlinear and a pain to deal with).
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.
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?
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.
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.
that doesn't necessarily mean it would hop
Great way to illustrate how published papers are part of a larger conversation and how they cite, support, and refute each others' claims.
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)
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?
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.
@@sorenlily2280 No explosions? :-(
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.
Adding a pupper chasing the yeeted hoop is no cap a poggers strat to catch Steve.
4head kekw
This would be enjoyed by those who enjoy jokes 5-10 years ago
Hm. Thumbs down!
Sadly, Google translate doesn't support this language.
@@SlenderSmurf yeet
Matt, an excellent video. Thanks so much, helps a lot!
Surely the rigidity of the hoop will play a role in the hop/no-hop outcome. The more flexible tyre might be storing some elastic potential energy and releasing it at the right time as a bigger hop?
I had been thinking of the lower-rigidity case as locally (slightly after the point of contact) acting like a lever in the event that the wheel deforms. Not sure if that makes any sense, but possibly that's a part of the mechanism that could force it upwards in the less-rigid case?
@@nikkiofthevalley true yeah I guess I can see what you mean. Just wanted to raise this point though that I think there will definitely be some effect the rigidity will produce and probably worthwhile adding to the physics model :D
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"
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.
Your videos make my heart smile!!
More footage of your dog chasing the hoop please!
Yes
"If you yeet something, it shall hop."
Inspirational
the motion of these objects is unnatural and disturbing to watch, like a physics glitch in an open world video game
Reality has several physics glitches. Check out "laser cooling" where you shoot lasers at something and it gets colder.
I conceptualize this using orbital mechanics - for any given hoop there is an orbital speed of the point mass for which it is stable. Given normal gravitational acceleration in the first cycle this orbital speed can never be exceeded. But if you introduce additional energy into the system (yeeting the hoop) you introduce additional energy into the orbiting body which causes it to orbit at a higher “altitude” from the center of the hoop. If this altitude is greater than the hoop’s radius the imbalanced centrifigal forces on the hoop result in a pulling/lifting force that causes the hop (the point mass trying to ‘orbit’ a point other than the center of the hoop and pulling the rest of the hoop with it)
I’m not sure how to describe this concept mathematically but it seems consistent with all the behavior described
Someone needs to roll these over a giant load-cell to measure the down-force accurately. I like the idea of "skimming" - where the vertical component of the resultant force on the ground is zero, but there remains a small horizontal component.
I'm having a job visualising the extrapolation of that into hopping though... \o/
@Stand-up Maths
Why are we assuming that if the wheel has no mass then the mass would follow the parabola? (9:00)
We aren't discussing a mass-only object. The object is a mass+wheel system. It has mass (mass if the mass + mass of the wheel - some+zero). Ergo it has both inertia of the mass and rigidity of the wheel shape. Hence it should have friction regardless of the mass of the wheel alone.
Hence it should be able to hop (skid) no problem regardless of the fact that the wheel alone has no mass.
I would actually enjoy seeing Ben Sparks on the channel sometime, he's a really great maths communicator aside from being a geogebra wizard!
Such a fun topic and review of the papers! Thanks, Matt!
The wheel hopped after about 270 degrees when the weight was going upwards - isn't that what you would expect? the wheel is pulled in the direction of the centrifugal force, in this case up - did I miss something?
I agree.
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.
Those physics students definitely had fun on this one. I love these sorts of ideas!
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.
"Yeet" is still perfectly cromulent terminology imho.
For a mass constrained to a wheel, this all makes sense. But consider any unbalanced mass on constrained to any spinning surface. This creates oscillating outward forces as felt in vibrating motors (rumble packs etc). When this forces exceed those which bind it (whether the weight of the wheel under gravity, the material strength of a disk, or the strength of an axel) the mass will break free of its constraints (Hop under gravity, shatter a disk, snap an axel).
Man, I wish back in High School and College I wrote down "By inspection" for my proofs lmao
Proved by intuition to be non-intuitive
i actually used "as one easily sees" in one of my proofs in university... though to be fair, i dont remember if i got points for that exercise or not 😆
You can get away with it sometimes. For instance, if an equation has a few obvious trivial solutions, you could say "by inspection, f(x) = 0 and f(x) = -x are solutions" or something. As a step in a proof of course, not the whole thing. Some textbooks like to use that phrase a lot.
never thought I would be so excited to see a circular disk jump
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!
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'".
@Standupmaths. Why are the controversies about BetterHelp being shadowbanned on your channel? Is this your work? Hope not.
12:23
“We’ve tried to go for no friction…”
I believe you tried to go for infinite friction!
My question is, if you have a wheel or radius r, how far back do you need to put the camera to see the demonstrations of it rolling properly? 😜
20:57 - "Contradicts Ref. 3" Lol XD
If you put a large mass on an actual bike tire and tried to ride it, would it keep trying to wheelie you?
I would absolutely do this if I wasn't scared of breaking my neck
for a half rotation yes, for a half rotation it will try to make you stoppie
"The hoop lifts off the ground when the radius vector to the weight becomes horizontal." On a horizontal line bisecting a circle, there are two points of contact, and when the circle is in motion those points are the leading edge of the circle and the trailing edge. The hoop will lift off the ground (if it has sufficient speed to overcome the force of gravity) when the weight passes the horizontal plane at the trailing edge of the hoop.
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.
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!
6:10 correction: yote
I loved the giant hop the tyre did when Matt ran into frame after it and excitedly pointed. Very clear it was a hop and not a bounce.
9:30 So to counter the Mass of the wheel, to make it "weightless" you need to apply Force. That's physics F=MA. So mathematically if you cancel out the weight of the wheel by applying force, you can achieve the "weightless" wheel and it should hop.
Better help shares your private information with Facebook. From what I know they can't be trusted, please do some research into the sponsors you pick. Otherwise I loved your video
RUclips needs to add a LOVE button for videos like this