The 56-Year Argument About a Hopping Hoop

Here I have a hoop it's actually a bike tyre but  you get the idea and we've taped some very heavy   ball bearings in the side here so the question  now is if I was to roll this hoop with a point   mass substantially heavier than the mass of the  rest of the hoop when it gets to here will it   just accelerate down or will this coming down  under Gravity lift up the rest of the hoop and   give us a Hopping action well the first person  to ask this question at least in writing was   our Littlewood a British mathematician at  Cambridge who in 1953 wrote a book called A   Mathematician's Miscellany and the introduction  basically says that Littlewood could not be   bothered coming up with some kind of theme or  general concept to hold all the bits in the book   together I Aspire for that  kind of uh Reckless abandon   when it comes to writing popular maths books  however a few pages in he presents this exact   concept uh here we go a weight is attached to a  point of a rough weightless hoop which then rolls   in a vertical plane starting near the position of  unstable equilibrium which is to say the weight is   right at the top what happens and is it intuitive  well I thought I thought I'd give it a go [Music] and according to Littlewood this hoop will hop  If released to rotate and they don't do a lot   of mathematics there's like one extra paragraph  a little bit here where they go through it but   several decades later Tadashi Tokieda who uh you  may know from their fantastic Numberphile videos   wrote a quick uh two-page bit of mathematics  this was in 1997 showing that yes the hoop   should hop when that mass is on its way down  but then these people disagreed so before we   get into what several papers which I mean I'm  not saying that mass is the best indicator for   mathematical value but a lot bigger than what was  happening here before we go into those I thought   we'd just roll some hoops we'll see if they hop so  we got this one which has got the weights inside a   bike tyre we also got this one here so that's some  metal weights and we've got the Sandpaper on the   outside it's got to be a rough hoop and as you  can see I mean if you roll it slowly like this   weird kind of lurching action we can do a lot  better we've also got this one it's got teeth   so it digs into whatever surface it's on to stop  it from slipping and there's a very heavy mass   uh built in there so we're going to take all  of these out into the Hall see what happens [Music] yes the tyre has a bit of a bounce to it but don't  worry the wooden one does hop see that's a hop [Music] foreign before we get down to hoopness I want to  mention that this video is part of the Stand-up  Maths Mental Health Season sponsored by  betterhelp.com that's h-e-l-p they have over 10   000 licensed therapists that you can contact to  help you with your mental health but the point of   the Stand-up Maths Mental Health Season is not  that you necessarily have to go to better help   there's loads of other places where you can get  support in this video I'm going to be highlighting   the fact that if you are a university student or  maybe you work or study somehow at a university   there may be free mental health support available  that was one of the times I got therapy myself   when I was an undergraduate at the University of  Western Australia they provided therapy sessions   for undergraduates I took advantage of that and  it was super helpful so if you're at a university   do see what support they provide if you're not  at a university you may have other access to   mental health support and of course betterhelp.com  h-e-l-p help if you go on there you can answer a   few questions about what you have hoping to get  out of therapy and they will match you up with a   licensed therapist anywhere in the world and you  can start communicating with them online through   either video chat or messages thank you very much  to betterhelp.com stand up Maths for sponsoring   this mental health season finally better help is  not a crisis mental health support service I will   link to some of those below if you want to check  those out and that's it now it's time to get back   to better hoop we managed to get these hoops to  hop out in the hallway the question now is does   that count I'm not gonna lie we really had to give  them some serious angular momentum eat them as the   kids would have said about five to ten years ago  anyway the point is we had to move them pretty   fast to get them to actually hop and my friend  Lisa Matha made these for me and we were actually   doing a geometric installation at a festival Last  Summer she brought along some of the prototypes   we realized we moved them slowly they absolutely  did not hop but if we gave them a real angular   momentaring as the kids will say from now on uh  they did hop and we were pretty pleased with that   um until sometime later Katie's tackles who  was there you may remember from previous   videos of mine sent me a Hopping hummus because  an unbalanced container of hummus would indeed hop   which leads to my first conjecture are there any  unbalanced wheel if heated fast enough will hop   the wording is not final but the point is I think  if any kind of disc has a imbalance in the mass   there's a speed such that it will hop which is not  the original question that Littlewood was trying   to raise to recap Littlewoods set up we've got a  point weight here on the top of a rough which is   taken to mean uh infinite friction not going to  slip which we've done by using the little kind of   saw teeth and we've got this non-slip mat on the  table today so it means this is never going to   slip weightless which means all the masses here  the technically no mass down here and it starts   near the position of unstable equilibrium and the  position of unstable equilibrium is that one there   because technically I could balance it right there  oh it's an equilibrium but it's so unstable like   if it's even slightly off away it goes and so the  idea is you start with it here you gently release   it and then you allow it to roll so you know what  we've got it all set up here let's do some slow   motion of that happening see if we can catch  the Hop In Action starting from the standstill foreign [Music] says here the hoop lifts off the ground when the  radius Vector to the weight becomes horizontal   which is that position there and you will have  seen in the slo-mo that that puzzle seam was   directly down when it should have been hopping  and they say uh the motion is equivalent to the   weight sliding smoothly under Gravity on the  cycloid it describes pretty intuitive um and   it will sooner or later leave that huh why doesn't  clear that up at all thank goodness Tadashi was on   the case there are two shapes we need to follow  Tadashi's argument one of them is the cycloid   that's the shape you get if you follow a point  on a rotating wheel so if you imagine if I start   that there you've got let's take the middle one of  these three as I rotate it down it becomes a line   coming down like this it hits the ground and stops  and then goes so straight back up again as such   right and this shape here this is a cycloid and  you can carry that on and so any point on a wheel   will Trace out this kind of bouncing pattern not  a parabola but the parabola is our second shape   so if I had an object I'm going to use this dice  this is going to represent again just the middle   one of these masses this is a point mass if there  was no wheel at all and this mass was moving it   would follow a parabola and that's what the shape  would do if there was no wheel however the wheel   has no mass so we've kind of got two conflicting  shapes here the point is technically on a wheel   so it should follow a cycloid but yet the wheel's  got no mass so it should be following a parabola   we need to plot and compare both those shapes  it's jojoba time so we've got a massless hoop   in blue and a point mass at p and you can see  there's some angle Theta we're measuring that   from the vertical Direction and I can actually I  can move Theta around manually to make the hoop   roll or I can just start animating it and there it  is rolling as if gravity oh did it hop who knows   um we'll find out in a second so I'm going to  reverse that back up again and so if you think   about the path that P is tracing out from before  that's our cycloid actually I can turn that on   there it is and now as I animate the who going  backwards and fours you can see p always stays   on that line now actually we're not going to  see it hop in this animation because this is   not a simulation it's a diagram so it's not like  we're numerically simulating what will happen   we're actually just animating the circle moving  backwards and forwards so we're not going to get   any additional insight into the actual mechanics  from what the circle does the Insight is what   happens if we turn on the parabola because you can  imagine at any point in the rotation of this hoop   if the Hoops suddenly vanish this would continue  moving along a parabola so actually at all points   in time there exists a parabola that this is  on that shows the motion if it was a projectile   to be at the same place and the same velocity  as if it's when it's rolling on the hoop so the   parabola is what would happen if the hoop suddenly  vanished and I can turn that on here there it is   that green line is the parabola so that's like  the Phantom Parabola if there suddenly was no   hoop the part but the particle's still there  it would carry on along that so if I move it   backwards and forwards you can see the parabola  changes as the hoop rolls it's very cool and the   question now is does that Parabola always stay  inside the cycloid and if you know I'm going to   animate it through so you'll watch the parabola  and oh did it or did it flip outside if I go a   little bit further forward look at that it's  outside and so that means at that exact point   here the projectile path is outside the cycloid  motion on a wheel path because if it's inside   basically means it's just pushing into the  wheel that's not going to happen but if it's   on the outside well this is what Tadashi argues  the moment that Parabola is outside the cycloid   it means we're going to switch from Rolling  wheel motion to projectile motion and that   means the hoop is going to hop now we didn't see  it because obviously this is not zero mass this   is made of wood and there's the friction we've  tried to go for no friction but the point is our   physical model wasn't a perfect enough Recreation  of the pure mathematics but what's mentioned first   of all by Littlewood and then done in detail  like Dashi is that in theory the hoop should   hop nope at least according to at a minimum this  paper hopping Hoops don't hop so this was not the   final word in 1997 we have a lot more mathematical  Publications about the hopping hoop all of these   are from 2000 onwards I'm going to put them aside  for now and look at these two from 1999 which are   pretty much have the same argument so I'm going  to pick the one that actually says hopping Hoops   don't hop but we have also got the hopping hoop  Revisited which came out like a month because   ones like July this one's no June July in  99 there's one's from the August September   publication so the general argument here is that  the hoop has to slip so they're saying that's all   well and good to have your argument here where the  mass comes down and then the parabola goes outside   the cycloid and hop but they're like well hang  on how is it going to hop like without slipping   so to hop assuming infinite friction it would have  to go like straight up somehow but there's always   going to be some element of slipping and I'm very  much paraphrasing some long involved mathematical   arguments but they say no if we retain Newton's  Laws which we probably should that's kind of the   whole point here and in impenetrable flaw which  means the hoop can't go below the surface it's   rolling on then the no slip condition must be  violated when n equals zero so that's like when   it would hop implying that the hoop scares so  they're saying when you get to the point here   where it should hop it can't hop that doesn't  work like in Newtonian physics the only thing   that can happen is it has to Skid there's no  way you're going to get a non-skidding hop   or so they say I will say there's one final  conclusion this paper that I absolutely love   at the very end they've got a final theorem the  behavior of hopping Hoops is not intuitive which   is what Littlewood originally raised and uh they  they state that as a theorem and their proof is   by inspection if there's anything we can agree on  so far it's that final three bits of mathematical   research that came out between the year 2000  and 2009 and this is when people start to take   the physics a bit more seriously and they're  like look instead of assuming that the hoop   is massless that's ridiculous let's look at the  ratio of masses so the hoop has got some mass   and there's some ratio which they used a gamma to  represent that's here is the point mass so that's   just an unbalanced hoop instead of having infinite  friction let's just have a coefficient of friction   what's wrong with mu bring mu back in again so now  they were able to Model A unbalanced rolling on a   surface with some friction and it turns out first  of all if we put hopping aside for a moment which   I know will make some people hopping, hopping mad  little joke there for all the joke fans for the   maths fans however there are actually three things  a hoop can do when it's rolling you've got the   actual rolling action that's kind of the default  Behavior you've also got slipping so if there's   not enough friction you might get a slipping  effect where it rotates this way and interestingly   the friction force is acting in that direction if  it's slipping this way you've also got skidding   now that's different to slipping skidding is like  the comedy coming to a halt that's the effect in   which case now the friction goes that way so  slipping and skidding are distinguished by the   different directions that the friction is acting  and are in a more realistic Universe where we do   have some non-infinite friction it turns out both  those things can happen when you release a hoop   with different amounts of initial angular momentum  but we've got a lot of variables now we're going   to need some diagrams some phase diagrams phase  diagrams so each one of these individual plots   is for a specific combination of mass ratio and  coefficient of friction in fact the top four all   have the same mass ratio where three quarters of  the entire system mass is in the point mass and   then they gradually ease up on the coefficient of  friction and then on the vertical axis we have the   amount of initial angular velocity so you pick a  point on the vertical axis for how fast you want   to start this thing going and as you move across  horizontally that's the rotation Theta as it's   rotating and it tells you what happens if it's  hashed it means that it's rolling normally if   it hits a d region that means it's skidding if it  hits an S region that means it's slipping and so   if you look at this you can pick your starting  point for whatever combination of the physical   setup you want and then you move across and  you're like oh it's rolling and now it's   slipping or skidding or doing whatever and then  eventually something interesting might happen it   might repeat so if it gets to the far side with no  blank spaces it means nothing interesting happened   if it hits a z that's the zero line that means it  grounds to a halt and if it hits an H that means   it hops there you are those are the Hop regions  and as you may have noticed they're all the top   which means High initial angular velocity  which means if you eat something it shall hop   still working on the wording now I just need to  bring this video gently rolling to a satisfying   conclusion and I will say that this kind of  exploration of the actual physics is super   interesting at the very end they've got a table  of all the different combinations of things the   hoop can do like you can see here it can roll and  then slide and then roll and then skid and then   roll again and then slide then roll then skid  then right it's so interesting all this going   on and I'm not the first person to like make  hoops and see what happens uh Tadashi way back   here did that they say at the very bottom here  um we taped a battery on the hula hoop I mean   that's pretty much what we've done here I'll be  honest and rolled it along the 12th floor hallway   in fine Hall that's at Princeton and says uh it  actually hopped I suspect that's a case of just   you know the Yeet situation and then over here in  one of these papers they've got a fantastic there   is at the back there's a shot they got a hoop  they put a mass on it they rolled it and they   showed once again if you give it a bit of speed it  will hop and the very last paper from 2009 this is   pretty cool they got a hoop actually not this  similar where's mine this kind of a setup and   they put LEDs on it so they could more closely  track what was going on and they released on a   variety of different slopes because the increase  in slope meant that you'd have that extra speeding   up which would cause it to hop and they found  that sometimes for very severe slopes it would   hop on the first rotation and for there's like  a range of middle slopes where it wouldn't hop   the first time the masses like the point mass  came down but it would hop on the second one   um very cool and actually this validated a lot  of the kind of the physics papers that came out   between 2000 and 2009 and it dissed these from the  conclusion here they say first of all our results   support the predictions made in reference five and  their reference five is the 2000 paper so this was   theoretical maths they then did the experiment  and showed that it all lines up very good however   they do say the experimental confirmation that the  hoop Hops at and then a range of angles supports   equation 11 and contradicts reference three that's  uh this one the hopping hoop Revisited so this   physics they're like No Deal what a what a what a  maths smackdown those are some harsh words look at   that contradicts reference three please language  from the conclusion of this paper they also say   that the 99 Theory doesn't work and they say that  what Tadashi Tokieda and Littlewood were saying   doesn't work either so uh from here therefore  and this is the situation for the massless who   like originally set up by Littlewood therefore  the hoop will hop as suggested by Littlewood and   Tokieda only if there is a discontinuous  increase in the angular acceleration   which is a fancy way of saying it doesn't work  in terms of like physics as we understand it they   actually introduce A New Concept called skimming  which is where the mass has switched from the   cycloid to the parabola but the hoop is now kind  of levitating along it's skimming not skidding   it's skimming along with it so it doesn't  actually hop so I mean the grand conclusion is   I don't know they are we don't know someone else  wants to make a bigger better hoop let me know or   some more maths occurs let me know or I've missed  something please do let me know but uh I mean the   Yeet Theory is correct these papers do prove if  you've got an unbalanced uh wheel and you move   it fast enough it will hop and to Littlewoods  credit back in 1953 they did originally put the   very last line in their setup in actual practice  the hoop skids first so even Littlewood from the   very beginning knew that you probably wouldn't  actually see this with a hoop because in reality   it's gonna skid so uh although we since renamed  that slipping ah what are you gonna do uh point   is we had a lot of fun building the hoop so huge  thanks to Lisa Mather and uh Joel who helped out   with uh their 3D printing this and putting these  hoops together thanks to Ben Sparks who did the   judge profile that I'm showing you uh before  and thank you to everyone who watches these   videos I hugely appreciate it and in exchange  for you obviously liking and subscribing come   on I got Steve Mould to catch uh I'm going to  show you 20 seconds of slow motion footage of   a Hopping hoop being chased by my dog please  uh watch it and take that time to subscribe [Music] thank you

Channel: Stand-up Maths

Views: 438,809

Keywords: maths, math, mathematics, comedy, stand-up

Id: ETRpkp03stQ

Length: 23min 55sec (1435 seconds)

Published: Mon Feb 27 2023