The Perfect Road for a Square Wheel and How to Design It

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  • Опубліковано 24 тра 2024
  • How do you design a road that a square wheel will roll smoothly over? And what about other wheel shapes? How do you even approach such a problem?
    =Chapters=
    0:00 - Intro
    1:36 - The Dynamics of Rolling
    4:05 - Vertical Alignment Property
    7:16 - Stationary Rim Property
    8:29 - Describing the Road and Wheel
    13:04 - The Road-Wheel Equations
    17:02 - The Perfect Road for a Square Wheel
    22:40 - Building the Road Visually
    25:54 - Wrap Up
    ===============================
    Many of the ideas in this video came from, or were inspired by, "Roads and Wheels," an article by Leon Hall and Stan Wagon that appeared in Mathematics Magazine, Vol. 65, No. 5 (Dec 1992). If you want a deeper dive (or if you want spoilers for the next video), I encourage you to read it yourself. As far as math papers go, it's fairly easy to read:
    web.mst.edu/~lmhall/Personal/...
    ===============================
    CREDITS
    ► The song at the beginning of this video is "Rubix Cube" and comes from Audionautix.com
    ► Thumbtack icon comes from Mister Pixel of the Noun Project.
    ===============================
    Want to support future videos? Become a patron at / morphocular
    Thank you for your support!
    ===============================
    The animations in this video were mostly made with a homemade Python library called "Morpho".
    I consider it a pretty amateurish tool, but if you want to play with it, you can find it here:
    github.com/morpho-matters/mor...

КОМЕНТАРІ • 830

  • @morphocular
    @morphocular  10 місяців тому +130

    There seem to be a lot of comments questioning the practicality/usefulness of square wheels, particularly whether you can turn side-to-side with them. The short answer is there's likely not much practical use for them and you can't turn side-to-side. To be clear, this video was mainly meant to be an interesting application of math and geometry to a fun problem and was not meant to be practical in the slightest.

    • @nyxalexandra-io
      @nyxalexandra-io 10 місяців тому +1

      yea

    • @bopcity5785
      @bopcity5785 8 місяців тому +1

      I hope you've seen the Cody dock rolling bridge which now applies this math

    • @lukcurious
      @lukcurious 7 місяців тому

      7ýyy ÿt5vg 22:25

    • @lukcurious
      @lukcurious 7 місяців тому

      Ÿtyfus y😅ÿyÿy y yt 24:55

    • @lukcurious
      @lukcurious 7 місяців тому

      24:58

  • @kindoflame
    @kindoflame Рік тому +1617

    I was going to mention that a second requirement for a smooth ride is that when the rotational speed of the wheel is constant, then horizontal speed of the axle is also constant. Otherwise, you could have a 'smooth ride' where the car constantly speeds up and stops short even when the wheels are not accelerating. However, the equation dx = r*d(theta) very simply shows that the only shape that could satisfy this new condition is a circle.

    • @kfawell
      @kfawell Рік тому +201

      I thought of the same thing as I watched. And I imagined what it would be like to ride in such a car that's constantly jerking you forward and backwards. It made me laugh out loud. I think we would be used to a bumpy road going up and down. It would be somewhat tolerable at least. We experienced that walking and jogging for example. On the other hand, having our head jerked back and forth would be hilariously unpleasant or at least irritating. For example, as though somebody has grabbed our collar and is shaking us back and forth. I don't want to detract from the video. It was very enjoyable and solves the smoothness problem as defined.

    • @fghsgh
      @fghsgh Рік тому +81

      @@kfawell I've tried out one of those square wheel cars in a museum before. It was exactly like that.

    • @kfawell
      @kfawell Рік тому +28

      @@fghsghI am laughing again thinking about that. Were you able to watch others doing that before you rode? If yes, I suppose you had to find out first hand. I just realized you had that memory while you watched the video. I wonder how you reacted when you saw the word smooth. I really appreciate that the creator specifically defined smooth. Thank you for telling me.

    • @fghsgh
      @fghsgh Рік тому +20

      @@kfawell I mean, you had to pedal yourself forward, and it was pretty slow so not too bad. It also mostly felt like variable resistance, not so much speed (because that's how inertia works). But yeah it seemed like it would not be entirely smooth from seeing others too. This was also at least 8 years ago so although my memory is pretty good, I can't give an exact description of the scene ;).
      But anyway I thought the lack-of-smoothness was just from the physical thing being imperfect, until this comment said otherwise.

    • @Zildawolf
      @Zildawolf Рік тому +3

      Well now I’m wondering what’s the shape that’d make the most speed inconsistency possible lol

  • @AsiccAP
    @AsiccAP 2 роки тому +1126

    I feel like I gained brain cells despite not understanding a word

    • @aartvb9443
      @aartvb9443 Рік тому +94

      You didn't gain brain cells, you gained connections between brain cells ;)

    • @ANormalLemon
      @ANormalLemon Рік тому +40

      *Brain.exe has stopped working.*

    • @wildcard_772
      @wildcard_772 Рік тому +7

      Same

    • @EverythingLvl
      @EverythingLvl Рік тому +11

      It's an illusion, still super dum

    • @BloonMan137
      @BloonMan137 Рік тому +1

      @@aartvb9443 🤓

  • @cambridgehathaway3367
    @cambridgehathaway3367 4 місяці тому +24

    We live in an astoundingly amazing age. One person is able to singlehandedly write, animate, narrate and publish such a polished, professional, easy to understand, and intriguing video. not to mention doing all the math and even providing a formal proof they crafted themselves. Such incredible talent has existed in past ages (rare tho it may be), but never before has the common man been able to so easily and readily benefit from it. I am astounded and humbled and grateful.

  • @thomasrosebrough9062
    @thomasrosebrough9062 Рік тому +160

    22:12 super hype to see my favorite curve show up in this video!! A Catenary Curve is also very commonly used in architecture for its even distribution of weight/pressure.
    The most famous catenary curve is the St Louis Arch which is over 600ft tall! It differs from the identity curve by having 0.01 in each exponent of e, as well as multiplying the entire equation by -68.8, resulting in a curve almost exactly as wide as it is tall!

    • @ethansmith876
      @ethansmith876 Рік тому +1

      Saarinen my beloved

    • @csar07.
      @csar07. Рік тому +10

      You ascend to a new level when you get your own favourite mathematical curve

  • @pulli23
    @pulli23 Рік тому +153

    I'm late: but there's a single also important point to make a "square wheel" work. The very point that needs to stay at the height also needs to be the center of mass. Otherwise a wheel would give a force rolling back/forward during part of it's movement.

    • @mujtabaalam5907
      @mujtabaalam5907 Рік тому +57

      We can assume a powerful motor is spinning the wheel on a fixed gear system so the wheel's mass doesn't effect the motion

    • @whoisgliese
      @whoisgliese Рік тому +19

      @@mujtabaalam5907 epic lateral thinking thanks

    • @gcewing
      @gcewing Рік тому +36

      You can always achieve that by weighting the wheel appropriately, so it's not a constraint on the wheel's shape.

    • @johnmount5487
      @johnmount5487 Рік тому +12

      That “force” exists even if the axle is at the center of mass. If the wheel is rotating at a constant angular speed the horizontal speed is by definition not constant (changing by a factor or r).the effect is exaggerated as the axle is moved away from the axle as the extremes of the bounds of the radius get larger. The wheels horizontal speed, speeds up and slows down constantly throughout its travel for any shape other than a circle

    • @aaaab384
      @aaaab384 Рік тому +2

      its*

  • @juanroldan529
    @juanroldan529 2 роки тому +346

    Awesome video! It's been just a few days since I have fallen in the rabbit hole of differential equations. I must say that I love your videos and that they inspire me to keep on improving and learning. Thank you!

    • @morphocular
      @morphocular  2 роки тому +49

      That's great! I'm so glad you found these videos so valuable. One of my hopes for this channel was to inspire others to learn and love math, so it pleases me deeply to be succeeding in that.
      I wish you the best on your continuing studies :)

    • @redtortoise
      @redtortoise Рік тому +1

      @@morphocular first

    • @maxwellhavoc6996
      @maxwellhavoc6996 10 місяців тому

      @@redtortoise I am confused by what you are trying to say.

  • @ineedtogetoutmore1848
    @ineedtogetoutmore1848 Рік тому +2

    that “Pivotal Role” pun at 11:14 was painful, well done

  • @alriktimo644
    @alriktimo644 Рік тому +3

    When I watched this video, I just realised that my intuition is strong that without even a mathematical description I can jump to the right conclusion, but at the same time I realised I lacked the ability to articulate since I didn't understand it mathematically or completely realising the fact that how this is so or 'How come?' in simple terms.
    I need to strengthen my mathematical comprehension of data into equations and other methods.
    Thanks 👍

  • @meade6291
    @meade6291 Рік тому +83

    The flaw in this is a vehicle with a continuous force applied through is engine to the axle wouldn't experience bumps in the x axis, but it would experience lurches and lags in it's movement on the y axis. Therefore it still would not be a comfortable drive unless the wheels rotational speed was constantly adjusted.

    • @eventhisidistaken
      @eventhisidistaken Рік тому +13

      Sure, but 'continuous force' was not specified. Yes, I'm an engineer.

    • @ob_stacle
      @ob_stacle Рік тому +4

      and if there's any wheelspin at all you'll be on the worst road in existance

    • @meade6291
      @meade6291 Рік тому +4

      @@ob_stacle holy shite I hadn't thought about that

    • @afoxwithahat7846
      @afoxwithahat7846 Рік тому +7

      I think you switched the axis, the axles aren't moving vertically at all.

    • @meade6291
      @meade6291 Рік тому +3

      @@afoxwithahat7846 yep, and I teach coordinate plane. Shame on me

  • @enbyarchmage
    @enbyarchmage Рік тому +41

    This video made me love catenaries even more, and I already considered them one of my favorite curves of all time! 🤩
    I like catenaries bc they appear everywhere, from the Brachistochrone problem to architecture. For instance, Catalan architect Antoni Gaudi took pictures of carefully arranged sets of hanging chains and turned them upside down to model the structure of the most famous church he designed, bc upside-down catenaries make EXTREMELY stable arches. Isn't that beautiful? 🥰

  • @RFVisionary
    @RFVisionary 2 роки тому +37

    Great tutorial. Good didactic structure. Instructive, helpful and optically "super nice" to look at.

  • @sozo8537
    @sozo8537 Рік тому +1

    The dopamine hit i got when i successfully calculated the equation of the road was something else. I thank you for presenting this problem.

  • @drmathochist06
    @drmathochist06 Рік тому +22

    Maybe you get to this later, but the "stationary rim property" also follows from the pivot principle. When the point in the wheel is the contact point itself, then any line through that point can do for the reference line in the orthogonal motion property. Only one possible velocity could be orthogonal to every line: 0.

  • @SophiaBrouchoud-se1ht
    @SophiaBrouchoud-se1ht 4 дні тому

    Who needs to spend thousands of dollars on therapy when you have this guy and his wheels? This genuenly sooths my brain and I love to learn things like this so yippy!

  • @danelyn.1374
    @danelyn.1374 Рік тому

    I feel like I've stumbled onto a video about a question that I never had in mind, and, along with an amazing explanation of the entire problem, has given me a solution that I am really satisfied by and solves that problem?
    plus the explanation is amazing so like, mad props

  • @mullactalk
    @mullactalk 2 роки тому +1

    This channel is a hidden gem of maths UA-cam

  • @amaarquadri
    @amaarquadri Рік тому +1

    Great video! You took an idea that seemed complicated at first and explained it so well that it seemed almost obvious in hindsight.

  • @AJMansfield1
    @AJMansfield1 Рік тому +38

    19:45 It seems like the road shape depends on how you parameterize the wheel's rotation then -- the function I always instinctively reach for when parameterizing straight lines in polar coordinates is the secant function, and I'd have written that line as { r(t) = sec(t), θ(t) = t }

    • @AJMansfield1
      @AJMansfield1 Рік тому +3

      (In fact, you can choose *any* θ(t) parameterization you want, and just use r(t) = sec(θ(t)) to get a straight line for whatever speed you rotate the wheel at.)

    • @ChariotduNord
      @ChariotduNord Рік тому +2

      This is interesting. I suppose you can get from your parameterization to his by the change of variables t → tan(t'). I wonder if this freedom of parameterization has any physical meaning.

    • @AJMansfield1
      @AJMansfield1 Рік тому +2

      @@ChariotduNord I went and simulated it, and the resulting road curves *are* actually different from each other.

    • @ChariotduNord
      @ChariotduNord Рік тому +14

      @@AJMansfield1 Oh, how did you simulate it? On my end, starting with your parameterization, I ended up doing the standard integral of sec(t) which is ln(|sec(t)+tan(t)|). I then plotted this parametrically on Desmos (typing in
      "(ln(|sec(t)+tan(t)|),-sec(t))"
      on the first line) with the domain [-π/2,π/2] for t. It already looked close to the catenary shape. But to make sure, on the 2nd line I put in his solution of y=-cosh(x), and the curves stack on top of each other rather exactly.

    • @dyld921
      @dyld921 Рік тому +7

      The parametrization of the road would change, but the shape (x-y relationship) wouldn't.

  • @LoganCralle
    @LoganCralle Рік тому +2

    Incredible video. I just took a dynamics course at university and I learned so much. This is an incredible application of maths. Bravo 👏

  • @saintgermain6694
    @saintgermain6694 2 роки тому +2

    I never expected it to be that intuitive! Thanks for the really really great video.

  • @dj_laundry_list
    @dj_laundry_list Рік тому +1

    What the hell is this? It's awesome. I think it would be more complete/satisfying to state that the vertical alignment property relies on shapes being convex, but honestly this is one of the best math(s) videos i've seen for a while

  • @bloomp7999
    @bloomp7999 Рік тому +1

    I deeply agree with your channel description and the Poincaré quote, i'm in for what you do, keep the good work !

  • @DitieBun
    @DitieBun Рік тому +1

    4:15
    This is the most insane wheel I've ever seen, and I'm here for it

  • @RFVisionary
    @RFVisionary 2 роки тому +1

    great tutorial. good didactic structure. instructive, helpful and optically "super nice" to look at.

  • @H3xx1st
    @H3xx1st Рік тому +1

    You explained that beautifully! I am definitely looking forward to your future videos.

  • @TheCynicalOne
    @TheCynicalOne Рік тому +5

    I want an entire video, or at least a short, dedicated to the orthogonal movement principle. It’s a mess and I want to dive in with full understanding! Great video about the wheels too. I feel like many of the wheels shown would slip a lot on their roads, so I guess the dream of bumpy square wheeled roads is a long shot lol.

    • @DonkoXI
      @DonkoXI Рік тому

      The proof he gave is actually pretty clean all things considered. If you are interested in understanding it, I highly recommend looking through it and trying to understand his reasoning one step at a time. You can ignore the algebraic details at first, but try to understand the concepts in the argument. If you understand the way complex numbers work well enough, it should all be pretty intuitive with some time. If you don't feel very comfortable with how complex numbers work, then stopping and thinking about each detail of this proof will actually be a pretty good way to get a better understanding of how they work.
      What feels clean to me is of course subjective though.

  • @philosophymikebill
    @philosophymikebill Рік тому +41

    Do you mind if I ask what programs/language/code you used to make this video? I'm attempting to learn this sort of simulation, but I'm not sure where to start.
    Thank you for making these videos. I've been trying to figure out this topic in my head for several years and this is the first meaningful insight I've come across in a good long while.

    • @morphocular
      @morphocular  Рік тому +27

      I actually used my own homemade software to make the animations in this video. You can find the software here if you want to play with it:
      github.com/morpho-matters/morpholib
      However, it's still largely just a personal project and the documentation is rather sparse. A more well-established and popular tool for making similar animations is called Manim, which you can find here:
      www.manim.community/
      Hope this helps :)

    • @philosophymikebill
      @philosophymikebill Рік тому +5

      @@morphocular I really appreciate the advice and even sharing your program! Thank you for getting back to me

    • @alexv1129
      @alexv1129 Рік тому +3

      @@morphocular Math is interesting and fun - but I am subbing because of this right here. Amazing of you to be so kind and helpful. Good luck, creator!

    • @ianhickey3423
      @ianhickey3423 Рік тому +3

      @@morphocular This is so unbelievably cool

  • @TerrifyingBird
    @TerrifyingBird Рік тому +1

    This problem (or rather a simpler version of the problem) ended up in an italian high school final exam, in 2017. It is to this day one of the most iconic problems to ever appear on the test.

  • @phlapjakz
    @phlapjakz Рік тому +1

    it always amazes me how e manages shows up everywhere even when the problem looks like it has nothing to do with it

  • @the25thdoctor
    @the25thdoctor Рік тому +1

    What I love about this is, it has a simple answer. Think gears, and a gear rack. But is far more complicated to preform

  • @brucea9871
    @brucea9871 Рік тому +5

    Very interesting video and analysis. I'll be watching more of your videos. This one reminds me of an old comic strip. It was called BC and based in prehistoric times. Their only form of transportation (other than walking) was what they called the wheel. It was a circular wheel with an axle through the centre and they stood on the axle to ride the wheel. (How they propelled it - especially uphill - is beyond me.) In one of the strip's comics (presumably before they thought of using circular wheels and hence only had square wheels) one character declares to another he has derived an improvement to the square wheel and produces a triangular wheel. "Improvement?", the second character says, confused. The first character replied "It eliminates one bump". But of course if they designed their roads as you specified they could actually have square or triangular wheels with no bumps. (Somehow I think it would be easier to come up with a circular wheel.)

  • @tsar_asterov17
    @tsar_asterov17 Рік тому +1

    This video is amazing, and all of his videos, ngl are basically 3b1b on light mode

  • @vikn331
    @vikn331 Рік тому

    This is the perfect example of "I have no idea what this man is talking about, but I like it"

  • @gergonagy846
    @gergonagy846 Рік тому

    I'm safe to say, that this is the most engaging video that I've ever watched.

  • @stuartl7761
    @stuartl7761 Рік тому +2

    6:10 I love that the first and last terms cancelled happily :D
    Loved the proof too, I must remember to check through if complex numbers might help when I come across a problem.

    • @morphocular
      @morphocular  Рік тому +3

      A good hint that complex numbers might help is if your problem involves 2D rotation or 2D rotational symmetry. That's where complex numbers often come in handy!

  • @user-wv1in4pz2w
    @user-wv1in4pz2w Рік тому +6

    I am pretty sure you can easily derive the pivot principle from the fact the contact point is stationary:
    observation 1: the wheel is a 2D rigid body, so its motion is fully described by horizontal speed, vertical speed, and rotational speed, so it has 3 degrees of freedom.
    observation 2: the constraint that the contact point is stationary restricts 2 degrees of freedom, thus leaving 1 degree of freedom.
    observation 3: pivoting motion satisfies the stationary contact point constraints and has 1 degree of freedom.
    therefore pivoting motion is the only possible way to satisfy the stationary contact point constraint.

    • @kindlin
      @kindlin Рік тому +1

      When he said it was _really hard to prove_ I was confused, as this is the only motion available due to the no-slip-condition and the rigid body motion.
      But honestly, the statement of the question itself is almost the proof of the question. You want to figure out how to prove that all points on the wheel move periductular around the contact point, well, proof by exhaustion, there are no other ways it could move around the contact point but to pivot, and the definition of pivoting, as noted in this video, is perpendicular motion about a point.

  • @deathpigeon2
    @deathpigeon2 Рік тому +8

    While a flat ride is certainly an important thing for a smooth ride, I'm not convinced it's sufficient. It seems reasonable to describe a jerky ride as also a non-smooth ride. That is to say, given a constant torque applied to the wheel, the third derivative (the jerk) of the forward motion produced by the wheel spinning should be precisely equal to zero.
    Put another way, a linear acceleration of the rotation of the wheel should produce linear forward acceleration for the whole system.
    Now, I think the stationary rim principle should be sufficient to ensure that this is the case because it ensures that the rim speed and the axle speed are equal, but I think it'd be insufficient to consider only the flatness of a ride to determine if it's properly a smooth ride.

    • @klikkolee
      @klikkolee Рік тому +5

      We are used to vehicles which are propelled by the wheels. However, if the vehicle is moved by means unrelated to its wheels, then the criterion in the video is sufficient. For vehicles which are wheel-propelled, unless a fanciful control system regulates the wheel speed, your additional criterion is required to make the vehicle feel subjectively smooth to a real human occupant.
      The no-slip condition (stationary rim principle in this video) does *not* guarantee your criterion. The r in the no-slip equation is a function of t. Your criterion is only consistent with the no-slip condition if the radius is constant -- meaning a circular wheel.

    • @deathpigeon2
      @deathpigeon2 Рік тому

      @@klikkolee ...Right. I was thinking it'd ensure 0 jerk because it ensures that the rotational velocity at the touching point and the forward velocity at the axel are the same, but, for constant torque, the velocity at the touching point would be in part a function of the distance from the axel so you *need* at least some slipping to ensure a smooth ride unless you have a constant distance from the axel (ie being a circle as you said).

    • @Nuclear868
      @Nuclear868 Рік тому +1

      What if, in case of a car, we make the distance between the front and the rear wheels such that front and rear wheels are offset - when the front wheels have the highest angular speed, the rear wheels have the lowest angular speed? Yes, they will not cancel out completely, but will reduce the 'jerk' feeling.

    • @eventhisidistaken
      @eventhisidistaken Рік тому +2

      Who said the torque had to be constant? Stop trying to impose your roundism on the rest of us.

    • @klikkolee
      @klikkolee Рік тому

      @@eventhisidistaken It would be a substantial engineering challenge to create a vehicle where the torque applied by the wheels varies in perfect concert with the road shape. Without that perfection, a wheel-propelled vehicle can't have a smooth ride on an extreme road without slipping.

  • @duncanhw
    @duncanhw 2 роки тому +5

    Great video! Love how you started by making the equations and then deriving the shape from them! Can't wait for the next video.
    also, wouldn't the wine glasses in the thumbnail be knocked forward/backward due the second law of road-wheel motion?

  • @adrianmisak07
    @adrianmisak07 2 роки тому +1

    fantastic video… cant even express how impressive this is to me, I try to do math recreationally after getting my masters in applied math…

  • @thirockerr
    @thirockerr Рік тому +5

    Nice video ! I would be interested to see how you would present the optimal road shape taking into account a specific mass for the wheel, the gravitationnal force.

  • @NoOffenseAnimation
    @NoOffenseAnimation Рік тому +1

    Great video, I like to wonder what this would look like in practise, if someone were to try this in the real world, but of course there would be a great deal of other things to consider

  • @mateuszbaginski5075
    @mateuszbaginski5075 2 роки тому

    I can't really point to what it is in your videos that makes them one of the best I discovered through 3B1B's SoME. Whatever it is, you are grokking it, man.

  • @blackboxlearning
    @blackboxlearning 5 місяців тому

    I was hoping to make a video on this exact topic, but I guess it has already been beautifully covered by this channel. While checking for that, I came across this channel and I love the animations and their interactivity. Already subbed. Expect a video soon covering more stuff, cus I'm not leaving the idea :)

  • @arlyu606
    @arlyu606 9 місяців тому

    I reeeally love your content. Thank you for all your videos :-)

  • @sriramn1809
    @sriramn1809 Рік тому

    First video ive seen on this channel.
    Wondering why youtube took so long to recommend me stuff from here.
    This channel is amazing!

  • @bitroix_
    @bitroix_ Рік тому +1

    This is an amazing video! Thank you.

  • @jursamaj
    @jursamaj Рік тому +10

    The horizontal motion of the axle is necessary for a smooth ride, but not sufficient. It needs to be *smooth* horizontal motion, not jerking forward and back. That, in turn, requires the wheels to rotate at highly variable speed. But that's not how driven axles tend to work.
    Additionally, when the wheel is moving up the slope, the wheel will be moving too fast at any given moment. Combined with the uphill configuration, you basically guarantee slippage. You face a similar problem on the downhill side, but inverse. Those novelty tourist attractions tend to reduce both these effects by having the front & back wheels be exactly a half wave out of phase. That way the slippage either way hopefully cancels out, and one axle can be speeding up while the other is slowing down.

  • @LunaAlphaKretin
    @LunaAlphaKretin Рік тому +16

    I'm curious what would happen if you impose the additional restriction of making the axle's horizontal speed (and, hence, velocity) constant (given constant rotation speed). I noticed the speed seemed to vary a lot with that particularly arbitrary-shaped wheel example at 4:18, which would probably be a disconcerting experience as a driver. Still I imagine the answer is that you can't have a road that does both - to prevent a change in horizontal speed you'd probably need a different road that causes vertical changes. What if we just say "constant velocity", allowing the vertical position of the axle to change as long as it feels like a smooth slope would for a circle-wheeled driver. I don't know how that would go, but it feels more likely to be possible.

    • @WaluigiisthekingASmith
      @WaluigiisthekingASmith Рік тому +4

      The second equation says dx/dx =rdtheta/dt. Differentiating a second time d^2x/dt^2= dr/dt dtheta/dt +r dtheta^2/dt^2. Given your restriction dr/dt dtheta/dt = -r dtheta^2/dt^2. Thus r'/r =u'/u. Doing what any good physicist would do and pretending we can just cancel our differentials like fractions, we get ln(r *dtheta)= c and thus dtheta/dt =c/r

    • @joaogiorgini1326
      @joaogiorgini1326 Рік тому +4

      Make velocity constant with constant rotational speed? In other words, dx/dt=cte and d0/dt=cte. Meaning, in the second equation, r must also be a constant.
      In other words, the only shape that satisfies a truly smooth ride is a circle.

    • @bears7777777
      @bears7777777 Рік тому +1

      I think the only way this would be possible would be to allow wheel slip. The amount of slip would be the fastest angular speed - slowest angular speed. The slip would have to occur when the point of contact is farther than the minimum. For the square, this would be when the point of contact tends towards the corners as they are farther from the center then the center of a side. I’m not sure that’s even solvable though

    • @scifiordie
      @scifiordie Рік тому

      Nobody cares bro get a life

  • @officiallyaninja
    @officiallyaninja 2 роки тому +2

    this video is so good. its criminal that you don't have hundreds of thousands of subs

    • @Happy_Abe
      @Happy_Abe 2 роки тому

      In time we’ll get this channel there

  • @convincingmountain
    @convincingmountain Рік тому +1

    very nice video, i really enjoyed the small steps taken each time to get to the answer. and even then, there's so much more to discover! well presented and paced, didn't feel like half an hour. your consistent use of both visual and verbal explanations for each new idea is great.

  • @lenskihe
    @lenskihe Рік тому +2

    Awesome 👍 I tried to solve this problem on my own once. I'm glad I watched this video, because now I know that I would never have been able to do it 😂

  • @karllenc
    @karllenc 4 місяці тому

    amaizing! just amazing video! Thanks

  • @miguelcabaero5843
    @miguelcabaero5843 Рік тому

    I love the production quality

  • @Adam-pj2qh
    @Adam-pj2qh Місяць тому

    thats so sick, finally some applied mathemathics!!!

  • @firiasu
    @firiasu 9 місяців тому

    So good explanation!

  • @pastadcasta
    @pastadcasta Рік тому +1

    I have a way I like to think about it, if you take the path that the axle takes when the shape is rolled continuously over a flat surface, and use that for the road surface, the shape will roll smoothly. It's cool to see the algebraic representation of that though.
    Very cool video! ^^

    • @steffahn
      @steffahn Рік тому +2

      A square wheel rolled over a flat surface will actually just pivot around each of the 4 corners. Thus, the axle would take a path composed of a series of arcs (i.e. sections of the perimeter of a circle), which is definitely *different* from the series of catenaries that are shown in this video to be the shape of road that you need.

  • @MF-dz7cp
    @MF-dz7cp Рік тому +2

    I'm a sophomore in high school so I have no clue what this video is talking about but it's still interesting

  • @iskallman5706
    @iskallman5706 Рік тому

    This is as good as mathematics vidéos get. The pinacle.

  • @WeeIrishLaddie1
    @WeeIrishLaddie1 Рік тому +2

    I'd be interested in a sister video where "smooth" was defined as "constant velocity" rather than "constant axel height", ie changing the axel height in the wheel as it rolls to keep it moving horizontally at constant speed

  • @vitorguilhermecoutinhodeba3253

    It is a nice video, even though I think some properties have different names in here. Instant center of rotation is the center (no pun intended) of all this procedure, and wasn’t mentioned. The animations were very good!

  • @redyau_
    @redyau_ Рік тому +1

    The way you use - I assume - MAnim is absolutely outstanding. I bet you come to understand every concept you explain in an incredible depth as you code these. Really impressive!

  • @g10royalle
    @g10royalle Рік тому

    The animations are so satisfying

  • @user-hd9oh9bk8b
    @user-hd9oh9bk8b Рік тому

    it's honestly fascinating how many titles and thumbnails this exact video's had. i've heard about this but never gotten to see it firsthand

  • @Danker1248
    @Danker1248 Рік тому

    this was a very enjoyable video

  • @k7iq
    @k7iq Рік тому +4

    This is fantastic ! 2 + 2 = 5 for large values of 2
    But would a square wheel do good in snow or maybe even ice ?

  • @ANZEMusic
    @ANZEMusic Рік тому +5

    This is a really good video. The math is fascinating, and you present it clearly with exceptional visuals, and I greatly appreciate it

  • @agy3256
    @agy3256 Рік тому

    This video is pure gold

  • @kinkinawesome
    @kinkinawesome 2 роки тому +2

    Exited for the next videos!

  • @nalat1suket4nk0
    @nalat1suket4nk0 Рік тому

    Awesome!!! You used math and physics so cool

  • @nerdsgalore5223
    @nerdsgalore5223 Рік тому

    This is a beautiful video.

  • @iamtraditi4075
    @iamtraditi4075 2 роки тому

    I know I'm late, but this is really good!

  • @Josephi_Krakowski
    @Josephi_Krakowski Рік тому

    These are the type of videos I watch at 3 AM

  • @Error-xl3ty
    @Error-xl3ty Рік тому

    Videos like this are why I love math

  • @Cesar-ey7wu
    @Cesar-ey7wu Рік тому +1

    You would actually feel "bumps" in a square wheeled car because for a constant speed, the rotational speed of square varies (you can see it in the video : it's accelerating when it gets to the side of the square and slowing down on the corner). So if your engine is putting a constant torque to the wheel, the car's acceleration would vary four times for each wheel rotation, which wouldn't be comfy at all.

  • @WAMTAT
    @WAMTAT Рік тому

    Great video. You've earned a subscriber

  • @archie1490
    @archie1490 Рік тому

    This was a nice brain teaser before I go off to do maths at uni. GL everyone off to uni in Septemember!

  • @darealmrog
    @darealmrog Рік тому +1

    Congrats! You just reinvented a train!

  • @abcdaiy
    @abcdaiy Рік тому +1

    KEEP IT UP BRO U ARE DOING GREAT WORK ❤❤❤❤❤

  • @benjaminstandfest6265
    @benjaminstandfest6265 Рік тому

    This is awesome

  • @jorgec98
    @jorgec98 Рік тому

    I'm kinda proud of myself I grasped the first analytical definition more easily than the second visual one

  • @nano_shinonome
    @nano_shinonome Рік тому

    so cool!!

  • @ParadoxProblems
    @ParadoxProblems Рік тому +1

    If you want a non circular wheel that moves with constant speed, you can give the wheel a non uniform mass density such that when the wheel would slow down, the part on the bottom that is moving slower is made more massive. It's momentum is transfered to the entire wheel body, maintaining a constant velocity.
    Most likely the mass distribution would be such that every dTheta slice around the axle has the same mass regardless of radius.
    (Constant moment of inertia)

  • @gabrielecusato4705
    @gabrielecusato4705 Рік тому

    Very nice and interesting video

  • @elitestryker5709
    @elitestryker5709 Рік тому

    Its so much more understandable than PHysics I and Technischemechanik at ETH together eith explaining all the concepts

  • @ZotyLisu
    @ZotyLisu Рік тому

    this should have way more views

  • @szeartur4813
    @szeartur4813 Рік тому

    great video, good job :D

  • @cut2000trees
    @cut2000trees Рік тому +1

    To demonstrate the stationary rim property, you could imagine a wheel with a notch cit out on the edge. If, say, a squirrel was in the road, and it positioned itself to line up with the notch, it would be safe. Because the notch wouldn't move, it would be stationary

  • @oskarandreasolsen495
    @oskarandreasolsen495 Рік тому +1

    When describing the small timesteps in the visual prrof at the end, maybe it would be more specific using a Δt->dt for smaller and smaller time steps :) But great video! really liked it!

  • @luish2161
    @luish2161 Рік тому

    Nice and fun video :)

  • @TomatoBulb
    @TomatoBulb Рік тому

    I have absolutely no idea what any of this means but I find it interesting

  • @RagingBadger68
    @RagingBadger68 4 місяці тому

    While I don’t understand 80% of what I’ve been told here, I did finally understand the purpose of imaginary numbers here.
    I’ve struggled through so many math classes which could never just explain it so effectively.

  • @manifaridi9200
    @manifaridi9200 Рік тому +2

    I hate mathematics but man... look how beautiful it is.

    • @WAMTAT
      @WAMTAT Рік тому +2

      You don't hate math, you just hate how it was taught to you.

    • @blackbeast9268
      @blackbeast9268 Рік тому +1

      Trust me nobody hates math. I used to have E and was on my way to F but then i moved school and my teacher was amazing and i got A because he explained everything so well and got me motivated. Math is a language with rules and if you're teacher doesn't explain the rules in details it will be boring because you rely on common methods and formulas instead of understanding why they work.
      It's very fun and i would argue chemistry or physics are much harder then any math expect super high level .

  • @Happy_Abe
    @Happy_Abe 2 роки тому +5

    Why isn’t x equal to the hyperbolic sin plus a constant because we integrated
    Where’s the +c

    • @morphocular
      @morphocular  2 роки тому +6

      Man! Nothing gets past you guys! :) Yes, technically a +C belongs there, but all it will do is shift the road forward or backward; it won't affect the actual shape of the road, which is what we were after. So I picked C = 0 in order to get the simplest and cleanest possible final answer.

    • @Happy_Abe
      @Happy_Abe 2 роки тому +2

      @@morphocular Thanks!
      Yeah I didn’t think it was a problem for the solution, I just thought it should have been addressed. Amazing video, can’t wait for your channel to explode in popularity!😊

  • @johnnyvishnevskiy8090
    @johnnyvishnevskiy8090 Рік тому +3

    I'm more interested in how this transfers over to 3 dimensions and how turning affects how the shape of the road is made.

  • @rebeccastevens2903
    @rebeccastevens2903 6 місяців тому

    Wheels are paradise. The wheel groups form paradise!

  • @festabrawl
    @festabrawl Рік тому

    Amazing

  • @tracy449
    @tracy449 2 роки тому +10

    Thanks for the video. I learned a lot. Also, I have a question: If the axle moves at a constant velocity, does the wheel rotate with a constant angular velocity?

    • @morphocular
      @morphocular  2 роки тому +17

      Thanks for watching! To answer your question: Not necessarily! The second Road-Wheel equation says the axle's velocity is dx/dt = r dθ/dt, where dθ/dt is the angular velocity. So the only way both the axle velocity and the angular velocity can be constant is if the wheel has a constant radius, meaning this will only happen for the case of a circular wheel.

    • @TheHuesSciTech
      @TheHuesSciTech Рік тому

      @@morphocular Fascinating -- I *believe* an involute rack and pinion has the property of dx/dt = k dθ/dt, where k is a fixed property of a given gear (the radius of the gear's "pitch circle", or half the "pitch diameter", perhaps?). This would appear to contradict the statement you made above, but I believe that might be because you're assuming no slippage between the wheel and road in your video, whereas an involute rack and pinion does have slippage?

    • @cheshire1
      @cheshire1 Рік тому

      @@TheHuesSciTech The equation you gave is approximately true, since a gear is pretty close to a circle.

    • @TheHuesSciTech
      @TheHuesSciTech Рік тому

      @@cheshire1 It's approximately true for all gears, yes. But I believe it's *precisely* true for an involute gear. (Neglecting real-world clearances and manufacturing tolerances, of course.)

    • @cheshire1
      @cheshire1 Рік тому

      ​@@TheHuesSciTech You may be right, involute gears do have slippage (and the contact point jumps around instead of staying on a vertical line), so the argument from the video doesn't work in their case.

  • @locryStudios
    @locryStudios 8 днів тому

    Epic!!!! ❤❤❤

  • @NoName-rd6et
    @NoName-rd6et 2 роки тому +8

    interesting