How hula hoops stay aloft

Welcome to Science Sessions, the podcast of the proceedings of the National Academy of Sciences, where we connect you with Academy members, researchers, and policymakers. Join us as we explore the stories behind the science. I'm Paul Gabrielson. For those skilled at hula hooping, a simple motion of the hips holds the hoop in a remarkable state of levitation. Although the hula hoop is a simple children's toy, the physics behind hula hoop levitation are somewhat complex. In a recent PNAS study, Olivia Palmer ink of New York University and colleagues studied the physics behind how hula hoops stay up, first with robotic models, and then with mathematical modeling. Body shape, they found, is key to counteracting gravity and keeping a hula hoop spinning. Olivia, where did the idea to study hula hoop motion come from? There's evidence of hula hooping in the human historical record as far back as 500 BCE. It shows up again and again throughout history, and in a myriad of cultures as a form of recreation, the logistic ceremony, exercise, you name it. And even today, artists and performers can do some really impressive stuff with hula hoops. And so you'd think that this problem has sort of been studied to death at this point, but it actually hasn't. So we dug into the literature at the start, purely driven by curiosity, and we found that hula hooping really isn't understood even at a basic physics level. Previous physical models for hula hooping have generally been restricted to a 2D cleaner problem rather than looking at the full 3D system. And so the 2D dynamics are very interesting in their own right, but they fundamentally ignore the most glaring question, which is really how does the hoop stay up in the first place? And so that's the question that we set out to answer. Why is the movement of a hula hoop fascinating? Also naively, it would seem that all of the forces exerted during hula hooping are directed horizontally. Your body seems like it's just pushing the hoop outwards as you swivel your hips. But the hoop obviously has mass, so it's being pulled down by gravity, and it's not really immediately clear how the hula hoop is levitating then, because something has to be counteracting that downward gravitational force. And so quantifying that mysterious something turns out to be very interesting and actually involves some rather subtle and pretty elegant physics. Did you have a hula hoop as a child? I did have a hula hoop as a child. I definitely enjoyed it growing up, and we have a hula hoop in the lab actually. So once or twice while we were kind of working on this study, I think I did take it out and gave it a spin. Pun intended. Tell us about your experimental setup. How did you construct robotic hula hoopers? We used experiments early on in the research to build intuition about the various factors which may or may not contribute to stable hoop levitation. So we sort of guessed at the outset that the shape of the body, executing the hula hooping might be important. And so to start playing around with this, we 3D printed several different axi-symmetric body shapes to test. These were all just plastic, maybe about the size of your hand, and the 3 main ones were a cylinder, so just straight up and down, a cone pointy tip up, and what's called a hyperboleid, which you can think of as just being an hourglass shaped body. And so these were rubberized so that a plastic hoop could roll along the body without slipping. And then each body shaped in turn we mounted on a vertical axle, which was then gyrated. By gyrating there, I mean translating in a circle without rotating. So we drove a constant gyration speed robotically, and then just by hand tossed plastic hoops around the gyrating bodies to initialize hula hooping. And then from there the experiment is hands off and we just watch the dynamics of the hoop as it continues to traverse this gyrating body. How did a mathematical model help you understand the experimental results? To begin by looking at just the planar dynamics, so this is sort of a top-down view of the hula hooping dynamics. And did a pretty extensive study of the necessary physics that's happening during that gyration with rolling of the hoop. We, from there, identify that there's this universally attracting steady state that is reached in two dimensions. And so then to build in the third dimension of dynamics, the height, the z-axis, we use that 2D steady state that we know is attained as the base state to then construct a three-dimensional model. What forces create a successful hula hoop motion? In order to successfully hula hoop, you basically need three conditions. First, very simple, you need your initial speed of the hula hoop to be sufficiently fast. And there isn't really a too fast. You just toss it quickly and you're good to go. And this matters for the 2D planar dynamics. Basically, a sufficiently fast throwing speed ensures that your hoop will remain in constant contact with the body throughout as time goes on. The next kind of condition that you need is that your body needs to be sufficiently sloped. So this slope is what's generating an upward local force on the hoop at the contact point. And that's the force that's balancing gravity. So that's how the hoop actually levitates. There's a third condition that you need your hoop to actually stay in position. You need it to be trapped at that particular height, which is giving rise to a levitational force. So in order for this to occur, your body shape needs to have sufficient curvature at that spot. So fast enough, sloped, curved are the three things that you need. How does the shape of the gyrating body affect hula hoop success? So recall, we tested three body shapes, a cylinder, a cone, and a hyperboloid, or an hour glass. So a cylinder has no slope and it has no curvature. Right? Sides are straight up and down. And so it turns out, indeed, in experiments and in our model, we show that a cylindrical body can never support stable hula hooping. No matter how fast it gyrates, the hoop always descends downwards because there's no slope, there's no upward force that can balance gravity. So the cylinder is kind of a no go. A cone on the other hand has slope, its sides are angled, but it has no curvature. The problem is that it can't trap the hoop in place. So in experiments, if you start the hoop too far near the top, the pointy end of the cone, there's almost too much lift and your hoop will just ascend until it hits the top and flies off of the body. If you start your hoop too low, similarly, there's not enough lift now and it'll descend down until it falls off the body. So there's no trapping that happened. But the hourglass shape has slope and it has curvature. The hoop gets trapped right below the narrowest point on the hyperboloid. It's like it was right below your waist. And so this particular sweet spot on a hyperboloid has the requisite slope and the requisite curvature to lift a hoop and keep it trapped in place. What can we do now with these findings? I'm not an engineer and not a roboticist. So I don't want to make too strong a claim about applications. But I guess beyond the scope of just tool hooping, our findings provide a general model for gyration-induced contact forces which can give rise to levitation. We have through this work developed an understanding of how to hold something without ever actually gripping it. So that's kind of cool in and of itself. And I could envision applications in robotics or maybe manufacturing, thinking about items that could be maybe transported vertically, up or down, simply by spinning them around an axle without ever actually picking them up in a traditional manner. What are the caveats or limitations of the study? In this work we're not addressing every aspect of the full 3D problem. We're treating the 2D dynamics as somewhat independent from the vertical dimension. So the fully 3D problem would probably involve couplings among each of the different dimensions, some of which we are neglecting here. So a future model could address these couplings and try to understand the mechanic, maybe of exactly how the hoop climbs or falls along a body, which we do not go into if it all in this work. There are also some other potentially relevant physical effects such as air resistance, which are neglected in this work and could be considered in a future iteration of this model. 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