A team of mathematicians at NYU has provided the first detailed explanation of the dynamics behind hula hooping, using robotics and 3D printing to test various body shapes and motions.
They discovered key physical traits that make some people better at maintaining a hoop’s elevation, offering potential applications in engineering and robotics.
Unraveling Hula Hooping Mechanics
Hula hooping is such a familiar activity that we often overlook the intriguing questions it poses: “What keeps a hula hoop up against gravity?” and “Do certain body types make hula hooping easier?” A team of mathematicians set out to answer these questions, uncovering insights that not only explain the mechanics of hula hooping but also suggest new ways to harness energy and enhance robotic systems.
This groundbreaking research marks the first comprehensive explanation of the physics and mathematics behind hula hooping.
“We were specifically interested in what kinds of body motions and shapes could successfully hold the hoop up and what physical requirements and restrictions are involved,” explains Leif Ristroph, an associate professor at New York University’s Courant Institute of Mathematical Sciences and the senior author of the paper, which appears in the Proceedings of the National Academy of Sciences.
To answer these questions, the researchers replicated, in miniature, hula hooping in NYU’s Applied Mathematics Laboratory. They tested different shapes and motions in a series of experiments on robotic hula hoopers using 3D-printed bodies of different shapes (e.g., cylinders, cones, hourglass shapes) to represent human forms at one-tenth the size. These shapes were driven to gyrate by a motor, replicating the motions we take when hula hooping. Hoops approximately 6 inches in diameter were launched on these bodies, with high-speed video capturing the movements.
This movie is taken from high-speed video of the experiments on a robotic hula hooper, whose hourglass form holds the hoop up and in place. Credit: NYU’s Applied Mathematics Lab
Experimental Insights and Findings
The results showed that the exact form of the gyration motion or the cross-section shape of the body (circle versus ellipse) wasn’t a factor in hula hooping.
“In all cases, good twirling motions of the hoop around the body could be set up without any special effort,” Ristroph explains.
However, keeping a hoop elevated against gravity for a significant period of time was more difficult, requiring a special “body type”—one with a sloping surface as “hips” to provide the proper angle for pushing up the hoop and a curvy form as a “waist” to hold the hoop in place.
This animation shows the top-down view of the hoop twirling motions from a computer simulation. Credit: NYU’s Applied Mathematics Lab
Implications for Engineering and Robotics
“People come in many different body types—some who have these slope and curvature traits in their hips and waist and some who don’t,” notes Ristroph. “Our results might explain why some people are natural hoopers and others seem to have to work extra hard.”
The paper’s authors conducted mathematical modeling of these dynamics to derive formulas that explained the results—calculations that could be used for other purposes.
“We were surprised that an activity as popular, fun, and healthy as hula hooping wasn’t understood even at a basic physics level,” says Ristroph. “As we made progress on the research, we realized that the math and physics involved are very subtle, and the knowledge gained could be useful in inspiring engineering innovations, harvesting energy from vibrations, and improving in robotic positioners and movers used in industrial processing and manufacturing.”
Reference: “Geometrically modulated contact forces enable hula hoop levitation” by Xintong Zhu, Olivia Pomerenk and Leif Ristroph, 30 December 2024, Proceedings of the National Academy of Sciences.
DOI: 10.1073/pnas.2411588121
The paper’s other authors were Olivia Pomerenk, an NYU doctoral student, and Xintong Zhu, an NYU undergraduate at the time of the study.
The work was supported by a grant from the National Science Foundation (DMS-1847955).
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