Mechanical systems with moving contact points, like rolling, sliding, and impacts, are standard in engineering and daily life. Analyzing these systems becomes more complex when objects interact with the changing surfaces of moving structures, as seen in activities like hula hooping.
A new study from New York University explores the physics and mathematics behind how a hula hoop can defy gravity, revealing that certain body types may be better suited for hula hooping.
The research identifies hula hooping as mechanical levitation driven by rolling points of contact and influenced by body shape. Mathematicians involved in the study also highlighted potential applications for improving energy efficiency and robotic positioning. This is the first study to explain the physics and math of hula hooping.
Leif Ristroph, an associate professor at New York University’s Courant Institute of Mathematical Sciences and the senior author of the paper, said, “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.”
To explore the physics of hula hooping, researchers at NYU’s Applied Mathematics Laboratory conducted miniature experiments with robotic hula hoopers. Using 3D-printed bodies of various shapes—cylinders, cones, and hourglasses—representing human forms at one-tenth size, they replicated the gyration motions of hula hooping.
How do animals stay ‘in shape’?
A motor drove the gyrations, while about 6 inches in diameter hoops were launched and captured on high-speed video. The results revealed that neither the exact gyration motion nor the body’s cross-sectional shape (circle vs. ellipse) affected the ability to hula hoop.
Ristroph explains, “In all cases, good twirling motions of the hoop around the body could be set up without any special effort.”
Keeping a hula hoop elevated for a prolonged period proved more challenging. It required a specific “body type”—one with sloping “hips” to push the hoop upward and a curvy “waist” to hold it in place.
People with these traits may find hula hooping easier, while others may struggle more. The study’s authors also used mathematical modeling to derive formulas explaining these dynamics, which could have broader applications beyond hula hooping.
Ristroph said, “We were surprised that an activity as popular, fun, and healthy as hula hooping wasn’t understood even at a basic physics level. 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.”
Journal Reference:
- Xintong Zhu, Olivia Pomerenk et al. Geometrically modulated contact forces enable hula hoop levitation. Proceedings of the National Academy of Sciences. DOI: 10.1073/pnas.2411588121
