Olivia Pomerenk, New York University – Hula Hoop Levitation

On New York University Week:  How does a hula hoop work?

Olivia Pomerenk, Ph. D candidate in mathematics, looks at the science.

Olivia Pomerenk is a fifth-year graduate student at the Courant Institute of Mathematical Sciences at NYU, working towards a Ph.D. in mathematics after receiving a bachelor’s degree in applied mathematics from Caltech. Olivia is interested in the dynamics and interactions of fluids and solids.

Hula Hoop Levitation

Humans have been hula-hooping for millennia. The practice of hula-hooping as recreation, exercise, or even religious ceremony appears in historical records as early as 500 BCE, resurfacing across myriad cultures throughout history. Despite its enduring presence, the dynamics of hula-hooping have not been thoroughly studied. The most fundamental question remains unanswered: how does the hoop stay up and “levitate” against gravity?

Using a combination of modeling and experiments, we set out to finally resolve this mystery. We 3D-printed different small plastic “body shapes”: a cylinder, a cone, and a hyperboloid (an hourglass). Each shape was driven to gyrate, mimicking the motion of a hula-hooping human, and small plastic hoops were twirled around them. Results revealed striking differences. The cylinder, with straight sides, was completely ineffective: no matter the speed of gyration, the hoop inevitably slid down and fell. The cone, with sloped sides, fared slightly better but still failed. The hoop either spun up and flew off the point, or it slid downward and fell. Only the hourglass shape successfully sustained hula-hooping. The hoop consistently settled below the hourglass’s narrowest point, or “waist,” and remained indefinitely trapped there as long as gyration continued.

Using a mathematical model, we confirmed that body shape is the key to successful hula-hooping. Two conditions must be met: the body must have sufficient slope (to provide an upward force counteracting gravity) and sufficient curvature (to trap the hoop in place). On a human, this corresponds to hips (for slope) and waist (for curvature). These findings resolve the mystery of hula-hooping, shedding light on an enduring human pastime, and offer broader insight into the mechanics of “gripping” an object via gyration, which could be useful in harvesting energy and improving robots.