Are you in the right shape for hula hooping? Mathematicians prove it →

Compiled by: Gong Zixin

Mechanical systems with moving contact points (including rolling, sliding, and impact) are common in engineering applications and daily experience. The challenges of analyzing such systems become more complex when objects dynamically explore the complex surface shapes of moving structures, just like the familiar but not well understood hula hoop. While enjoying the fun, we may overlook some interesting issues it brings:

Why can the hula hoop defy gravity? What body type is better at hula hooping?

A team of mathematicians at New York University has explored and answered these questions, explaining for the first time the physics and mathematics of the hula hoop. Their findings also point to new ways to better utilize energy and improve robot positioners. The results of the study were published in the Proceedings of the National Academy of Sciences.

"We were particularly interested in what kinds of body movements and shapes could successfully support a hula hoop, and what physical requirements and constraints would be involved," says Leif Ristroph, an associate professor at NYU's Courant Institute of Mathematical Sciences and senior author of the paper.

To answer these questions, researchers at the New York University Applied Mathematics Laboratory simulated a hula hoop. They tested different shapes and movements in a series of experiments with robotic hula hoops, using 3D-printed bodies of different shapes (e.g., cylinders, cones, hourglass shapes) to represent 1/10 the size of a human body. These shapes are rotated by motors to mimic the movements of people when spinning a hula hoop, and a ring with a diameter of about 6 inches (about 15 cm) is projected onto these bodies, and high-speed video can capture these movements.

The results showed that the exact form of the rotational motion or the cross-sectional shape of the body (circular or elliptical) was not a factor in hula hoop movement. "In all cases, it did not require any special effort to establish a good rotational motion around the body with the hula hoop," Ristroph explained.

However, keeping the hula hoop elevated against gravity for a considerable period of time is difficult and requires a special "body shape" - "hips" with sloping surfaces to provide the proper angle to propel the hoop, and a curved "waist" to hold the hoop in place.

"People have many different body shapes, and some have sloped and curved hips and waists, while others don't," Ristroph said. "Our findings may explain why some people are naturally good at hula hooping, while others seem to have to work harder."

The researchers mathematically modeled these dynamics to derive formulas that explain the experimental results. These calculations not only explain the physics of hula hoop movement, but can also be used in other fields.

Schematic

The researchers expressed surprise that an activity as popular, fun and healthy as hula hooping was not understood at a basic physics level.

"As we progressed in our research, we realized that the math and physics involved in hula hooping are very subtle, and the knowledge we gained could help inspire engineering innovations, harvest energy from vibrations, and improve robotic positioners and movers used in industrial processing and manufacturing," Ristroph said.