Getting the ball rolling

How gravity causes a perfectly spherical ball to roll down an inclined plane is part of elementary school physics canon. But the world is messier than a textbook.   

Scientists in the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS) have sought to quantitatively describe the much more complex rolling physics of real-world objects. Led by L. Mahadevan, the Lola England de Valpine Professor of Applied Mathematics, Physics, and Organismic and Evolutionary Biology in SEAS and FAS, they combined theory, simulations, and experiments to understand what happens when an imperfect, spherical object is placed on an inclined plane.

Published in Proceedings of the National Academy of Sciences, the research, which was inspired by nothing more than curiosity about the everyday world, could provide fundamental insights into anything that involves irregular objects that roll, from nanoscale cellular transport to robotics.

“We go about the world seeing just about what everyone else sees,” Mahadevan said. “But if we choose to pause and wonder even as we wander, we learn about the world, and perhaps even about ourselves. Drawing connections between different fields of mathematics and physics by exploring this simple problem was fun – who knows, it might even turn out to be useful one day.”

The authors started with simulations of slightly irregular objects (either spheres or cylinders) rolling down various degrees of incline, noting that an irregularly shaped object does not always roll, whereas a uniform object will just roll along. The steeper the ramp, the more likely the object rolls; as the ramp flattens out, the more likely the object stops. The transition from not-rolling-to-rolling, which happens at a critical angle of inclination, is where some interesting physics is seen, said first author Daoyuan Qian, a former research fellow in Mahadevan’s group.

“Indeed the behavior of the object near the transition angle, or a critical point, has the features of a phase transition, or bifurcation, which separates two qualitatively distinct   states – rolling and not rolling,” Qian said.

Near the phase transition, the terminal rolling speed serves as a simple measure of  “order,” and the authors found that the rolling speed changes depending on factors like the dimensions of the object and its inertia. For example, they showed how the time period of rolling diverges, or increases to infinity, near the transition, and how the system settles into a stable rolling motion away from the critical point. Cylindrical objects were predicted to behave differently from spherical objects because there many ways for a sphere to roll, but just one way for a cylinder to roll. Think about the difference between how a baseball would roll down an incline versus a paper towel roll.

To test their calculations, the authors took to the lab, observing irregular rolling cylinders and spheres on different inclines, and they showed that their results match their calculations for the behavior near the onset of motion.

While experimenting with irregularly shaped spheres, they saw some things they didn’t expect, “but retrospectively should have,” Mahadevan said. Watching a sphere roll jerkily forward, much like a dung beetle rolls its jagged bounty to its destination, makes it seem like the trajectories would be completely random and require a complex mathematical description.

But when the researchers mapped out the motions of the balls as distinct trajectories, an undeniable pattern emerged: No matter how irregular the sphere, its motion was periodic – that is, it repeated itself indefinitely once it reached steady state. What’s more, they found that the ball rolls over itself twice in each period of motion before going back to the same state.

“This was something we did not see coming at all,” Qian said.  

The results provide vivid physical manifestations of  topological theorems that mathematicians have long known, including a demonstration of the “Hairy Ball Theorem” that says, colloquially, “that you cannot comb the hair on a sphere without a cowlick,” according to Mahadevan, “here seen in how the rolling trajectories look on the surface of the sphere.” The experiments also serve to illustrate Dirac’s Plate Trick, which posits that a rotating object with strings has to rotate twice to return to its original state.

“It’s quite interesting how we can see these kinds of abstract mathematics made visible with this simple experiment,” said co-author and postdoctoral fellow Yeonsu Jung.  “And then the question could be, ‘What else can we do?’ … Maybe we could explore something that hasn’t been studied by mathematicians  yet.”

The study was funded by Transition Bio Ltd, Cambridge University, the National Research Foundation of Korea, the Simons Foundation, and the Henri Seydoux Fund.