Many a pub crawler has engaged in the time-honored tradition of throwing beer mats—those round cardboard coasters that are ubiquitous in bars—as if the mats were frisbees, often competing to see who can throw one the farthest. But unlike frisbees, beer mats tend to flip in the air and fly with a backspin. Now physicists at the University of Bonn have come up with a theoretical model to explain the phenomenon, according to a new paper submitted to the physics arXiv preprint server.
Thanks to their natural curiosity, physicists are fascinated by the physics of beer, and lead author Johann Ostmeyer is no exception. A couple of years ago, he became intrigued by the physics of so-called “beer tapping”: a common prank where the perpetrator, holding an open bottle of beer, finds a target in the bar holding another open bottle. The prankster strikes the top of the target’s bottle with the bottom of their own, then savors the sight of beer explosively foaming all over the target’s hands and clothes.
Back in 2013, Javier Rodríguez-Rodríguez, a physicist at Carlos III University of Madrid, and several colleagues presented experimental and computer-simulation findings of why beer cans foam up so much after being shaken. They concluded that the foaming-over stemmed from a series of waves. Apparently the physics is similar to the development of the cloud in an atomic bomb, although the source of the “explosion” is very different.
The initial tap creates a shockwave that travels from the top to the bottom of the target’s bottle. That shockwave’s energy is transferred to the beer, sending a second shockwave bouncing back and forth between the bottom of the bottle and the beer’s surface. All that activity releases tiny pockets of gas trapped in small imperfections in the bottle’s glass, creating clouds of smaller bubbles from the beer’s dissolved CO2. As they rise to the surface, those bubbles get bigger and speed up, eventually creating the shower of foam that makes beer tapping such a popular pub prank.
Ostmeyer submitted his own analysis on the beer tapping issue to the arXiv last year, specifically focusing on why the lower beer bottle foams over and the upper one does not. The initial tap creates a low pressure in the lower bottle, and the CO2 bubbles expand and collapse into fragments.
More gas gets diffused into the cluster of bubbles, rising up to create the foam. Something different happens in the top bottle. There is a higher initial pressure, so there are only moderate oscillations in the bubbles, and the eventual bubble collapse doesn’t happen—or at least not to the same extent. Thus, you don’t get the same rapid growth cloud turning your entire beer into foam. “The prankster leaves the scene dry,” Ostmeyer wrote.
Now Ostmeyer, inspired by a 2017 trip to Munich for a German physics outreach demonstration show, has turned his attention to the pressing question of beer-mat flight trajectories. Beer mats (at least the round variety) are basically thin, flat disks with a given radius and mass. A frisbee is also a thin flat disk, and the physics surrounding its flight is well-known.
A frisbee’s distinctive curved upper surface is essentially the same shape (when viewed from the side) as an airfoil. Throw it with sufficient force to overcome the downward pull of gravity, and it will generate lift. Of course, there is also air resistance, or drag, acting on the frisbee as it flies through the air. The resistance depends on the angle of attack. If the angle is negative, it pushes the air up and forces the frisbee down, while a positive angle will push the frisbee up. To keep wobbling to a minimum, professional frisbee players know to add a spin when they throw, ensuring their frisbees stay on a single plane as they travel.
But beer mats don’t have the frisbee’s distinctive airfoil-like edge, and that affects their aerodynamics. Ostmeyer et al. began by assuming a horizontal flight and an axis of rotation perpendicular to the direction it is moving. Toss a beer mat like a frisbee, and the rotation will initially stabilize it.
However, the pull of gravity will soon cause the mat to drop, which will alter the angle at which it attacks the air. You’ll still get lift, but it will be focused close to the leading edge rather than the mat’s center of mass. So the mat will start to precess, forcing it to flip on its side so that it is traveling vertically. The researchers’ model predicted that this should happen at about 0.45 seconds into the flight. Furthermore, the model showed that the mat could have either a backspin (which is stable) or a topspin (which is unstable). Thus, the mat will be more likely to have a backspin.
Every good theoretical prediction must be experimentally tested, of course, and randomly tossing beer mats around in the local pub wouldn’t be scientifically rigorous. Thrown by hand, the timing of the tilt appears to be random, but the model predicts otherwise. So Ostmeyer and his team built their own makeshift beer-mat launcher out of two electric motor-powered treadmills. The launcher enabled the researchers to control the horizontal velocity and rate of rotation of each launch.
The researchers placed the launcher on a table, then placed a beer mat between the treadmills and launched the mats at various speeds, marking where they landed on the floor. The flight trajectories were recorded with high-speed cameras, and the team used a program called Tracker to extract the coordinates of the beer mat at any point along its trajectory. The results of the researchers’ subsequent analysis showed very good agreement with the theoretical predictions.
Ostmeyer et al.’s theoretical model can also be used to predict the flight trajectories of other kinds of flying discs. For instance, their model predicts that a CD will flip after 0.8 seconds, while a larger and heavier discus will flip after 16 seconds. A flying playing card has the shortest trajectory of all, flipping after just 0.24 seconds. “Even Rick Smith Jr., the world record holder for farthest card thrown, or a playing card machine gun, cannot avoid their cards flying a curve and ending up with backspin after much less than a second,” the authors wrote in a footnote.
The model’s prediction for a frisbee—without taking into account any aerodynamics due to its curvature—is a surprising 0.8 seconds. “Of course, the wing form of a frisbee allows it to remain stable for a much longer time when thrown professionally,” they wrote. “The reason is that frisbees have their aerodynamic center very near to their center of mass and, thus, experience much less torque.”
We raise a glass to Ostmeyer and his colleagues for their entertaining and informative work, which was driven purely by scientific curiosity.”Our sincere apologies to everyone hit by a beer mat,” the authors noted in their acknowledgements, “be it through inaccurate aim or due to us instigating others to perform silly experiments.”