Inspired by a video on YouTube, two researchers have uncovered new insights into the physics of toppling dominoes. Through an extensive set of simulations, David Cantor at Canada’s Polytechnique Montréal, together with Kajetan Wojtacki at the Polish Academy of Sciences, showed that the speed of a wave of falling dominoes is affected by two types of friction, as well as the spacing between the dominoes.
In 2017, Destin Sandlin, host of the YouTube channel SmarterEveryDay, posted a video called “Dominoes – HARDCORE Mode” – where he used a high-speed camera to film a chain of toppling dominoes. He noticed that the speed of the resulting wavefront – which propagates as each domino falls and strikes its neighbour – was affected by the friction between the dominoes and the surface they were placed on.
On smooth, low-friction hardwood, each domino appeared to backslide as it fell – in contrast to high-friction felt, where the bottom of each domino largely stayed in place. On low-friction hardwood, a domino struck its neighbour further down, slightly lowering the speed of the wavefront. Yet under the limitations of his experiment, Sandlin soon realized that the problem was far more complex than he had anticipated – leading him to admit: “this has broken me. I do not understand dominoes”.
Toppling simulations
In a new study, Cantor and Wojtacki delved further into the problem by simulating the toppling of 200 evenly spaced dominoes. Across 1210 simulations, they examined a wide range of spacings between dominoes, while also varying surface friction, and the friction between neighbouring dominoes.
The duo discovered that when dominoes are spaced apart by half their thickness, increasing the friction between them causes the wave to slow down, since the dominoes absorb more of its energy. In contrast, increasing domino-surface friction increased the wave speed in some cases, for the same reasons that Sandlin highlighted in his video.
Yet for spacings between 1.5–5 times the dominoes’ thickness, the simulations showed that domino-surface friction had little effect on the wave speed. This suggested that each domino gains more kinetic energy as it falls, making its neighbour less likely to backslide, regardless of surface friction.
Backsliding to a halt
For spacings larger than three times the dominoes’ thickness, Cantor and Wojtacki showed that the wave could become unstable when both domino–domino friction is high, and domino–surface friction is low. This combination would cause the dominoes to backslide too far to reach their neighbours, bringing the wave to a stop.
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One other interesting result is that once the coefficient of domino–domino friction reaches 0.4, any further increase in friction does not seem to affect the wave’s propagation speed. This could be because the motion was no longer significantly affected by the dominoes sliding against each other. This same saturation effect is commonly found in related systems such as the friction between grains in a pile of sand – where friction places an upper limit on the steepness of the pile.
Based on these results, Cantor and Wojtacki constructed a law to predict wave propagation speeds. This incorporates domino spacing, as well as both types of friction. This law is in close agreement with past experiments – but more work is needed to fully uncover the physical mechanisms responsible.
The research is reported Physical Review Applied.
