Physicists unlock the secret of a child’s swing

For many children, swinging on a playground set feels like second nature. But what a child intuits, grown scientists struggle to understand in detail. Now, a new mathematical model captures how a swinging child subtly shifts technique as a swing's motion increases and helps explain what makes the ubiquitous playground equipment work.

"The model is simple, but seems complete," says Mont Hubbard, an engineer emeritus at the University of California, Davis, who has studied the mechanical aspects of myriad sports.

A swing is basically a pendulum: a mass (the rider) sits on the seat that hangs from an overhead bar by a pair of chains. When the seat is pushed away from its equilibrium position—hanging straight down from the bar—it moves outward, but also slightly upwards. Gravity then pulls the swing back toward its initial position, which the swing then overshoots. Once the swing has swung outward in the opposite direction, gravity once again pulls it back under the bar. This incessant pull back toward the center is what causes the swing to oscillate back and forth.

A standard pendulum requires an external force to set it swinging—think of a baby needing periodic pushes to keep swaying. But by about age 6, a typical child has learned how to propel themself by shifting their weight at just the right moments in the pendulum's motion. When the swing reaches its highest point backward, the child leans back and stretches their legs, shifting their weight so that instead of lying in line with the chains, it lags behind them as the swing moves forward. At the highest point in a forward swing, the rider tucks their legs and leans forward, now putting their weight in front of the chains. At all times, the rider's unconscious goal remains constant: to shift the position of their center of mass in such a way that it adds angular momentum to the swing and increases the amplitude of swinging.

Although it's child's play in practice, capturing the essential physics in a model isn't easy. Researchers must include enough details to accurately describe the system, but not so much that they render it intractably complex. One model put forth in 1990 assumed riders rock backward and forward at a constant frequency in simple sinusoidal motion—meaning the movement makes the shape of a sine wave in time. This model works well enough for lower swinging amplitudes, but it falls apart as the amplitude of a swing increases. That's because as a child swings higher and higher, their swinging frequency decreases. If a rider kept pumping at a fixed frequency, their motion would eventually fall out of synchronization with the swinging and they would lose the ability to pump energy into the system and maintain its motion.

Other models assumed that a child subconsciously feels this shift in frequency and adjusts the timing of their bodyweight shifts accordingly. But these models also assume such shifts occur instantaneously and jarringly at the highest points of the swing, when in reality children employ smooth, continuous motions.

Now, Chiaki Hirata of Jumonji University and his colleagues have found a compromise between the two approaches. They modeled a swinging child as a three-component system comprising a torso, seat, and lower legs. The torso and lower legs each move relative to the seat in an oscillatory fashion, but the frequency of that oscillation changes to remain optimal for pumping.

Based on this model, the researchers found that when a swing is just starting, the optimal pumping strategy is to lean all the way back just as the swing passes its equilibrium position on its way forward. But as the amplitude increases, the optimal timing shifts to favor leaning back earlier, when the swinger is at the highest point in her backswing. The scientists tested their model using real humans swinging in a sort of playground laboratory and found that it was a good match with real life, they report in a paper in press at Physical Review E.

The analysis strikes the right balance between simplicity and accuracy, says Paul Glendinning, an applied mathematician at the University of Manchester. "I’d certainly incorporate their model into [my students’] projects in the future," he writes in an email. However, Andy Ruina, a mechanical engineer at Cornell University, says the new model doesn't resolve how a child uses information from their environment to appropriately shift the frequency and timing of their motion--so-called active feedback. To design a robot capable of pumping a swing, for example, applying the principles of active feedback is the more natural approach, Ruina says.

Applications aside, Hirata says the work made him realize the playground is really an applied physics lab for children. "It changed my view of watching kids on swings," he says, "They're not just playing, they're interacting with the laws of physics."