Nobody has to teach a child how to pump a swing. Within a few tries they work out that small shoves, well timed, add up to a huge ride, while big shoves at the wrong moment get you nowhere. That instinct is resonance, and it is the same idea behind tuning a radio and the ring of a wine glass.
The swing has a natural period of 3.5 s. Each tap is the same small shove. The dot on the graph shows how big a swing that rhythm can build.
Tap PUSH a few times at random and the swing barely moves, because half your shoves meet it coming back. Now watch the swing and push only as it starts to move away from you. Each shove now feeds energy in, and the arc grows. Switch on the auto-pusher and slide the rhythm: far from 1.00 the swing jiggles and stays small, while close to 1.00 the same gentle pushes pile up into a big arc.
A swing is a pendulum, and a pendulum takes a fixed time per swing. For small swings the period is about 2π√(L/g), where L is the chain length and g is 9.8 m/s². A 1 m pendulum takes about 2.0 s, and one with 3 m of chain about 3.5 s. Notice what is missing: the mass. A heavier child swings with the same rhythm. Swing height barely matters either; even a swing out to 30° is only about 2% slower than a tiny one.
That fixed rhythm is the natural frequency. A pendulum, a guitar string, a playground see-saw and a glass all have one (or several), set by their shape and stiffness.
Pushing is work: force times distance. If your shove goes in the direction the swing is already moving, you add energy. If it goes against the motion, you take energy out. A rhythm that matches the natural beat means every push lands at the same helpful moment in each swing, so the gains add up instead of cancelling. A rhythm that does not match drifts in and out of step with the swing: sometimes it helps, sometimes it hurts, and the average gain is small.
Something always drains energy: air drag and friction in the hooks. The swing grows until the energy you add each cycle equals the energy lost each cycle. Light damping means a tall, narrow peak on the graph: the swing is very sensitive to rhythm but rewards a good match with a big response. Heavy damping means a low, wide one. Engineers sum this up with the quality factor Q: at resonance, the swing's size is about Q times how far a steady push would hold it. The swing in the demo has Q of roughly 6; a good wine glass or a quartz crystal has a Q in the hundreds or far higher.
A loud, steady note that matches the ring pitch of a wine glass can make the glass vibrate so strongly that it cracks. It takes a very pure tone, a lot of volume and a thin glass, which is why most singers cannot do it unaided. A radio does the opposite trick: its tuning circuit has its own natural frequency, and turning the dial moves that frequency until it matches the station you want, so that station is amplified enormously while the others barely register.
One famous example is usually told wrongly. The 1940 collapse of the Tacoma Narrows Bridge is often blamed on simple resonance with the wind, but engineers now attribute it to aeroelastic flutter, in which the twisting bridge deck and the wind feed each other. It is a close cousin of resonance, not the textbook case.
The lesson of the swing holds for all of them: you do not need strength, only the right beat.