What if a swing never slowed down?

Give a swing one push and let go. It swings and swings.

But each swing is a bit smaller. Soon it stops. What if it never slowed down at all?

Pull a swing back, let go and don't touch it again. At first it swoops high, but every swing is a little lower than the one before, until it hangs still. OpenStax, a free physics textbook, points out that to keep swinging on a playground swing, you have to keep pushing.

So what is slowing the swing down? And what would happen if nothing slowed it at all? A model can answer that, because in a model we can switch off things we can never switch off in real life.

OpenStax observes that in the real world oscillations seldom follow ideal simple harmonic motion: friction of some sort usually damps the motion so it dies away, or more force is needed to keep it going. Completely undamped motion, it adds, is rare.

This lab uses the pendulum model's damping control, a linear stand-in for air drag and pivot friction, to compare a lightly damped swing (b = 0.05 s⁻¹) with the idealized undamped case (b = 0). The pendulum is 2 m long, released at 20° under Earth's gravity. What happens to the amplitude, and to the period?

In our world, air rubs on the swing as it moves. The swing's ropes rub where they hang.

So each swing is a little smaller. The swing slowly stops.

In the model, set the air and friction to "a little". The swing, 2 meters long, starts 20° out. After one swing over and back it reaches only about 19°, and after six swings, about 13°. The swings keep getting smaller.

Notice what stays the same: each swing still takes about 2.85 seconds. Damping makes the swing smaller, not slower in its timing. In real life, OpenStax notes, a guitar string stops vibrating a few seconds after being plucked, for the same kind of reason.

With b = 0.05 s⁻¹ the model's turning points fall from 20.00° to 18.61° after one period, 16.13° after three and 13.02° after six (17.11 s), close to a steady fractional loss per swing. The simulated period is 2.847 s, slightly below the undamped 2.859 s for a 20° release.

OpenStax notes that for a small amount of damping the period stays nearly the same as for undamped motion while the amplitude gradually decreases, exactly the pattern the model shows.

Now Olly switches off the air and the rubbing. This can only happen in a pretend lab.

The swing goes up to the same height every time. It never gets smaller!

Now set the air and friction to "none". This is a simplification: in the real world we cannot remove all air and friction. But in the model, the swing now rises to exactly 20° on every swing. Nothing takes its energy away, so nothing makes it smaller.

The model would keep this swing going forever; the run stops after six swings so you can see a result. Try "a lot" of air and friction too: after six swings the swing is down to less than 2°, almost still.

At b = 0 the model's turning points stay at 20.00° and the measured period equals the exact undamped value, 2.859 s; the bottom speed of 1.54 m/s returns every half period. The model's equation becomes θ″ = −(g/L) sin θ, which conserves mechanical energy exactly.

At b = 0.3 s⁻¹ the amplitude falls to 16.1° after half a swing, 13.0° after one and 1.54° after six, with a period of 2.844 s. Each run stops after six swings, as the model documentation states.

A swing has energy when it moves. Air and rubbing take a little energy away each time.

With nothing taking energy away, the swing keeps all of it. So it never stops!

A moving swing has energy. As it swings up, that energy becomes height; as it swoops down, the height turns back into speed. If nothing took any energy away, it would climb to the same height every single time.

But air pushes against the moving swing, and the ropes rub where they hang. OpenStax explains that these forces take away energy, usually turning it into heat. A little is lost on each swing, so the swings shrink. In our model, all of this is squeezed into one simple setting called air and friction.

OpenStax explains that a non-conservative damping force removes energy from an oscillating system, usually in the form of thermal energy, so the amplitude decreases with each oscillation. Without it, the exchange between kinetic and gravitational potential energy is lossless.

The model's simplification: damping enters as −bθ′, proportional to angular speed. Its documentation notes that real air drag grows faster than linearly with speed, so the decay is approximate, and that the bob is a point on a weightless rigid rod. Setting b = 0 is an idealization; it does not describe any real swing.

Real swings always slow down. That is why someone has to push!

When you pump your legs, you give the swing a little energy back.

No real swing goes on forever. OpenStax says that completely undamped motion is rare, which is why you must keep pushing, or pumping, to keep a playground swing going. Each push puts back the energy the air and friction took away.

Scientists have known about this loss for a long time. A University of Virginia lecture describes one of Galileo's pendulum experiments: the pendulum swings back up to almost the same height it started from, apart from a slight loss caused by air resistance.

OpenStax's damped-oscillations section uses a playground swing to illustrate that counteracting damping forces requires continued pumping; it also notes that damping is sometimes wanted, as in car shock absorbers. In both cases a non-conservative force turns mechanical energy into heat.

The idea is old. Michael Fowler's University of Virginia lecture describes Galileo's experiment with a pendulum and a nail from Two New Sciences: the bob swings back up to almost the same height it started from, if we neglect a slight loss caused by air resistance. Galileo used that near-equality to argue that the speed a ball gains rolling down a ramp of a given height does not depend on the ramp's slope.