# Spring Pendulum

Bodies · mechanical systems

```
elastic
l=1  m=1  k=9.81κ  a=0.05  s=0.8
```

[Open in the app](https://www.wavelace.com/app#p=156) · [This page](https://www.wavelace.com/presets/spring-pendulum)

### The setup

A bob of mass 1 hangs from a pivot on a spring of natural length 1 and stiffness `k = 9.81κ`, so the slider counts the stiffness in units of the bob's weight. At `κ = 3` the weight stretches the spring by a third. The pivot sits at that resting length, 1.33, so the bob at rest touches the floor, and the pivot moves as the slider does. The bob starts pulled 0.8 further down and tilted by 0.05 radians, which puts it 0.09 off the vertical.

### Two rhythms

Straight up and down, the bob bounces with the period `2π/√k ≈ 1.16` seconds at Speed 1. Swinging, it is a pendulum of the stretched length `4/3`, with the period `2π√(L/g) ≈ 2.32` seconds for `L = 4/3`. That is exactly two bounces to one swing, and `κ = 3` is the stiffness that makes it so. Bouncing this deep, both run a little faster, at about 1.04 and 2.24 seconds.

### The trade

The spring's pull rises and falls with every bounce, twice in each swing, which is the rhythm that pumps a swing. So the bounce feeds the swing. The sideways reach grows from 0.09 to 0.87 by 4.4 seconds. Then the swing hands the energy back, and at 9.8 seconds the reach is down to 0.088 and the bob mostly bounces again. The trade repeats about every 10 seconds. Two runs started a millionth of a radian apart stay within 0.00004 of each other for a minute, so this is regular motion, not chaos. At the bottom of each bounce the bob dips 0.47 under the floor.

### History

Aleksandr Vitt and Gabriel Gorelik published the first analysis of the swinging spring in 1933. Leonid Mandelstam suggested it to them as a mechanical twin of the Fermi resonance, which Enrico Fermi had used in 1931 to explain the Raman spectrum of carbon dioxide. Peter Lynch took it up in 2002 as a simple model of balance in the atmosphere.

### Try

- Drag `κ` to 1. The bounce is out of step with the swing and never feeds it: the reach stays at its starting 0.09.
- Drag `κ` to 2. The trade still happens but only reaches 0.26, first at 7 seconds, then every 13 to 14 seconds.
- Drag `κ` to 5. The swing grows wider, to 1.18, and the energy goes back and forth every 5 seconds.
- Set `a=0` on the line. With no tilt there is nothing for the bounce to pump, and the bob bounces straight up and down for good.
- `Front` in the deck shows the swing face on.

### Read more

- [Elastic pendulum](https://en.wikipedia.org/wiki/Elastic_pendulum)
- [Parametric oscillator](https://en.wikipedia.org/wiki/Parametric_oscillator)
- [Fermi resonance](https://en.wikipedia.org/wiki/Fermi_resonance)
- [Normal mode](https://en.wikipedia.org/wiki/Normal_mode)