## Springs and Pendulums

Periodic motion is motion that is repeated at regular time intervals. The number of cycles in a given time period determine the frequency of the motion.

Examples of periodic motion include springs, pendulums, and waves. Below, we explore springs and pendulums.

Sorry, there are no calculators here for these yet, just some simple demos to give an idea of how periodic motion works, and how it is affected by basic parameters.

### Springs

The demo below shows the behavior of a spring with a weight at the end being pulled by gravity. There is no damping included, which is unavoidable in real systems.

Hooke's Law states that the amount of force needed to compress or stretch a spring varies linearly with the displacement:

$$F = - k\,x $$The negative sign means that the force opposes the motion, such that a spring tends to return to its original or equilibrium state. The factor \(k\) is the spring constant, and is a property of the spring. Higher \(k\) means that a spring is harder to stretch and compress.

Move the slider to change the spring constant for the demo below. In this case the force on the spring is the gravity pulling the mass of the ball: \(F = mg \). Since the force is constant, the higher values of k lead to less displacement.

At the equilibrium point (no periodic motion) the displacement is \(x = - m\,g\, /\, k\)

### Pendulums

For small amplitudes the period of a pendulum is given by

$$T = 2\pi \sqrt{L\over g} $$

and for larger amplitudes, the approximate period is given by$$T = 2\pi \sqrt{L\over g} \left( 1+ \frac{1}{16}\theta_0^2 + \frac{11}{3072}\theta_0^4 + \cdots \right)$$

^{2}on earth), and \(\theta_0\) is the initial angle.

Try changing length of the pendulum to change the period.

original spring code from html5canvastutorials

### Double Pendulum

Double pendulums, at certain energies, are an example of a chaotic system, in the sense that future behavior determinable, but it depends very highly on the initial conditions. Try running the pendulum with one set of values for a while, stop it, change the path color, and "set values" to ones that are almost the same (minimum step is 0.1), then start again. See what happens to the new path.

Chaotic motion can be seen typically for larger starting angles, with greater dependence on "angle 1"

original double pendulum code from physicssandbox

A few notes on the real world: Everything is more complicated than simple harmonic oscillators, but it is one of the few systems that can be solved completely and simply. Damping is always present (otherwise we could get perpetual motion machines!). Energy is inevitably lost on each bounce or swing, so the motion gradually decreases.