# Motions of a Simulated Damped Harmonic Oscillator

Motions of a Simulated Damped Harmonic Oscillator

Consider a marble free to move inside a bowl. You release the marble at rest from an initial position on one of the walls. This Demonstration determines the subsequent trajectory of the marble. If there is no friction or air resistance, the marble continues to oscillate forever (undamped). If there is resistance, however, the oscillation eventually dies out. How fast it dies out and what the trajectory looks like depend on the properties of the medium.

Here the medium is assumed to behave like a Newtonian fluid, thus resistance is a linear function of velocity. If the medium is air, the marble oscillates, but air resistance causes the oscillation to damp out with time. For water, the oscillation damps out sooner (underdamping). For sugared water, the viscosity increases, and with more sugar, at some point the oscillation decelerates so quickly that it does not complete even a single oscillation (critical damping). If the viscosity is increased even further, for example with glucose/fructose syrup, the marble slowly sinks into the fluid and slowly undergoes exponential decay, until it reaches its equilibrium position (overdamped).