Harmonic Oscillator

The "Harmonic Oscillator" applet facilitates the investigation of a second-order differential equation governing the position of a spring-mass system.  The user can vary the parameters of the ODE and view the effects on the position of the mass.

The form of the second-order equation that models the harmonic oscillator is:

y = the position function

m = the mass

k = the spring constant

b = the damping coefficient

Development of the second-order equation

Using the Applet:

In the applet, below, the user can vary m, b, k, y₀ (the initial position), and v₀ (the initial velocity). When the parameters and initial conditions are changed, the applet plots the position of the mass as a function of time. Furthermore, the applet classifies the behavior and period of the motion (when the period exists), and it also determines the eigenvalues.

Select the "Oscillator Analysis" button for an analysis of the harmonic oscillator.

Points of Interest:

• the relationship between the value of the parameters m, b, and k, and the classification

• the relationship between the value of the parameters m, b, and k, and the eigenvalues

• classification of a harmonic oscillator

• graphical analysis of a harmonic oscillator

• long-term behavior of a second-order, linear, homogenous equation

Exercises 

Use Applet (author Garrett Heath)