The damped harmonic oscillator
The set up is a damped oscillator governed by a differental equation of the form ay'' + by' +cy =0, where a,b,c are arbitrary constants ( for the case of a mechanical oscillator then a=mass, b= the damping constant and c is the magnitude of the spring constant). You can move the sliders to change the constants and see how the displacement varies with time (the blue line) and how the velocity varies with time (the dashed line). Experiment with the constants by moving the sliders. Notice how if the damping is strong compared with the spring constant (b^2 > 4ac) we have overdamped motion, but if the damping is small (b^2 < 4ac), the motion becomes oscillatory. You can also change the initial conditions (the velocity and displacement when t=0) by sliding the relevant starting points up and down the y-axis.
Other useful commands are a right-click to change the zoom on the graph or the relative scales along the y- and t-axes. Use the arrow button on the tool bar to move the points on the sliders; use the cursor button on the toolbar to move the plot around on the screen. Find the solutions for the following differential equations, describe the character of the solution (e.g. oscillatory, underdamped, critically damped, exponentially growing etc.) and *then* use the applet to check your answers (i) y'' + 2y' + y = 0, where y=1 and v=0 when t=0 (ii) y'' + 2y' + 10y =0, where y=1 and v=0 when t=0 (iii) y'' + 9y = 0, where y=0 and v=1 when t=0 (iv) y'' - y + 6y =0, where y=1 and v=0 when t=0