Periodic Oscillations move in predictable patters that can be described mathematically in terms of a simple sinusoidal equation.

A Damped oscillator moves in a modified periodic fashion so that its amplitude gets smaller and smaller with each cycle.

Examples: An empty rocking chair as it comes to rest after being pushed, a diving board after the diver jumps

A graph used to model this motion is y=f(x)cosBx where f(x) is y=ab^x
damped.gif
Periodic Motion

After collecting the appropriate data, one can begin to create a formula.

To find the period, take the coordinates from two consecutive peaks and subtract the lesser value from the greater. Then use B to find the period using the formula period= 2pi/B.

Now, to model the exponential equation y=ab^x, one must use the y-intercept coordinates as well as the coordinates from Peak 2 (above). Plug the y-intercept coordinates into the equation to find a.

Use the Peak 2 coordinates and plug in your a value to solve for b. One will need to use a logarithmic equation.

Now, put all of the data together so that the equation is in the form y=(ab^x)(cosBx)

Simple Harmonic Video

Simple_harmonic_oscillator.gif
damped.gif