2014-09-07

Harmonic oscillations


Mathematics which explains the distance
at time for harmonic oscillations.
Harmonic oscillations and the representation of distance, speed and acceleration over time. So, if we derive the distance over time, we get the speed. If we derive that, we get acceleration.

This distance at time is composite function which has inner and outer parts and constant. Which parts in the formula are constants? Amplitude, angular frequency and phase shift. Firstly we see that its a multiplication of constant (amplitude) and composite function (frequency, time and phase within the sine). So we leave amplitude intact and will multiply it with derivative of outer function which keeps the inner function. So, cosine becomes negative sine, and then multiply it with derivative of inner function. Inner function is the first order polynome, so phase dissapears and frequency loses time. So, the derivative of inner function is pure angular frequency. So, recall, how far we are - we multiply amplitude with negative sine of intact inner function and multiply it with derivative of inner function - angular frequency. To get rid of negative sign, we can change sine to cosine by adding pi/2 inside of  this trigonometric function. Remember that sine of angle is equal to cosine of angle plus pi/2.

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