# Harmonic Oscillator

## Contents

## Introduction[✎ edit | edit source]

Harmonic oscillatory movement describes movement in which an object that has been displaced from its point of equilibrium is acted upon by a restoring force directly proportional to the amount of displacement. The classic model of such a movement is a weight attached to a spring, oscillating between the two extremes of maximum compression and maximum extension. In a simple (ideal) system as described, no energy is lost and the weight will oscillate for an infinite amount of time between the two extremes
^{[1]}. Based on these principles, the position, velocity, and acceleration of such a weight can be modeled by sinusoidal equations, as demonstrated below ^{[2]}.

## Equations[✎ edit | edit source]

Equations used to model simple harmonic oscillation in the case of a spring include:

- Restoring force:
*F*= -*k*·*x*

*k* is the spring constant; *x* is the displacement from position of equilibrium; *F* is the restoring force ^{[1]}

- Position of the body:
*y*=*A*·sin(*ω*·*t*+*φ*)

*Y* is position relative to the point of origin; *A* is maximum elongation; *ω* is angular frequency (given by *ω*=2·π·*ƒ*=(2·π)/*T*); *T* is the time to make one full oscillation; *φ* is the phase difference which gives the position of the body at *T* = 0 and its direction ^{[2]}

- Maximum velocity:
*V*max =*A*·*ω*^{[3]} - Instantaneous velocity:
*V*=*V*max·cos(*ω*·*T*)^{[3]}

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### Bibliography[✎ edit | edit source]

- ↑
^{a}^{b}University of Winnipeg.*Simple Harmonic Motion*[online]. ©1997. The last revision 1997-09-10, [cit. 2012-12-10]. <Simple Harmonic Motion>. - ↑
^{a}^{b}Proyecto Newton. MEC.*SIMPLE HARMONIC MOVEMENT*[online]. ©2012. [cit. 2012-12-10]. <Simple Harmonic Movement>. - ↑
^{a}^{b}AMLER, Evžen.*Acoustics*[lecture for subject Biophysics, specialization General Medicine, The 2nd faculty of medicine Charles University in Prague]. Prague. 2012-11-12. .