Editing Harmonic oscillator (section)

== Simple harmonic oscillator ==
{{main|Simple harmonic motion}}
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A simple harmonic oscillator is an oscillator that is neither driven nor [[Damping ratio|damped]]. It consists of a mass ''m'', which experiences a single force ''F'', which pulls the mass in the direction of the point {{math|1=''x'' = 0}} and depends only on the position ''x'' of the mass and a constant ''k''. Balance of forces ([[Newton's second law]]) for the system is
<math display="block">F = m a = m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} = m\ddot{x} = -k x. </math>

Solving this [[differential equation]], we find that the motion is described by the function
<math display="block" qid=Q3299367> x(t) = A \sin(\omega t + \varphi), </math>
where
<math display="block" qid=Q834020>\omega = \sqrt{\frac k m}.</math>

The motion is [[Periodic function|periodic]], repeating itself in a [[sine wave|sinusoidal]] fashion with constant amplitude ''A''. In addition to its amplitude, the motion of a simple harmonic oscillator is characterized by its [[Frequency|period]] <math>T = 2\pi/\omega</math>, the time for a single oscillation or its frequency <math>f=1/T</math>, the number of cycles per unit time. The position at a given time ''t'' also depends on the [[phase (waves)|phase]] ''φ'', which determines the starting point on the sine wave. The period and frequency are determined by the size of the mass ''m'' and the force constant ''k'', while the amplitude and phase are determined by the starting position and [[velocity]].

The velocity and [[acceleration]] of a simple harmonic oscillator oscillate with the same frequency as the position, but with shifted phases.  The velocity is maximal for zero displacement, while the acceleration is in the direction opposite to the displacement.

The potential energy stored in a simple harmonic oscillator at position ''x'' is
<math display="block" qid="Q891408">U = \tfrac 1 2 kx^2.</math>