Editing Harmonic oscillator (section)

===Spring/mass system===
[[Image:Harmonic oscillator.svg|thumb|Spring–mass system in equilibrium (A), compressed (B) and stretched (C) states]]

When a spring is stretched or compressed by a mass, the spring develops a restoring force. [[Hooke's law]] gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length:
<math display="block">F(t) = -kx(t),</math>
where ''F'' is the force, ''k'' is the spring constant, and ''x'' is the displacement of the mass with respect to the equilibrium position. The minus sign in the equation indicates that the force exerted by the spring always acts in a direction that is opposite to the displacement (i.e. the force always acts towards the zero position), and so prevents the mass from flying off to infinity.

By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation:
<math display="block"> F(t) = -kx(t) = m \frac{\mathrm{d}^2}{\mathrm{d} t^2} x(t) = ma, </math>
the latter being [[Newton's laws of motion#Newton's second law|Newton's second law of motion]].

If the initial displacement is ''A'', and there is no initial velocity, the solution of this equation is given by
<math display="block"> x(t) = A \cos \left( \sqrt{\frac{k}{m}} t \right).</math>

Given an ideal massless spring, <math>m</math> is the mass on the end of the spring. If the spring itself has mass, its [[Effective mass (spring-mass system)|effective mass]] must be included in <math>m</math>.

====Energy variation in the spring–damping system====
In terms of energy, all systems have two types of energy: [[potential energy]] and [[kinetic energy]]. When a spring is stretched or compressed, it stores elastic potential energy, which is then transferred into kinetic energy. The potential energy within a spring is determined by the equation <math display="inline"> U = \frac{1}{2}kx^2. </math>

When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring. By [[conservation of energy]], assuming the datum is defined at the equilibrium position, when the spring reaches its maximal potential energy, the kinetic energy of the mass is zero. When the spring is released, it tries to return to equilibrium, and all its potential energy converts to kinetic energy of the mass.