Editing Harmonic oscillator (section)

===Simple pendulum===
[[Image:Simple pendulum height.svg|thumb|A [[simple pendulum]] exhibits approximately simple harmonic motion under the conditions of no damping and small amplitude.]]

Assuming no damping, the differential equation governing a simple pendulum of length <math>l</math>, where <math>g</math> is the local [[Gravitational acceleration|acceleration of gravity]], is
<math display="block" qid=Q20702>\frac{d^2\theta}{dt^2} + \frac{g}{l}\sin\theta = 0.</math>

If the maximal displacement of the pendulum is small, we can use the approximation <math>\sin\theta \approx \theta</math> and instead consider the equation
<math display="block">\frac{d^2\theta}{dt^2} + \frac{g}{l}\theta = 0.</math>

The general solution to this differential equation is
<math display="block">\theta(t) = A \cos\left(\sqrt{\frac{g}{l}} t + \varphi \right),</math>
where <math>A</math> and <math>\varphi</math> are constants that depend on the initial conditions.
Using as initial conditions <math>\theta(0) = \theta_0</math> and <math>\dot{\theta}(0) = 0</math>, the solution is given by
<math display="block">\theta(t) = \theta_0 \cos\left(\sqrt{\frac{g}{l}} t\right),</math>
where <math>\theta_0</math> is the largest angle attained by the pendulum (that is, <math>\theta_0</math> is the amplitude of the pendulum).  The [[Sine|period]], the time for one complete oscillation, is given by the expression
<math display="block" qid=Q3382125>\tau = 2\pi \sqrt\frac{l}{g} = \frac{2\pi}{\omega},</math>
which is a good approximation of the actual period when <math>\theta_0</math> is small. Notice that in this approximation the period <math>\tau</math> is independent of the amplitude <math>\theta_0</math>. In the above equation, <math>\omega</math> represents the angular frequency.