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'''Simple harmonic motion''' is any motion caused by a [[force]] that acts to restore a body to [[equilibrium]] and has a magnitude proportional to the [[distance]] of the body from its equilibrium position. Mathematically, the [[acceleration]] of the body, <math>\ddot x</math>, may be related to the [[displacement]] of the body by:

'''Simple harmonic motion''' is any motion caused by a [[force]] that acts to restore a body to [[equilibrium]] and has a magnitude proportional to the [[distance]] of the body from its equilibrium position. Mathematically, the [[acceleration]] of the body, <math>\ddot x</math>, may be related to the [[displacement]] of the body by:

<math>

<math>

\ddot x(t) = -kx

\ddot x(t) = -kx

==Mathematics==

==Mathematics==

Simple harmonic motion is described by a second order [[differential equation]]:

Simple harmonic motion is described by a second order [[differential equation]]:

<math>

<math>

\ddot x(t) = -kx

\ddot x(t) = -kx

</math>

</math>

The solution of this equation is:

The solution of this equation is:

<math>

<math>

x(t) = A \sin{\omega t} + B \cos{\omega t}

x(t) = A \sin{\omega t} + B \cos{\omega t}

</math>

</math>

where <math>\omega</math> is <math>\sqrt{k}</math> and called the angular frequency. <math>A</math> and <math>B</math> are arbitrary constants that can be determined from the initial conditions of the problem. Since the solution involves only sines and cosines which oscillate, the solution itself will oscillate. Simple harmonic oscillators also have the property that the frequency of the oscillation is independent to its amplitude.

where <math>\omega</math> is <math>\sqrt{k}</math> and called the angular frequency. <math>A</math> and <math>B</math> are arbitrary constants that can be determined from the initial conditions of the problem. Since the solution involves only sines and cosines which oscillate, the solution itself will oscillate. Simple harmonic oscillators also have the property that the frequency of the oscillation is independent to its amplitude.

{{Math-h}}

{{Math-h}}

Suppose there is a potential, <math>V(x)</math>, that has a equilibrium point (minima) at <math>x_0</math>. <math>V(x)</math> can be approximated around the minima using a [[Taylor series]] as:

Suppose there is a potential, <math>V(x)</math>, that has a equilibrium point (minima) at <math>x_0</math>. <math>V(x)</math> can be approximated around the minima using a [[Taylor series]] as:

<math>

<math>

V(x) = V(x_0) + V'(x_0)(x-x_0) + \frac{V''(x_0)(x-x_0)^2}{2!} + \frac{V^{(3)}(x_0)(x-x_0)^3}{3!}+...

V(x) = V(x_0) + V'(x_0)(x-x_0) + \frac{V''(x_0)(x-x_0)^2}{2!} + \frac{V^{(3)}(x_0)(x-x_0)^3}{3!}+...

</math>

</math>

where <math>V'(x_0)</math>, <math>v''(x_0)</math> are the first and second derivatives of <math>V(x)</math> evaluated at <math>x_0</math>. As <math>x_0</math> is an equilibrium point, the gradient of the potential is 0 (at the equilibrium point, the net [[force]] is zero). Therefore the second term in the expansion is 0. Near to <math>x_0</math>, terms of cubic order or higher can be neglected and so the potential is approximately:

 

where <math>V'(x_0)</math>, <math>V''(x_0)</math> are the first and second derivatives of <math>V(x)</math> evaluated at <math>x_0</math>. As <math>x_0</math> is an equilibrium point, the gradient of the potential is 0 (at the equilibrium point, the net [[force]] is zero). Therefore the second term in the expansion is 0. Near to <math>x_0</math>, terms of cubic order or higher can be neglected and so the potential is approximately:

 

<math>

<math>

V(x) \approx V(x_0) + \frac{V''(x_0)(x-x_0)^2}{2!}

V(x) \approx V(x_0) + \frac{V''(x_0)(x-x_0)^2}{2!}

</math>

</math>

in the vicinity of <math>x_0</math>. The constant term corresponds can be neglected as it only affects where the zero of the potential is. Hence the potential is approximately [[quadratic]] and behave like a simple harmonic oscillator.

 

in the vicinity of <math>x_0</math>. The constant term corresponds can be neglected as it only affects where the zero of the potential is. Hence the potential is approximately [[quadratic equation|quadratic]] and behave like a simple harmonic oscillator.

===Example===

===Example===

An example of this is that of a simple pendulum. A simple pendulum has the equation of motion:

An example of this is that of a simple pendulum. A simple pendulum has the equation of motion:

<math>

<math>

\ddot \theta = - \frac{g}{L} \sin{\theta}

\ddot \theta = - \frac{g}{L} \sin{\theta}

</math>

</math>

where:

where:

:<math>\theta</math> is the angle of the pendulum to the vertical

:<math>\theta</math> is the angle of the pendulum to the vertical

Here, the small angle approximation, that <math>\sin{\theta} \approx \theta</math> can be made, so that

Here, the small angle approximation, that <math>\sin{\theta} \approx \theta</math> can be made, so that

<math>

<math>

\ddot \theta \approx - \frac{g}{L} \theta

\ddot \theta \approx - \frac{g}{L} \theta

This is of the form of simple harmonic motion above. Hence when the angle of oscillation is small, the period of the pendulum is

This is of the form of simple harmonic motion above. Hence when the angle of oscillation is small, the period of the pendulum is

<math>

<math>

P \approx 2 \pi \sqrt{\frac{g}{L}}

P \approx 2 \pi \sqrt{\frac{g}{L}}