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Created page with "'''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 bod..."
'''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>
\ddot x(t) = -kx
</math>
where <math>k</math> is a constant.
This results in the body performing oscillations about its equilibrium position. A body undergoing simple harmonic motion is also known as a simple harmonic oscillator. A potential of <math>\frac{1}{2}kx^2</math> will cause a body to undergo simple harmonic motion.
==Mathematics==
Simple harmonic motion is described by a second order [[differential equation]]:
<math>
\ddot x(t) = -kx
</math>
The solution of this equation is:
<math>
x(t) = A \sin{\omega t} + B \cos{\omega t}
</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.
==Uses==
The simple harmonic oscillator is a very important potential as it can be used to approximate the equilibrium points other potentials. This is one of the most common approximations used in physics, and is used in [[quantum field theory]].
===Derivation===
{{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:
<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!}+...
</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:
<math>
V(x) \approx V(x_0) + \frac{V''(x_0)(x-x_0)^2}{2!}
</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.
===Example===
An example of this is that of a simple pendulum. A simple pendulum has the equation of motion:
<math>
\ddot \theta = - \frac{g}{L} \sin{\theta}
</math>
where:
:<math>\theta</math> is the angle of the pendulum to the vertical
:<math>g</math> is the [[acceleration due to gravity]]
:<math>L</math> is the length of the pendulum<ref>http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c3</ref>
Here, the small angle approximation, that <math>\sin{\theta} \approx \theta</math> can be made, so that
<math>
\ddot \theta \approx - \frac{g}{L} \theta
</math>
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>
P \approx 2 \pi \sqrt{\frac{g}{L}}
</math>
==See Also==
*[[Newtonian mechanics]]
*[[Potential energy]]
==References==
[[Category:Mechanics]]
<math>
\ddot x(t) = -kx
</math>
where <math>k</math> is a constant.
This results in the body performing oscillations about its equilibrium position. A body undergoing simple harmonic motion is also known as a simple harmonic oscillator. A potential of <math>\frac{1}{2}kx^2</math> will cause a body to undergo simple harmonic motion.
==Mathematics==
Simple harmonic motion is described by a second order [[differential equation]]:
<math>
\ddot x(t) = -kx
</math>
The solution of this equation is:
<math>
x(t) = A \sin{\omega t} + B \cos{\omega t}
</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.
==Uses==
The simple harmonic oscillator is a very important potential as it can be used to approximate the equilibrium points other potentials. This is one of the most common approximations used in physics, and is used in [[quantum field theory]].
===Derivation===
{{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:
<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!}+...
</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:
<math>
V(x) \approx V(x_0) + \frac{V''(x_0)(x-x_0)^2}{2!}
</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.
===Example===
An example of this is that of a simple pendulum. A simple pendulum has the equation of motion:
<math>
\ddot \theta = - \frac{g}{L} \sin{\theta}
</math>
where:
:<math>\theta</math> is the angle of the pendulum to the vertical
:<math>g</math> is the [[acceleration due to gravity]]
:<math>L</math> is the length of the pendulum<ref>http://hyperphysics.phy-astr.gsu.edu/hbase/pend.html#c3</ref>
Here, the small angle approximation, that <math>\sin{\theta} \approx \theta</math> can be made, so that
<math>
\ddot \theta \approx - \frac{g}{L} \theta
</math>
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>
P \approx 2 \pi \sqrt{\frac{g}{L}}
</math>
==See Also==
*[[Newtonian mechanics]]
*[[Potential energy]]
==References==
[[Category:Mechanics]]