Simple harmonic motion

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, ${\ddot {x}}$, may be related to the displacement of the body by:

${\ddot {x}}(t)=-kx$

where $k$ 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 ${\frac {1}{2}}kx^{2}$ will cause a body to undergo simple harmonic motion.

Mathematics

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

${\ddot {x}}(t)=-kx$

The solution of this equation is:

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

where $\omega$ is ${\sqrt {k}}$ and called the angular frequency. $A$ and $B$ 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

${\frac {d}{dx}}\sin x=?\,$

This article/section deals with mathematical concepts appropriate for late high school or early college.

Suppose there is a potential, $V(x)$, that has an equilibrium point (minima) at $x_{0}$. $V(x)$ can be approximated around the minima using a Taylor series as:

$V(x)=V(x_{0})+V'(x_{0})(x-x_{0})+{\frac {V''(x_{0})(x-x_{0})^{2}}{2!}}+{\frac {V^{(3)}(x_{0})(x-x_{0})^{3}}{3!}}+...$

where $V'(x_{0})$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V''(x_0)} are the first and second derivatives of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x)} evaluated at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} . As Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} , terms of cubic order or higher can be neglected and so the potential is approximately:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x) \approx V(x_0) + \frac{V''(x_0)(x-x_0)^2}{2!} }

in the vicinity of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0} . 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ddot \theta = - \frac{g}{L} \sin{\theta} }

where:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} is the angle of the pendulum to the vertical

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} is the acceleration due to gravity

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} is the length of the pendulum

Here, the small angle approximation, that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{\theta} \approx \theta} can be made, so that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P \approx 2 \pi \sqrt{\frac{g}{L}} }

See also

• Newtonian mechanics
• Potential energy