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[parent] simple harmonic oscillator (Definition)

A simple harmonic oscillator is a mechanical system which consists of a particle under the influence of a Hooke's law force. The equation of motion of such a system is

$\displaystyle m {\ddot x} + k x = 0 $
It is typical to define the quantity $\omega = \sqrt{k/m}$ and write this equation as

$\displaystyle {\ddot x} + \omega^2 x = 0 $

Note that this equation is linear. Among other consequences, this means that the period of oscilltions does not depend on amplitude. It is rather simple to solve this equation in TEMs of trigonometric functions to obtain a general solution. This solution is typically written in one of two forms.

$\displaystyle x = v_0 \sin (\omega t) + x_0 \cos (\omega t) $

$\displaystyle x = A \sin (\omega t + \phi) $
Either of these solutions shows that the frequency of oscillation is $\omega$ (independent of the amplitude). The relation between the two solutions is provided by the angle addition law for the sine. One finds that the constants appearing in the two solutions are related in the following way:

$\displaystyle v_0 = A \cos \phi $

$\displaystyle x_0 = A \sin \phi $

$\displaystyle A = \sqrt{v_0^2 + x_0^2} $

$\displaystyle \phi = \arctan (x_0 / v_0) $
These constants have the following interpretation: $A$ is the amplitude of the oscillation. $\phi$ is the phase of the oscillation. $v_0$ is the velocity at time $t = 0$. $x_0$ is the position at time $t = 0$.



"simple harmonic oscillator" is owned by rspuzio.

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See Also: quantum harmonic oscillator and Lie algebra, resonance, time-dependent harmonic oscillators


This object's parent.

Cross-references: position, velocity, relation, functions, TEMs, motion, Hooke's law, system

This is version 3 of simple harmonic oscillator, born on 2005-08-14, modified 2009-03-05.
Object id is 58, canonical name is SimpleHarmonicOscillator.
Accessed 12108 times total.

Classification:
Physics Classification: 40. (ELECTROMAGNETISM, OPTICS, ACOUSTICS, HEAT TRANSFER, CLASSICAL MECHANICS, AND FLUID MECHANICS)
Pending Errata and Addenda
None.
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Discussion
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classification by bloftin on 2005-09-01 16:43:30
For now I need to change the classification from (40-XX) to (40.).  Having a top level *-XX classification prevents it from showing up in the browse by subject for encyclopedia.

Ben
[ reply | up ]

Testing some escape charachters for html category with a generator has an injective cogenerator" now escape ” with "