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Simple Harmonic Motion


Simple harmonic motion refers to the periodic sinusoidal oscillation of an object or quantity. Simple harmonic motion is executed by any quantity obeying the differential equation

 x^..+omega_0^2x=0,
(1)

where x^.. denotes the second derivative of x with respect to t, and omega_0 is the angular frequency of oscillation. This ordinary differential equation has an irregular singularity at infty. The general solution is

x=Asin(omega_0t)+Bcos(omega_0t)
(2)
=Ccos(omega_0t+phi),
(3)

where the two constants A and B (or C and phi) are determined from the initial conditions.

Many physical systems undergoing small displacements, including any objects obeying Hooke's law, exhibit simple harmonic motion. This equation arises, for example, in the analysis of the flow of current in an electronic CL circuit (which contains a capacitor and an inductor). If a damping force such as Friction is present, an additional term betax^. must be added to the differential equation and motion dies out over time.


See also

Damped Simple Harmonic Motion, Harmonic Addition Theorem, Simple Harmonic--Motion Quadratic Perturbation, Uniform Circular Motion

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Cite this as:

Weisstein, Eric W. "Simple Harmonic Motion." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SimpleHarmonicMotion.html

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