Motion which repeats after a fixed period of time, such as harmonic oscillation or orbital motion governed by a central inverse-square law force. This is an important class of physical situations that share mathematical features.
Motion which repeats after a fixed period of time, such as harmonic oscillation or orbital motion governed by a central inverse-square law force. This is an important class of physical situations that share mathematical features.
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Periodic motion repeats itself exactly after a given time, called the period T. The position, velocity, and acceleration are exactly the same at any time as they are a time T later.
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Everyone is familiar with periodic motion. The sun rising every day, a lion pacing back and forth in his cage, a swing in a park, the piston in an engine are all examples of motion that:
• recurs with a fixed period, T, and therefore
• has a frequency f=1/T where the units are cycles (or rotations…) per second.
Periodic motion is sometimes called harmonic motion, because it can be represented by a sum of sinusoidal functions with the lowest one at angular frequency \omeag_1 = \frac
, and all others at a higher harmonic (multiple) of this basic angular frequency, e.g. \omega_n = n \omega_1. The mathematics of doing this is called Fourier Analysis, and the resulting expression for x(t) typically looks like
X(t) = \Sigma etc…..
This type of expression, a sum of sinusoidal functions each of whose frequency is a multiple of some basic frequency is called a Fourier Series or sometimes a harmonic series. Fourier analysis is beyond the scope of this WIKItextBook. [but look up Wikipedia and give ref]