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|!Lissajous Figure.PNG!|
|Lissajous Figure
from Wikimedia Commons: Image by Peter D Reid|



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{excerpt:hidden=true}Image generated by a Pendulum with two natural Frequencies.{excerpt}

The figures commonly called *Lissajous Figures* were investif=gatedinvestigated  by Jules Antoine Lissajous in the 1850s, but had earlier been generated and studied by Nathaniel Bowditch of Salem, Massachusetts in 1815, who built a special pendulum to generate them. Later, a Scottish Professor named Hugh Blackborn would create a similar device, which would devlopdevelop into the "Harmonograph". The figures it generated would become a 19th century craze.

The basic Lissajous figure is made up of two sinusoidal oscillations along directions at right angles to each other, and when the periods of the osillationsoscillations along both directions are in whole-number ratio to each other (such as 1:2, or 2:3. The abioveabove figure has a ratio of 3:4). The simplest way to achieve this is by using a pendulum whose supportstrings hasform a "Y" shape. The Lissajous figures are the paths taken by the pendulum bob during its swings, if viewed from directly above or below. 



h4. Solution

{toggle-cloak:id=sys} *System:*  {cloak:id=sys}Each of the two major directions of oscillation can be independently treated as  [simple oscillator].{cloak}

{toggle-cloak:id=int} *Interactions:*  {cloak:id=int}Each direction is an independent case of Simple Harmonic Motion with Gravity and the Tension in the String acting as the Restoring Force.{cloak}

{toggle-cloak:id=mod} *Model:* {cloak:id=mod} Simple Harmonic Oscillator.{cloak}

{toggle-cloak:id=app} *Approach:*  

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{toggle-cloak:id=diag} {color:red} *Diagrammatic Representation* {color}

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First, consider the *Y* support for the pendulum:

!Lissajous Figure 1.PNG!

The pendulum effectively has length *L{~}2{~}* when swinging in the horizontal plane in and out of the page, but length *L{~}1{~}* along the horizontal direction _in_ the plane of the page. 

!Lissajous Pendulum 2.PNG!
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{toggle-cloak:id=math} {color:red} *Mathematical Representation* {color}

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We ignore the distribution of tensions in the upper cables, and simply view the pendulum as a simple pendulum along either the plane of the drawing or perpendicular to it. In the plane perpendicular to the drawing (where the mass oscillates toward and away from the reader) the pendulum length is *L{~}2{~}* and the angular frequency of oscillation is given by the formmula for the Simple Pendulum.


{latex}\begin{large}\[ \omega_{2} = \sqrt{\frac{g}{L_{\rm 2}}} \]\end{large}{latex}

Along the plane lying in the page, where the mass moves left and right, the pendulum length is the shorter *L{~}1{~}* and the angular frequency is

{latex}\begin{large}\[ \omega_{1} = \sqrt{\frac{g}{L_{\rm 1}}} \]\end{large}{latex}

the ratio of frequencies is thus:

{latex}\begin{large}\[ \frac{\omega_{2}}{\omega_{1}}= \frac{\sqrt{\frac{g}{L_{2}}}}{\sqrt{\frac{g}{L_{1}}}} = \sqrt{\frac{L_{1}}{L_{2}}} \]\end{large}{latex}

in order to have a ratio of 1:2, one thus needs pendulum lengths of ratio 1:4. In order to get a ratio of 3:4 (as in the figure at the top of the page), the lengths must be in the ration 9:16.

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