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Lissajous Figure |
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Image generated by a Pendulum with two natural Frequencies. |
The figures commonly called Lissajous Figures were investigated 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 develop 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 oscillations along both directions are in whole-number ratio to each other (such as 1:2, or 2:3. The above figure has a ratio of 3:4). The simplest way to achieve this is by using a pendulum whose strings form a "Y" shape. The Lissajous figures are the paths taken by the pendulum bob during its swings, if viewed from directly above or below.
Solution
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Each of the two major directions of oscillation can be independently treated as a case of |
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Each direction is an independent case of |
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First, consider the Y support for the pendulum:
The pendulum effectively has length L2 when swinging in the horizontal plane in and out of the page, but length L1 along the horizontal direction in the plane of the page.
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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 L2 and the angular frequency of oscillation is given by the formula for the Simple Pendulum (see Simple Harmonic Motion.
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{table:cellspacing=0|cellpadding=8|border=1|frame=void|rules=cols} {tr:valign=top} {td:width=310|bgcolor=#F2F2F2} {live-template:Left Column} {td} {td} |!Lissajous Figure.PNG!| |Lissajous Figure from Wikimedia Commons: Image by Peter D Reid| {composition-setup}{composition-setup} {excerpt:hidden=true}Image generated by a Pendulum with two natural Frequencies.{excerpt} The figures commonly called *Lissajous Figures* were investigated 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 develop 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 oscillations along both directions are in whole-number ratio to each other (such as 1:2, or 2:3. The above figure has a ratio of 3:4). The simplest way to achieve this is by using a pendulum whose strings form 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:* {cloak:id=app} {toggle-cloak:id=diag} {color:red} *Diagrammatic Representation* {color} {cloak:id=diag} 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! {cloak:diag} {toggle-cloak:id=math} {color:red} *Mathematical Representation* {color} {cloak:id=math} 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
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the
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plane
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lying
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in
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the
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page,
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where
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the
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mass
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moves
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left
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and
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right,
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the
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pendulum
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length
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is
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the
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shorter
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L
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1
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and
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the
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angular
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frequency
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is
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}\begin{large}\[ \omega_{1} = \sqrt{\frac{g}{L_{\rm 1}}} \]\end{large}{latex} |
the
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ratio
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of
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frequencies
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is
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thus:
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}\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
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order
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to
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have
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a
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ratio
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of
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1:2,
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one
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thus
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needs
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pendulum
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lengths
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of
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ratio
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1:4.
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In
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order
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to
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get
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a
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ratio
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of
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3:4
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(as
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in
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the
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figure
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at
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the
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top
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of
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the
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page),
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the
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lengths
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must
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be
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in
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the
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ration
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9:16.
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