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|!Mass Between Two Springs.PNG!|
|Mass on a frictionless surface between two springs|



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{excerpt:hidden=true}A case of [Simple Harmonic Motion].{excerpt}

 



h4. Solution

{toggle-cloak:id=sys} *System:*  {cloak:id=sys}   [Simple Harmonic Motion].{cloak}

{toggle-cloak:id=int} *Interactions:*  {cloak:id=int} [Simple Harmonic Motion] with theThe forces due to the compression or extension of the two springs acting as the [restoring force].{cloak}

{toggle-cloak:id=mod} *Model:* {cloak:id=mod} [Simple Harmonic Motion].{cloak}

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

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

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|!Mass With Two Springs Displaced.PNG!|

The mass *m* is subjected to force from the springs on each side. If you assume that the springs are in their relaxed state when the mass is at rest between them, then displacement of the mass to the right (as shown) compresses the spring on the right and extends the spring on the left. this results in [restoring force] to the left from _both_ springs. 

|!Mass With Two Springs Displaced with forces.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 formula for the Simple Pendulum (see [Simple Harmonic Motion].


{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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