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\begin{large}\[ \vec{L} = I \vec{\omega} \]\end{large}

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|!559px-Spinning_top.jpg!|
|Spinning Top
from Wikimedia Commons: Image by User:Lacen|



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{excerpt}The rapidly spinning Symmetric Top exhibiting Precession under the force of [Gravity] is a classic Physics problem.{excerpt}

We assume that we have a symmetric top that can easily rotate about an axis containing its [center of mass] at a high [angular velocity] ω . If the top tilts at all from the vertical, so that its center of mass does not lie directly above the top's point of contact with the surface it's resting on, there will be a [torque (one-dimensional)] exerted on the top, which will act to change its [angular momentum (one-dimensional)]. What will happen?

 



h4. Solution

{toggle-cloak:id=sys} *System:*  {cloak:id=sys}Each of the two major directions of oscillation can be independently treated as a case of  [Simple Harmonic Motion].{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 Motion].{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 formula for the Simple Pendulum (see [Simple Harmonic Motion].


{latex}\begin{large}\[ \omega_vec{2\tau} = \sqrtvec{\frac{g}{Lr} \times \vec{F_{\rm 2g}}} \]\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}\[ F_{\rm g} = mg \]\end{large}

\begin{large}\[ \mid \vec{r} \times \vec{F_{\rm g}} \mid = rmg\; sin(\theta) \]\end{large}

\begin{large}\[ L = I \omega \]\end{large}

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

the ratio of frequencies is thus:

{latex}

\begin{large} \[  \left| \frac{\Delta \omega_vec{2L}}{\omegaDelta t} \right| = L \Omega sin(\theta) = rmg \; sin(\theta) \]\end{large}

\begin{large} \[ \Omega = \frac{rmg}{L} = \frac{rmg}{I\omega} _{1}}= \frac{\sqrt{\frac{g}{L_{2}}}}{\sqrt{\frac{g}{L_{1}}}} = \sqrt{\frac{L_{1}}{L_{2}}} \]\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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