27.6: Stability of Top Spinning about Vertical Axis

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(from Landau) For \(\theta=\dot{\theta}=0, \quad L_{3}=L_{Z}, E^{\prime}=0 . \text { Near } \theta=0\),

\begin{equation}
\begin{aligned}
V_{\text {effective }}(\theta) &=\frac{\left(L_{Z}-L_{3} \cos \theta\right)^{2}}{2 I_{1}^{\prime} \sin ^{2} \theta}-M g \ell(1-\cos \theta) \\
& \cong \frac{L_{3}^{2}\left(\frac{1}{2} \theta^{2}\right)^{2}}{2 I_{1}^{\prime} \theta^{2}}-\frac{1}{2} M g \ell \theta^{2} \\
&=\left(L_{3}^{2} / 8 I_{1}^{\prime}-\frac{1}{2} M g \ell\right) \theta^{2}
\end{aligned}
\end{equation}

The vertical position is stable against small oscillations provided \(L_{3}^{2}>4 I_{1}^{\prime} M g \ell, \text { or }\), or \(\Omega_{3}^{2}>4 I_{1}^{\prime} M g \ell / I_{3}^{2}\)

Exercise \(\PageIndex{1}\)

Suppose you set the top vertical, but spinning at less than \(\Omega_{3 \text { crit }}\), the value at which it is just stable. It will fall away, but bounce back, and so on. Show the maximum angle it reaches is given by \(\cos (\theta / 2)=\Omega_{3} / \Omega_{3 \mathrm{crit}}\).