26.2: Precession of a Symmetrical Top
- Page ID
- 30538
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A more interesting case is the free rotation (zero external torque) of a symmetrical top, meaning \(I_{1}=I_{2} \neq I_{3}\).
We can take any pair of orthogonal axes, perpendicular to the body’s symmetry axis, as the \(x_{1}, x_{2}\) axes. We’ll choose \(x_{2}\) following Landau, as perpendicular to the plane containing \(\vec{L}\) and the momentary position of the \(x_{3}\) axis, so in the diagram here \(x_{2}\) is perpendicularly out from the paper/screen, towards the viewer.
This means the angular momentum component \(L_{2}=0\) and therefore \(\Omega_{2}=0 . \text { Hence } \vec{\Omega}\) is in the same plane as \(\vec{L}, x_{3}, \text { and so the velocity } \vec{v}=\vec{\Omega} \times \vec{r}\) of every point on the axis of the top is perpendicular to this plane (into the paper/screen). The axis of the top \(O x_{3}\) must be rotating uniformly about the direction of \(\vec{L}\).
The spin rate of the top around its own axis is
\begin{equation}
\Omega_{3}=L_{3} / I_{3}=\left(L / I_{3}\right) \cos \theta
\end{equation}
The angular velocity vector \(\vec{\Omega}\) can be written as a sum of two components, one along the body’s axis \(O x_{3}\) and one parallel to the angular momentum \(\vec{L}\) (these components are shown dashed in the figure)
\begin{equation}
\vec{\Omega}=\vec{\Omega}_{\text {precession }}+\vec{\Omega}_{3}
\end{equation}
The component along the body’s axis \(O x_{3}\) does not contribute to the precession, which all comes from the component along the (fixed in space) angular momentum vector.
The speed of precession follows from
\begin{equation}
\Omega_{\text {precession }} \sin \theta=\Omega_{1}
\end{equation}
and
\begin{equation}
\Omega_{1}=L_{1} / I_{1}=\left(L / I_{1}\right) \sin \theta
\end{equation}
so
\begin{equation}
\Omega_{\text {precession }}=L / I_{1}
\end{equation}
Note also the ratio of precession rate to spin around axis is
\begin{equation}
\Omega_{\text {precession }} / \Omega_{3}=\left(I_{3} / I_{1}\right) \sec \theta
\end{equation}
This means the precession rate and the spin are very comparable, except when \(\theta\) is near \(\pi / 2\), when the precession becomes much faster. Remember this is the body’s precession with no external torque, and is clearly completely different—much faster precession—than the familiar case of a fast spinning top under gravity.