Classical Mechanics (Goldstein) · Problem 6

The Rigid Body Equations of Motion — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · torque-free-symmetrical-top-precession

Problem

(a) Show that the angular momentum of the torque-free symmetrical top rotates in the body coordinates about the symmetry axis with an angular frequency

\Omega

. Show also that the symmetry axis rotates in space about the fixed direction of the angular momentum with the angular frequency

\dot{\phi} = \frac{I_3 \omega_3}{I_1 \cos\theta},

where

\phi

is the Euler angle of the line of nodes with respect to the angular momentum as the space

z

axis. (b) Using the results of Exercise 15, Chapter 4, show that

\boldsymbol{\omega}

rotates in space about the angular momentum with the same frequency

\dot{\phi}

, but that the angle

\theta'

between

\boldsymbol{\omega}

and

\mathbf{L}

is given by

\sin\theta' = \frac{\Omega}{\dot{\phi}} \sin\theta'',

where

\theta''

is the inclination of

\boldsymbol{\omega}

to the symmetry axis. Using the data given in Section 5.6, show therefore that Earth's rotation axis and the axis of angular momentum are never more than 1.5 cm apart on Earth's surface. (c) Show from parts (a) and (b) that the motion of the force-free symmetrical top can be described in terms of the rotation of a cone fixed in the body whose axis is the symmetry axis, rolling on a fixed cone in space whose axis is along the angular momentum. The angular velocity vector is along the line of contact of the two cones. Show that the same description follows immediately from the Poinsot construction in terms of the inertia ellipsoid.

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