Writing the absolute magnitude of the angular momentum as L, \(L_{2}=L \sin \theta\) (remember L is in the Z direction, and \(x_{1}\) is momentarily along ON ) so the rate of precession \(\dot{\phi}=L...Writing the absolute magnitude of the angular momentum as L, \(L_{2}=L \sin \theta\) (remember L is in the Z direction, and \(x_{1}\) is momentarily along ON ) so the rate of precession \(\dot{\phi}=L / I_{1}\). Finally, the component of \(\vec{L}\) along the \(x_{3}\) axis of symmetry of the top is \(L \cos \theta=I_{3} \Omega_{3}\), so the top’s spin along its own axis is \(\Omega_{3}=\left(L / I_{3}\right) \cos \theta\).
The inertial properties of a body for rotation about a specific body-fixed location is defined completely by only three principal moments of inertia irrespective of the detailed shape of the body. As ...The inertial properties of a body for rotation about a specific body-fixed location is defined completely by only three principal moments of inertia irrespective of the detailed shape of the body. As a result, the inertial properties of any body about a body-fixed point are equivalent to that of an ellipsoid that has the same three principal moments of inertia. The symmetry properties of this equivalent ellipsoidal body define the symmetry of the inertial properties of the body.
