In Section 5.6, the precession of Earth’s axis of rotation about the pole was calculated o...

In Section 5.6, the precession of Earth’s axis of rotation about the pole was calculated on the basis that there were no torques acting on Earth. Section 5.8, on the other hand, showed that Earth is undergoing a forced precession due to the torques of the Sun and Moon. Actually, both results are valid: The motion of the axis of rotation about the symmetry axis appears as the nutation of the Earth in the course of its forced precession. To prove this statement, calculate θ and ϕ as a function of time for a heavy symmetrical top that is given an initial velocity ϕ0, which is large compared with the net precession velocity β/2a, but which is small compared with ω3. Under these conditions, the bounding circles for the figure axis still lie close together, but the orbit of the figure axis appears as in Fig. 5.9(b), that is, shows large loops that move only slowly around the vertical. Show for this case that (5.71) remains valid but now

From these values of θ and , obtain ω1 and ω2, and show that for β/2a small compared with ϕ0, the vector ω precesses around the figure axis with an angular velocity

in agreement with Eq. (5.49). Verify from the numbers given in Section 5.6 that corresponds to a period of about 1600 years, so that ϕ0 is certainly small compared with the daily rotation and is sufficiently large compared with β/2a, which corresponds to the precession period of 26,000 years.