Chapter 18.3, Problem 18.123P

Using the relation given in Prob. 18.121, determine the period of precession of the north pole of the earth about the axis of symmetry of the earth. The earth may be approximated by an oblate spheroid of axial moment of inertia I and of transverse moment of inertia $I\text{'}=0.99671$ . (Note: Actual observations show a period of precession of the north pole of about 432.5 mean solar days; the difference between the observed and computed periods is due to the fact that the earth is not a perfectly rigid body. The free precession considered here should not be confused with the much slower precession of the equinoxes, which is a forced precession. See Prob. 18.118.) If the earth were a sphere, the gravitational attraction of the sun, moon and planets would at all times be equivalent to a single force R acting at the mass center of the earth. However, the earth is actually an oblate spheroid and the gravitational system acting on the earth is equivalent to a force R and a couple M. Knowing that the effect of the couple M is to cause the axis of the earth to precess about the axis GA at the rate of one revolution is 25800 years, determine the average magnitude of the couple M applied to the earth. Assume that the average density of the earth is 5.51 g/cm³, that the average radius of the earth is 6370 km, and that $\bar{I}=\frac{2}{3}m{R}^2$ . (Note: This forced precession is known as the precession of the equinoxes and is not to be confused with the free precession discussed in Prob. 18.123.)