An elementary treatment of the theory of spinning tops and gyroscopic motion, by Harold Crabtree.

CHAPTER VII.
GENERAL MOTION OF A TOP.
97. General motion of a top spinning on a fine fixed point.
Hitherto we have only discussed the motion of a top when
its axle makes a constant angle a with the vertical. We will
now consider the case where this angle may be a variable
quantity 0 throughout the motion.
It should be particularly remembered throughout this chapter
that all frictions are neglected except where otherwise stated.
Let 0 be the angle which the axle makes with the vertical
through 0 (Fig. 53), w the spin of the top, and, the azimuthal

".si
t sin 0

/
/
/
/
/
/
Y //

FIG. 53.

velocity, V being the angle the azimuthal plane makes with
some plane fixed in space and passing through OZ.
Since there are no forces acting about the axle, with regard
to which the top is symmetrical, w will remain constant.*
* A more rigorous proof of this is given in Art. 128, where it is shown that the
condition of symmetry is essential.
The reader is reminded that, as in Art. 82, X is equal to cw, + cos 0, where
wt is the (now variable) velocity relative to the azimuthal plane.
G