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

106

PATH OF THE HEAD OF THE TOP

in each case start in the invariable plane (Art. 103). In Fig. 60
00, the upper limit, is zero.
Other cases can be represented in a similar manner. In
Fig. 61, where 00 is acute, an initial velocity v has been given
at S to the head of the top in the azimuthal plane. At this
SI i f R. ~ ~ I I X
H R, R HR,
FIG. 58. FIG. 59. FIG. 60.
position 4 is zero, and therefore it will change sign as H
returns with velocity v through the same position in the
azimuthal plane. The path consequently has a loop as shown..,o -- -_.., ' "x
(2 z R
FIG. 61. FIG. 62. FIG. 63.
In Fig. 62 00 is acute, and an initial velocity has been given in
the direction of precession, causing the top to rise, though not
necessarily, to a vertical position.
In Fig. 63 00 is zero, but again an initial velocity v has been
given to H in any direction. H therefore returns through
its starting point with the same velocity, thus describing a series
of loops.
109. Motion of any solid of revolution spinning on a smooth
horizontal plane. Considering the case where G has no initial
motion, we see that since the plane is smooth, the spin w of
the body about its axle remains constant, and G only moves
in a vertical line.
Let 0 be the inclination to the vertical at any time of the
axle of the body, GN= z, GL =, PL = y (Fig. 64).
We can consider the whole motion of the body as comprised
independently of the translational motion of G and the rotational
motion about axes through G.