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

CENTRE OF GRAVITY INITIALLY IN MOTION

105

The condition that 0 should be zero gives a cubic in cos 0,
having one real root between cos 0= -1 and cos 0= cos 00, and
another real root between cos 0 = cos 00 and cos 0 = 1. The third
root is inadmissible, being greater than unity.
If D is the initial angular
momentum about the verti- Z
cal, which remains constant L
throughout the motion, we
have in this case
A sin20+C cos 0 = D,
or, if we write D = Cwb cos 0
where b and 00 are constants,
+. Cw(bcos00 - cos0) A / Cet
AVAsin20
Also tan X A sin0 t
AlsoCtan O=)^ - Al/rsin0 ',
b cos 00 - cos 0
sin 0 FIG. 57.
Hence the resisting torque becomes
C22(b cos 00- cos 0)(1 - b cos 00 cos 0)
A sin3 0
and the oscillations in the azimuthal plane are given by
C2 ' (b cos 00- cos 0)(1 - b cos 00 cos 0)
AO = ga sin 0 -30
the integral of which equation gives the equation of energy.
108. Path in space of H, the head of the top. We can now
discuss the general appearance of the path of H in space when
viewed from the point Z vertically above the origin. The paths
will vary according to the initial conditions of motion; but in
all cases, where ow is sufficiently large to prevent the top from
falling altogether, the head will oscillate between the two
positions R, R2, in the azimuthal plane, while the azimuthal
plane itself rotates about the fixed vertical. The actual velocity
of H in space is that resulting from these two motions.
Since 4r is a function of 0 only, it is the same for all angular
positions of H equidistant from Z.
It is also clear that H passes and returns through any one
position in the azimuthal plane with the same velocity.
We will first consider the case in which G has no initial
velocity.
Here the starting point S coincides with R1, the upper limit
of H. In Fig. 58 00 is an acute angle, in Fig. 59 it is obtuse.
Since G (and therefore H) has no initial velocity, H must