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

ANGULAR MOMENTA

83

If w, is the angular velocity of the top about GO relative to
the azimuthal plane, then the total angular velocity of the top
about GC is given by w = a++ Q cos a, and the total angular
velocity about GA is Q sina.
Thus the angular velocity of the top is the resultant of
(i) the spin w about the axle GC;
(ii) the spin Q sin a about the axis GA.
About an axis perpendicular to GC and GA there is no
angular velocity, since a is constant.*
The momenta. Let the moments of inertia about GC, GA
be C, A, respectively.
Since GC, GA are principal axes, the angular velocity o about
GC gives an angular momentum
component Cw about that axis (and z
none about any other axis); so the
angular velocity Q sin a about GA
gives a component A2 sin a about
GA.
Thus the angular momentum of
the top is the resultant of CO and
A 2 sin a about GC and GA respec- //
tively. A
83. Velocity axis. The axis of
component angular velocity lying /
in the azimuthal plane we shall call // ' sin a
the velocity axis. It is clear (Fig. C
42) that it makes an angle 3 with '
~A sin a
GC given by tan p=. The FIG. 42.
axis of total resultant angular velocity is called the instantaneous
axis of rotation.
84. Momentum axis. The axis of component angular momentum lying in the azimuthal plane we shall call the momentum
z
* The motion will perhaps be more easily
realised if we consider a small roller such as
is sometimes used for mounting photographs.
In the accompanying figure, Z'CH, which H
corresponds to the azimuthal plane, is turn-
ing about Z'C with velocity t~, which has r
about GO a component velocity 92 cos a; the /
roller corresponds to the top and turns about /
its axle GC with velocity w, relative to the /
frame, but with total velocity c, where
w = w, + i cos a. When we watch a 'top
spinning, we do not think of its spin rela- C
tive to the azimuthal plane, but of its total
spin about its axle. C