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

APPENDIX

135

On this system we shall suppose a couple whose axis is YOY'
to act in such a direction that it tends to turn A and B out
of the paper towards the observer, and C and D into the paper
away from the observer.
We shall show that any such rotation set up by this couple
results in a set of forces being called into play which tend to
make the system precess about the third perpendicular axis OZ.
Z
OA ^B
FIG. B.
By reason of its rotation about OX the particle A is increasing
its distance from the axis OY'.
Thus, if it has any rotation about this axis due to the couple,
there will be a tendency to diminish the speed of this rotation
in consequence of the increasing distance of the particle from
this axis.
This retarding tendency may be represented by a force acting
on the particle A perpendicular to the plane of the paper and
away from the observer.
Again, the particle B is getting nearer to the axis 0 Y of the
couple and so must be increasing its rate of rotation about this
axis. This acceleration of rotation may be supposed to be
caused by a force through B perpendicular to the plane of the
paper and towards the observer.
Similarly, the effects on the particles C and D may be supposed
caused by forces acting on C, D perpendicular to the plane of
the paper and respectively towards and from the observer, since
the rotation about YOY' is causing C and D to move away
from the observer.
We see that we thus have forces at A and D away from the
observer, and at B and C towards the observer.
Clearly these forces will make the system precess about
the third axis OZ.
Thus, when the body is rotating about OX, the application
of a couple about YO makes the body turn about the third
axis OZ.