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

REPRESENTATION OF ANGULAR VELOCITY

33

that we have a thin wire rod, OA, on which a small bead P
is moving with velocity Ac relative to the end 0 of the rod
from which it started, while the rod moves with velocity v
in space, keeping parallel to itself, in the direction OB. The
total velocity in space is then along a definite straight line OD
passing through the original position of 0 in space.
B D
f yII. Al
FIG. 15.
The same total velocity would have been obtained by taking
the velocity v of the particle relative to a rod OB, while
OB is moved parallel to itself in the direction OA with a
velocity u.
22. It will be noticed that at any instant the total velocity
of P in the direction O'A' in space is equal to that relative to
O' + that of O' =u+vcosa,
which, if a= 90~ (namely if OA, OB are at right angles to each
other) =
Hence in this case the total velocity of P in the direction
O'A' in space, is the same as the velocity relative to the rod; so
that, when we compound two velocities at right angles, each
component is the total velocity of the particle in the direction
considered. Similarly, if a particle possesses three simultaneous
velocities u, v, w not in one plane, we must further suppose the
whole frame AOB to move in space with velocity w keeping
parallel to itself, in a direction OC not in the plane of the
paper; and if the three directions OA, OB, OC, are mutually
at right angles, each of the components is the total velocity of
the particle in the direction considered.
23. We will now extend the above observations to the case of
angular velocities or rotations.
Simultaneous Rotations. Definition. If a rigid body is
rotating with angular velocity wo about some axis OA, fixed in
the body, relative to a frame in which that axis is fixed,
while the frame rotates about some fixed axis OB with angular
velocity W2, then the total angular velocity of the body is said
to be compounded of the two simultaneous rotations Wo, Wo2 about
the axes OA, OB.
c