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

14

ROTATION ABOUT A FIXED AXIS.. the rate at which this angle is described is w - O-.in0.'. the angular velocity of P about 0 is v sin 0
OP
2. The angular velocity of a line in a plane is the rate of
increase of the angle it makes with some fixed line in that
plane.
For instance, if a stick is so thrown that its motion is in
a vertical plane, its angular velocity is the rate of increase
of the angle it makes with some vertical line in the plane
of motion (Fig. 3).
Notice that we speak of the angular velocity of a point
about a point, but of the angular velocity "of a line,"
simply-not "about" anything.
Fied Plae.
xFied Plane.
sI 0 X
FIG. 3. FIG. 4.
3. Angular velocity of a plane. The angular velocity of a
plane is the rate of increase of the angle it makes with any
plane fixed in space.
For instance, in Fig. 4, if 0 is at any instant the angle
between the fixed and revolving planes, the angular velocity,
de
w say, is Wt
4. Rigid Body. By a rigid body or system we mean one in
which the particles composing the body or system never alter
their positions relative to each other.
An ordinary walking stick is an approximate instance of a
rigid body: an umbrella is not.
5. Angular velocity of a rigid body. The angular velocity
of a rigid body is the angular velocity of any plane fixed in
the body.
In the accompanying figure the body is represented by a
cylinder. If 0 is at any instant the angle between the two
planes, then the angular velocity of the cylinder is
dO
dt