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

38

PRECESSION

We will consider the frame to be so light that we are only
concerned with the motion of the wheel itself.
Let OX, OY, OZ (Fig. 20) be three mutually perpendicular
lines, OZ being vertical. Suppose the gyroscope rotating with
angular velocity w about OX, its axle, whilst the latter rotates
about OZ with angular velocity Q. The component angular
momenta of the gyroscope are Ice and I'T2 where I, I' are the
moments of inertia about OX and OZ. Let OA, OC represent
these components in magnitude and direction.
After time st the angular momenta are Ic and I'T, about
OA' and 0C, where the angle AOA'=Q6t.
Then if A'B be parallel to OA, OB represents the change of
angular momentum in time 6t, for if compounded with OA and
OC it yields OA' and OC.
But OB = AA'= OA x (QS6t) = Io28t;. the change of angular momentum in time dt is about OB and
= Iw6st;. the rate of change of angular momentum is about OB and
= I2.
But this requires for its production a torque IwQ2 about OB.
Thus the rotation of the axle about the vertical requires for
its maintenance the application of a torque K, say, perpendicular
to il? vertical, and to the axle of the gyroscope, such that
K=IWc2.
35. In the gyroscope of Fig. 19, the necessary torque is
supplied by the weight M1g of the system, and the equal and
opposite vertical reaction at the point of support, which together
have a moment Mga, where a is the distance of the centre
of gravity from the point of support.
Hence, in this case, Mga = IcQ-2.
Dimensions. Mcga = [-T]2 [L]
Io2Q = [M][L]2 [
It should be noticed that the centre of gravity of the system
describes a horizontal circle with uniform angular velocity Qt
under the action of the horizontal reaction at the point of
support. (See Art. 43.)
36. Precession. The rotation of the axle OX of the gyroscope, by a torque about 0Y, round the third perpendicular axis
OZ we shall call a precessional motion, and the axle will be said
to precess about OZ, the axis of precession.
The equations of steady motion of a gyroscope or top with
its axle inclined to the vertical are given in Chapter VI.