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

62

OSCILLATIONS

been destroyed by friction, an angular velocity Q is reached
consistent with the values of K and Io, i.e., when
K
Steady motion will then follow; but any attempt to alter this
existing state of steady precession will be followed by oscillations
similar to those observed at the starting of precession.
It should be noticed that during the motion above described,
the gyroscopic resistance IwQ (Art. 45) is alternately greater
and less than the applied couple.
It is clear, then, that the "dip" and consequent oscillations are
entirely due to what we may call the inertia "about OZ" of
the frame, attached weight, and wheel; but if they possessed
the imaginary property of having no inertia "about OZ" as
suggested in Art. 59, there would be no dip and no oscillations.
In fact the gyroscope would precess exactly as it does in the
case of steady motion where there is no inertia to be overcome.
The reactions at the bearings of the wheel contribute slightly
to the oscillations, in the same way that the inertia does, but
are not the primary cause. The prim s,_eof the ocitions
is the inertia ofJLPsa teiy.,
63. The reason will now be seen for the phenomenon mentioned in the Introductory Chapter, that if a downward pressure
is applied at X (Fig. xxii.), when the gyroscope is spinning
slowly, the axle XX' dips appreciably before precession takes
place, but a sudden removal of the pressure will leave the
system oscillating violently: while if the spin is very fast,
the dip is hardly perceptible, and the oscillations are scarcely
more than a "shiver." For, in order that steady motion may
be arrived at, 0 (which equals -) must be large when ce is
small, and consequently the impulsive actions and reactions due
to the inertia will be large before the required value of Q is
reached; whereas, when o is large, f2 is small, and in this case
the dip and oscillations will be small also.*
A similar explanation to the above accounts for the oscillations
or ntutatiorns of a top. These are especially conspicuous just
before ts fall, when it goes through a sort of reeling motion.
The spin has by this time been considerably diminished, and
*The reason for this may also be partly seen from the parallelogram of
velocities. For, in the accompanying figure, if w' represents the velocity created
oy
by the applied couple about its own axis, it is clear that when w is large the
axis of resultant angular velocity is only slightly displaced; and vice versd.