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

OSCILLATIONS

61

61. The above phenomenon of the dipping of the gyroscope
illustrates also the second principle laid down in Art. 57, i.e.
the torque can only produce angular momentum about its
own axis. For it is clear that when the system is precessing
its kinetic energy is greater than before it was released; since,
as far as the spin of the wheel is concerned, it is the same, and
the wheel, frame, and weight are now precessing. The fact is,
this excess of kinetic energy is provided by the work done by
the torque in turning the system about its own axis, and
causing it to dip. Hence the dip is necessary for conservation
of energy as well as for conservation of angular momentum.
62. Oscillations. This precession will not at first be steady:
for we have seen that whether precession is taking place steadily
or not, the angular momentum about the vertical of the frame,
weight, and wheel (regarded as not spinning) must be cancelled
by the angular momentum about the vertical due to the spin
of the wheel-which necessitates the dipping of the axle.*
Hence the precessional velocity depends on the dip, and alters
if the dip is altering. There will however be a steady value
of Q2, consistent with IcL and K, and also a steady angle at
which the axle dips to the horizontal: and this steady or
mean value of the "dip" and of the precessional velocity Q,
is arrived at in the following manner by a series of impulsive
jerks and oscillations which are eventually destroyed by
friction.
The particles which at first, by their inertia, retard the
precession, as soon as "dip" takes place acquire, by impulsive
action, a velocity of precession greater than the mean velocity []
of the axle, and now, owing to their momentum, have the effect
of hurrying the precession. The next instant the precessional
velocity of the axle will be greater than that of the particles,
and their inertia will again retard precession; and so on.t But
every time the precession of the axle is hurried or retarded the
centre of gravity of the system rises or falls (Art. 47). The
result is that, in addition to the backward and forward
"wobbling" the gyroscope oscillates up and down about the
mean position of "dip," until finally, the oscillations having
* By hurrying or retarding the precession the angle made with the horizontal
by the axle of the gyroscope can be diminished or increased: but if the system is
left to itself it must precess with a small " dip."
Although the axle is precessing at an inclination to the horizontal we may
still employ the equation K=Iw&2 since the angle in question is small. In
Chapter VI. this equation is discussed for all inclinations.
t The successive retarding and hurrying effect of the inertia can here again be
most clearly realised by observing the suspended weight, which swings backwards
and forwards in jerks about a mean position.
An analogy to this starting of precession may be found in the motion of two
railway trucks coupled together at rest, when one receives a sudden impulse.
Each in turn assists and hinders the other's motion, until the same steady motion
is arrived at for both.