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

124

OSCILLATIONS

Making the same substitution in (ii) we have
A = [sin a + cos a. e] [Mga - CoG(i + X)
+ A (Q2 + 2M2X)(COS a - sin a. e)]
=- X sin a(Cw - 2AQ2 cos a) - A22 sin2a. e, employing (iii),
[(MI-ga- cosa)2 + A2 sin ].. + [A224- 2Mga 2 cos a + M2g2a2] =
i.e. +y -— J;
27r.the period is
where p = A- I / A2Q4 - 2MgaA 2 cos a + M2g2a2.
Since the quantity under the radical can be written
(AQ2 - Mga cos a)2 + (Mga sin a)2,

it is always positive,
Q
t \
\ i
\ I
\d rl

and we see that the position of steady
motion is stable, a result already established in Art. 104.
It should be noticed that the node P
(Fig. 74) travels uniformly round the
circle of steady motion with angular
velocity SQ;
PR 27r. the periodic time is -- = -;.the length of PR=2
N -, ~~~~~~~~~~~~~~~~~~~In.. ----^ _..... __ 27rA f 22
FIG. 74.,/A2Q4 - 2MgaAQ2 cos a + M2g2a2'
These oscillations have already been mentioned in Chapter IV.,
where they were accounted for by the inertia of the top
alternately resisting and hurrying the precession. Where the
disturbing force is being continually increased, as for instance
when a top is gradually sinking, these oscillations are not
regular but instantaneous only about the instantaneous position
of steady motion.
135. Case 2. The sleeping top.
Definition. A top is said to be asleep when the whole motion
of its axle is in the neighbourhood of the vertical, but not
necessarily passing through it.
If a small displacement be given to a sleeping top, two
independent vibrations are set up, namely the oscillations of
0 and f about their mean values; but since V is a function of
0 only, it follows that the periods of these two oscillations are