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

PRECESSION

39

37. The preceding results may be stated more generally as
follows, without any reference to vertical or horizontal (see
Art. 85):
If a bod yyhich has angular momentum Ico about an axis
OX T under The acineo or uqe eodicuhe_- IatfI
akis 0 Y, then the an EMOar m nomen whil be rtcatec aboui
ethird er endicular it ty L
minne bythe e uatjpn a. = IS.
38. Rule for direction of precession. It will be clear from
the above discussion that: If the rotated angular momentum
and the torque axis are drawn in the same sense, then the
angular momentum sets itself towcards the torque axis.
39. It should be noticed that although the gyroscope in
Fig. 19 is rotating about the vertical, yet the torque does not
create angular momentum about the vertical. It creates angular
momentum about its own horizontal axis, which combining, as
fast as created, with the angular momentum about the axle,
causes the axle to take up successive positions in the plane XY.
We have already said (Art. 34) that our investigations refer
to the caintenance of an existing motion: otherwise our
equation K = Iwo2 involves the paradox that if w =0, i.e. if
the body is not spinning, then Q2, the precessional velocity, is
infinite! This is, of course, not true, but it is true that a wheel
spinning with a small velocity precesses with a large velocity,
and vice versa.
A full discussion of all that occurs when precession is being
started, is given in Chapter IV., where it is shown that
oscillations are set up which are quickly destroyed by friction,
and in many cases are scarcely visible. In instances given
before Chapter IV., where, strictly speaking, the starting of
precession is in part involved, the preliminary phenomena will
either be assumed or neglected.
[8.] Explain what effect the couple due to the action of gravity on a
Diabolo spool will have, if the string is not quite under the centre of gravity,
(i) when the spool is not spinning; (ii) when it is.
40. It is clear that the investigation of Art. 34 is equally true
when K varies and consequently Q2 as well.
For let 0 be the angle which the axle makes at any instant
with its initial position.
Then 60O=AOA' and AA'=Io680.
Hence the increase of angular momentum in time 6t is about
OB and is equal to Ic630: namely, the rate of change of angular
momentum is equal to
dO
Ieo. -dt= K.