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

86

FINE PEG

moment about the axle, and therefore diminish the spin (see
Art. 128). Let the directions of rotation be those marked in
Fig. 45.
The external forces acting on the top are:
(i) its weight Mg at G;
(ii) a vertical reaction R at 0;
(iii) a horizontal reaction F at 0.
This last must act in the direction marked, since G describes
a horizontal circle about OZ, and F, the force causing it to do so,
must be parallel to the inward radius.
From Art. 82 we see that the momenta are CO about GO, and
AM2sin a about AO, where A is the moment of inertia about
the axis through 0.
These give component momenta AQ2 sin a cos a - CO sin a about
OX, and a component about OZ, the latter of which need not be
considered, as it is not rotated and no torque therefore is required
on account of it.
The former is rotated with angular velocity -Q about OZ,
and this requires a torque
(AO2 sin a cos a - Cw sin a)(- Q)
about OY (Art. 37).
But this torque is Mgasina, where OG= a.
Hence, CwQ sin a - A22 sin a cos a = Mgca sin a,
whence either sin a = 0,
or ceQ- A Q2 cos a = Mga.
The former gives a = 0 or 7r, both of which are possible angles
for steady motion.
The latter gives a quadratic for 2, i.e.
A cos a. Q2- C. Q+l Mag= 0,
showing that there are, in general, two possible angular velocities
of precession, i.e.
w Cw /OC2W2- 4A cos a. Mag
2A cos a
If cos a is negative, i.e. G below O, 2 is always real.
If cos a is positive, then for real values of Q2 we must have
C22 > 4A cos a. Mag,
2./AMag. cos a
or > -
If w has a smaller value than this, the top cannot spin steadily
(compare Art. 102).
The larger or the smaller value of Q2 will be maintained
according to the particular initial conditions of motion. Thus if