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

GYROSCOPIC RESISTANCE

87

the top is started at an angle a with angular velocity a, and
either of the above values of S2, it will continue to spin steadily.
From the above quadratic equation we see that during steady
motion CoQ -Mag
cos a= A2
showing that if C is increased a diminishes,
and if A,, a increases;
hence it appears that a top with a longer leg spins at a greater
angle to the vertical.
87. When the majority of rotations in a problem under
consideration are right-handed it will be found convenient to
work throughout with right-handed axes. Whichever rotation
is employed, the precessional velocity must be considered
negative if it does not turn the angular momentum rotated,
towards the torque axis both being drawn in the same sense.
88. Equation deduced from gyroscopic resistance. The
equation of the preceding article may be obtained by considering
that the torque Mga sin a is tending to turn about its axis the
two (right-handed) components of angular momentum Cw, and
AQ~ sin a; but these components, instead of being turned about
O Y, are precessing about OA and O G with velocities Q sin a
and 2 cos a respectively.
Hence, the gyroscopic resistances to being turned are respectively CCQ sin a and AQ2 sin a cos a (Art. 45).
If the torque axis is drawn right-handed we see that OG
sets itself towards the torque, while OA sets itself away from
the torque.
Thus, Cac, sina must be considered a positive resistance,
AQ22 sin a cos a must be considered a negative resistance.
Since there is no change in angular momentum about the
torque axis we have
Mlga sin a - CwQ sin a +Af22 sin a cos a = 0,
or Mga = Cw2 - A Q2 cos a, as before.
The general equations of which the above are a particular case
are given in Art. 125. 1: Pt-$89. The equation of Art. 86 might also be obtained by
rotating the momentum axis instead of the horizontal component of angular momentum.
In Fig. 46, let A be at any instant the axis of precession*
* It should be noticed that the term n" axiLf Lresion," which is suggested
by astronomy, is not strictly in accordance: it astronomical usage, where the
axis of the earth is said to precess about a line perpendicular to the plane of its
orbit, and not to its own axis. In the same way a top spinning at an inclination
to the vertical can be said to precess about the vertical; but we are taking the
axis of precession to be the axis perpendicular to the rotating torque and
the angular momentum rotated.