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

GYROSCOPIC RESISTANCE

115

In Fig. 68 let us suppose the system to be in motion under
the action of external couples K1, K2, K3 about OX, OY, OZ
respectively.
It is clear that the angular momentum
h1 is precessing about 0 Y with velocity 02,
and,,,, OZ,, 03.
Similarly for the angular momenta h2 and h3. Z
It follows that when the external couple K2 acts about O Y
the resistance offered is
due to h1, measured by h103,
and,, h3,,, -h301,1
the negative sign being taken, / X
since h3 sets itself away from the
torque-axis. y /
Hence, from moments about 0 Y, FI. 68.
we get
K2-hl3 + h31 = h2,
or K2 = h- h301 + ha03,
and two similar expressions as before.
126. It will be seen that our fundamental equation K=IwQl
(Art. 37) is a special case of the general equations, namely, when
01=02=0, 03 =,
h1=I/w, constant, h2= 0, h3 =C2,
K =K3=0, K2=K.
For in this case the first and third of the general equations
vanish, and the second becomes
h103 = K2,
or Iw2 =K.
127. Applications of moving axes.
Steady motion of a top on a fine fixed point. We will
now employ the above expressions to determine the motion of a
top spinning steadily as in Fig. 69, taking moving axes as
marked in the figure, which is entirely in the azimuthal plane,
axis (2) being perpendicular to (1) and (3) and towards the
reader. We have
01= -- sin a; wi= -2 sina;
02= =; = 0;
03 = Q cos a; 3 = ).

Also

hl=Aw; h2=A =w; h3=Cw.