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

THE SLEEPING TOP

129

With the usual notation, let the axes (1), (2), (3) be drawn so
that OZ and (3) are nearly coincident, and the angle between
OX and (1) is Mew, that between Y and (1) being (- v)~
The angular velocities about (1), (2), (3) are
- 4 sin 0, 0, o respectively.
The angular momenta about (1), (2), (3) are
-AAi sin 0, A, CO respectively,
or - A AO, AO, Cw.
The angular momentum about OX
=(-Ar0)cos *+A 0cos(, +,) + Cw.
= -A; + CO.
Similarly about 0 Y
=A + Co)).
The torques about these axes are - Macyi, Mag$ respectively.
Hence, since the rate of change of momentum is equal to the
torque, and OX, 0 Y are fixed in space,
-Aii + Co7w= -Magi,
A+ (Ca,,w = Mag;
whence e+ 2xA, - (X2 - x) 2)= 0,
and il - 2X- (Xl2 - 2) = o.J
The equations can both be satisfied by putting
= i cos(Xt + e) and 1 = sin(Xt + ),
provided that X2 - 2XlX + I\2 -\2 = 0,
i.e. (X- h)2 =X2,
or X = XAl X2,
which shows that the general solution of S and I is
= r( cos IX + X2. t + el} + i2 {cos X - 2. t + e2},
= r1 sin{X +X. t+ el} +-r2{sin XI - 2. t+ e2}.
Hence the motion is the combination of two oscillations of
27Tr
periods -
141. Geometrical interpretation of the two oscillations.
The coordinates of the projection of H on the plane XY are
proportional to e and q.
Referring now to Fig. 56, p. 104, and substituting NX, X2 for
wo and a2 respectively, we see that P describes a circle uniformly
with velocity XA + X, while the image of P in the line ZR2