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

120

STABILITY OF ROTATION

which is a real quantity, and the motion is stable. Similarly
if the spin 2 is about the axis of wc; but if it is about the axis
of W2, the period becomes
27Q / B A -)
S2 (Bi-A)(2- )'
which is imaginary.
Hence the motion is stable for rotation about OA or O0
but unstable about OB.
132. Diabolo which will not spin. The reader will remember
that when he is beginning to spin a Diabolo spool by means of
the string, unless the two portions of the string are absolutely
in the same plcane and that at right angles to the axle of the
spool, then, besides the spin about the axle which he means to
communicate, there is also a residuum "wobble" about a
transverse axis; but as he continues his spinning this wobble
begins to disappear and the spool settles down to a steady
rotation. The reason for this is that the axle of the spool is in
general an axis of minimum moment of inertia (though if it is
one of maximum moment the same argument applies) and the
instantaneous axis tends to revert to the position of the axle
of the spool. If we construct a spool which is dynamically
equivalent to a sphere we shall find that it is impossible to
spin it; for, since the moments of inertia about all axes are
equal, the spool has no tendency to revert to its axle as instantaneous axis of rotation. The result is that the axle wanders
about indefinitely in space, though with considerable rapidity.
If this Diabolo were already spiqning about its axle, it would
possess a certain degree of stability; for, the spin about the
axle being large compared to that generated by any disturbing
force about another axis, the instantaneous axis would deviate
only to a very slight degree from the axle of the spool.'
(See Art. 63, footnote.)
A heavy conical sheet projecting equally on either side of the
vertex whose semi-vertical angle is equal to tan-'1/2 (Fig. 71),
has the dynamical properties of a sphere; but such an ideal
construction is impracticable. Since, however, the addition of
matter inside the angles AOB', A'OB would increase the moment
of inertia about the axis of the cone, while the addition of matter
inside the angles AOB, A'OB' would increase the moment of
inertia about a transverse axis, we can on such a skeleton cone
form a material double hollow cone, in the form of a Diabolo
spool which is dynamically equivalent to a sphere; and this
Diabolo we shall find almost impossible to spin.
- It is for this reason that in the game of Cup and Ball, when it is required to
jerk the ball so that the axial hole catches on the wooden peg, it is necessary
to first give the ball a spin about the axis containing the hole and the point
where the string is fastened to the ball; for then a lateral jerk of the string does
not appreciably displace the axis of rotation.