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

CONSERVATION OF ANGULAR MOMENTUM

57

Let us regard the cat as made up of a fore part and a hind
part, whose moments of inertia I1, I2 are equal when the legs
are fully extended at right angles to the body. The photographs* given in Plate II. show that it first contracts its fore
legs (thereby making I1 less than I2) and then turns its fore
part round. This latter action necessitates the hind part being
turned in the opposite direction (since the total angular
momentum about the axis is zero) but to a less extent, since
I2 is > I1. The animal then contracts its hind legs, extends
its fore legs, and gives its hind part a turn. This necessitates
the fore part being turned in the reverse direction but, again,
to a less extent, since I1 is now > 12. It will thus be seen
that by continued action of this kind the cat can turn itself
through any required angle, though at no time has it any
angular momentum about its "axis."
[23.] If the cat were allowed to fall on to a cushion, resting on a smooth
floor, would the fall of the cat move the cushion?
57. The foregoing remarks may be summed up in the following two principles connected with the motion of a gyroscope
such as we have been considering:
1. Since there is no torque about a vertical axis, the angular
momentum about the vertical through the point of support must
throughout the motion remain constant, i.e. equal to the value,
whatever it was, at the beginning of the motion.
2. The torque, which is about a horizontal axis, mLust produce
angular momentum about that axis, and it cannot produce
angular momentum about any other axis.
These are of the utmost importance in considering the motion
of a gyroscope.
An explanation of the phenomenon of precession by the
principle of conservation of angular momentum will be found
in the Appendix.
58. Why precession increases as a top dies. It will be
remembered that if a gyroscope is spun with one end of its axis
in the socket of the vertical stand, Fig. XVI., then towards the
end of the motion it precesses faster and faster, as it descends,
until it finally flies off the stand.
Let us apply the principle of conservation of angular
momentum to the consideration of this phenomenon.
We will suppose the wheel to be spinning initially with
angular momentum Ie, its axis being inclined at an acute
angle 00 to the vertical. If we neglect all friction,t then the
*These photographs are reproduced from Nature (Nov. 22nd, 1894), by the
kind permission of the proprietors.
tIn reality the friction cannot be entirely neglected. The friction at the
socket resists precession, and that at the bearings of the wheel causes a gradual
diminution of its spin, but these retardations are not enough to affect the
statement that the precession increases as the gyroscope descends. A further
discussion, taking some of the friction into account, is given in Art. 66.