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

64

OSCILLATIONS

about its axis is Ie. Then Ic cos 00 is the angular momentum
about the vertical through the centre of the socket, and this
angular momentum would remain constant throughout the
motion, but for the diminution owing to the frictional couple
at the socket on the frame. We can neglect the effect of
friction on the spin of the wheel, for it will be small.
Now we have seen that the work done by the couple due
to gravity
= K.E. of precession
+ K.E. of descent or ascent (i.e. "dip"),
and since the latter of these two contributions is not large, it
is clear that most of the work done shows itself in K.E. of
precession, and for this reason the gyroscope precesses faster
as it falls.
Again, the work done by the frictional couple at the socket
= K.E. of descent (corresponding to precession)
+ K.E. taken from existing precession (corresponding to " dip ").
Of these two contributions the former is the greater, and
therefore the gyroscope also descends more quickly as it falls.
The same principles govern the motion of an ordinary
spinning top which is gradually falling from a vertical position,
or one which has never risen to the vertical, but is descending
from its position of steady motion.
The gravity couple, by a large number of excessively small
"dips," gradually causes the top to fall, and, simultaneously
with this, produces the more appreciable increase of precession.
The friction couple at the toe (besides destroying the spin
of the top) resists the increasing precession-the result, about
its own axis, corresponding to "dip"-but contributes, more
appreciably, to the falling of the top-the result corresponding
to precession. A further discussion of this motion is to be
found in Chapters VII. and IX.
If the toe of the top is free to move instead of being constrained as in the above instance, the same principles apply;
for we can consider the centre of gravity of the top as a fixed
point for purposes of rotation, and discuss its translational
motion independently.
67. To prove that when the axle of the gyrostat has been
released from its horizontal position, as much energy is lost
as is communicated to the system.
Let the axle (Fig. 30) on being released descend, by oscillations, through an angle d0 before steady motion is reached.
Let the angular momentum about the axle be Io, supposed
constant throughout: we shall however take into account the
friction at 0, which assists in destroying the oscillations. Let
Iz be the moment of inertia about OZ of the frame and wheel,
and Q the final velocity of precession.