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

OSCILLATIONS

63

consequently the friction which in the earlier part of the motion
tended to "damp" the oscillations.
64. The above investigations may now be summed up as
follows:
Whenever we apply to the axle of a spinning gyrostat a
torque which does more than maintain existing precession,
this torque produces three results:
(a) Oscillations are set up and continue until destroyed by friction.
They may be so quickly destroyed, or so slight, that they are hardly
appreciable to the eye.
(p) The torque produces, about its own axis, an effect which is not
always appreciable to the eye, and corresponds to the dipping of the
gyroscope in the preceding articles. Thus we see that work is done
on the gyrostat.
We shall in future pages allude to this effect of a torque
about its own axis as "dip."
(y) It produces about an axis perpendicular to itself and to the
axle, an effect which is much more appreciable to the eye, and
corresponds to precession. Thus we realise the increased kinetic
energy due to the work done on the gyrostat.
This effect of the torque we shall allude to as precession.
Lastly, it must be remembered that there canb-e no change
of angular momentum about an axis round which no torque
has acted.
65. Explanation by the principle of energy of "Hurry the
precession, the top rises." We shall now see more clearly
(compare Art. 47) why a spinning top rises if we hurry its
precession. For we have seen that the "hurrying" force
produces only a small hurrying effect on the precession (corresponding* to "dip"), but the turning effect (corresponding to
precession) is much more appreciable, and raises the top.
We are, of course, doing work on the top by hurrying the
precession, and since the kinetic energy of precession is not
appreciably increased, the potential energy of the top must
be increased, i.e. the centre of gravity of the top rises. In
the case when friction at the toe alone hurries the precession,
the work which raises the top is done at the expense of the
kinetic energy of spin of the top.
When precession is retarded, work is done by the top, which,
since it does not lose much kinetic energy of precession, loses
potential energy, i.e. the centre of gravity descends.
66. Explanation by the principle of energy of the increase
of precession in a dying top. As in Art. 58, let us consider
that a gyroscope is placed in the socket of its stand with its
axis at an angle 00 to the vertical, when the angular momentum
* As the spin of the top diminishes the hurrying effect is greater. See Art. 63.