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

48

THE PHENOMENA

Now the only forces acting on the top (besides the resistance
of the air) are:
(i) its weight;
(ii) the reaction at the point of contact of the toe with
the table.
The weight acting vertically downwards cannot hurry or
retard the precession.
In Fig. 24 let K and A be respectively the centre and radius
of curvature of P's path. The reaction at the toe can be
resolved into two components, one S, perpendicular to the plane
on which the top is spinning, due to the stiffness of the plane:
G(
S/
R
FIG. 24.
and another, R, in the plane due to its roughness. The component of R in the direction PK provides the necessary normal
acceleration, while the component along the tangent to the path
is either hurrying or retarding the precession.
If it is hurrying the precession we see by the rule given in
Art. 38, that the axle of the top rises * to a more vertical
position, and vice versa. Let us consider the cases in detail.
Let w be the velocity of spin, and Q the angular velocity
of P about K, and let PN=a. The direction of the friction
at P depends on the direction of motion of P, for it will act
contrary to the direction of the latter. At the start there
must be skidding if the top is dropped on to the plane.
Now the actual velocity of P is that relative to N+that of
N; or, the actual velocity of P (measured into the paper)
=a w-AQ2.
* The rise or fall of a top is more fully discussed in Chapter IV. from considerations of conservation of angular momentum and conservation of energy,