This is effected automatically in the Weston centrifugal machine (fig. 10) used for the separation of water and Centrifugal machine. molasses, by the friction of the indiarubber cushions above the support; or else the spindle is produced downwards below the drum a short distance, and turns in a hole in a weight resting on the bottom of the case, which weight is dragged round until the spindle is upright; this second arrangement is more effective when a liquid is treated in the drum, and wave action is set up (The Centrifugal Machine, C. A. Matthey).
Similar considerations apply to the stability of the whirling bowl in a cream-separating machine.
We can write equation (1)
N = An2 sin θ − μ·CK = (A2n2 − KM·KN) sin θ/A,
(4)
so that N is negative or positive, and the axis tends to rise or fall according as K moves to the inside or outside of the hyperbola of free motion. Thus a tap on the axis tending to hurry the precession is equivalent to an impulse couple giving an increase to C′K, and will make K move to the interior of the hyperbola and cause the axis to rise; the steering of a bicycle may be explained in this way; but K1 will move to the exterior of the hyperbola, and so the axis will fall in this second more violent motion.
Friction on the point of the top may be supposed to act like a tap in the direction opposite to the precession; and so the axis of a top spun violently rises at first and up to the vertical position, but falls away again as the motion dies out. Friction considered as acting in retarding the rotation may be compared to an impulse couple tending to reduce OC′, and so make K and K1 both move to the exterior of the hyperbola, and the axis falls in both cases. The axis may rise or fall according to the direction of the frictional couple, depending on the shape of the point; an analytical treatment of the varying motion is very intractable; a memoir by E. G. Gallop may be consulted in the Trans. Camb. Phil. Soc., 1903.
The earth behaves in precession like a large spinning top, of which the axis describes a circle round the pole of the ecliptic of mean angular radius θ, about 23½°, in a period of 26,000 years, so that R/μ = 26000 × 365; and the mean couple producing precession is
CRμ sin θ = CR2 sin 23½° /26000 × 365,
(5)