N = sin θ (Aμ2 cos θ − G′μ + gMh)
= A sin θ cos θ (μ − μ1) (μ − μ2);
(2)
and hence, as μ increases through μ2 and μ1, the sign of N can be determined, positive or negative, according as the tendency of the axis is to fall or rise.
When G′ = CR is large, μ2 is large, and
μ1 ≈ gMh/G′ = An2/CR,
(3)
| Fig. 10. |
the same for all inclinations, and this is the precession observed in the spinning top and centrifugal machine of fig. 10 This is true accurately when the axis OC′ is horizontal, and then it agrees with the result of the popular explanation of § 2.
If the axis of the top OC′ is pointing upward, the precession is in the same direction as the rotation, and an increase of μ from μ1 makes N negative, and the top rises; conversely a decrease of the procession μ causes the axis to fall (Perry, Spinning Tops, p. 48).
If the axis points downward, as in the centrifugal machine with upper support, the precession is in the opposite direction to the rotation, and to make the axis approach the vertical position the precession must be reduced.