(1)
Again, the components of angular momentum about OC, OA′ are Cn, −A sin θ ψ̇, and therefore the angular momentum (μ, say) about OZ is
μ = A sin2 θψ̇ + Cn cos θ.
(2)
We can hence deduce the condition of steady precessional motion in a top. A solid of revolution is supposed to be free to turn about a fixed point O on its axis of symmetry, its mass-centre G being in this axis at a distance h from O. In fig. 83 OZ is supposed to be vertical, and OC is the axis of the solid drawn in the direction OG. If θ is constant the points C, A′ will in time δt come to positions C″, A″ such that CC″ = sin θ δψ, A′A″ = cos θ δψ, and the angular momentum about OB′ will become Cn sin θ δψ − A sin θ ψ̇ · cos θ δψ. Equating this to Mgh sin θ δt, and dividing out by sin θ, we obtain
A cos θ ψ̇2 − Cnψ̇ + Mgh = 0,
(3)
as the condition in question. For given values of n and θ we have two possible values of ψ̇ provided n exceed a certain limit. With a very rapid spin, or (more precisely) with Cn large in comparison with √(4AMgh cos θ), one value of ψ̇ is small and the other large, viz. the two values are Mgh/Cn and Cn/A cos θ approximately. The absence of g from the latter expression indicates that the circumstances of the rapid precession are very nearly those of a free Eulerian rotation (§ 19), gravity playing only a subordinate part.
| Fig. 84. |
Again, take the case of a circular disk rolling in steady motion on a horizontal plane. The centre O of the disk is supposed to describe a horizontal circle of radius c with the constant angular velocity ψ̇, whilst its plane preserves a constant inclination θ to the horizontal. The components of the reaction of the horizontal lane will be Mcψ̇2 at right angles to the tangent line at the point of contact and Mg vertically upwards, and the moment of these about the horizontal diameter of the disk, which corresponds to OB′ in fig. 83, is Mcψ̇2. α sin θ − Mgα cos θ, where α is the radius of the disk. Equating this to the rate of increase of the angular momentum about OB′, investigated as above, we find