(1)
so that
gM·Gm = Aμ2 sin α(KC/KC′),
(2)
| A | = | KC′ | g | Gm = GQ·Gm. | |
| M | KC | μ2 sin α |
(3)
The instantaneous axis of rotation of the case of a gyrostat would be OP; drawing GI parallel to OP, and KK′ parallel to OG, making tan K′GC′ = (A/C) tan IGC’1; then if GK represents the resultant angular momentum, K′K will represent the part of it due to the rotation of the fly-wheel. Thus in the figure for the body rolling as a solid, with the fly-wheel clamped, the points m and Q move to the other side of G. The gyrostat may be supposed swung round the vertical at the end of a thread PA′ fastened at A′ where Pm produced cuts the vertical AB, and again at the point where it crosses the axis GO. The discussion of the small oscillation superposed on the state of steady motion requisite for stability is given in the next paragraph.
12. In the theoretical discussion of the general motion General motion of a gyrostat rolling on a plane. of a gyrostat rolling on a horizontal plane the safe and shortest plan apparently is to write down the most general equations of motion, and afterwards to introduce any special condition.
Drawing through G the centre of gravity any three rectangular axes Gx, Gy, Gz, the notation employed is
| u, v, w, | the components of linear velocity of G; |
| p, q, r, | the components of angular velocity about the axes; |
| h1, h2, h3, | the components of angular momentum; |
| θ1, θ2, θ3, | the components of angular velocity of the coordinate axes; |
| x, y, z, | the co-ordinates of the point of contact with the horizontal plane; |
| X, Y, Z, | the components of the reaction of the plane; |
| α, β, γ, | the direction cosines of the downward vertical. |