The equation of motion
The equation of motion will first be established rigorously and in fairly advanced mathematical terms. The effect will then be described again but in a rather looser and more physical sense. Figure shows the axis of the gyro spinner placed with respect to the surface of the Earth.Aset of orthogonal axes is established with the origin at O, the i direction in the direction of gravity, the j and k directions are in the local horizontal plane with the k direction making a small angle, α, with the local direction of north.
j is at right angles to i and k such that the axes form a right-handed orthogonal set. φ is the latitude of O. i, j and k are unit vectors in the directions i, j and k. Ω is the rotational velocity vector of the Earth.
The velocity vector of the earth may be broken into its component parts in the i, j and k directions so:
If this triad of vectors is now rotated about the i axis with an angular velocity −˙α then the angular velocity of the axes may be described in vector terms by
If the spinner is now placed so that it rotates about the k axis with an angular velocity of n, then the total angular velocity of the spinner is
(a) Meridian section through the earth, (b) Plan view at 0
If A is the moment of inertia of the spinner about the i or j axis (they will be the same because of rotational symmetry) and B is the moment of inertia of the spinner about the k axis then the total angular momentum of the spinner L is