## Counter Gyroscopes

Good to hear from Franklin Amador. This is a very interesting paper. This apparatus and the apparatus by Laithwaite are special cases of the motion of a symmetrical top with one point fixed. (J. B. Marion and S. T. Thornton (MT), “Classical Dynamics of Particles and Systems” (Harcourt, New York, 1988, third edition, pp, 395 ff. Section 10.10). The problem was first solved by Lagrange in “Mecanique Analytique” (volume one 1811, volume two 1815) using the lagrangian method. By reference to Figure 10-13, page 396, the Laithwaite experiment is described by the figure with theta = pi / 2 radians initially. It is much easier to use the force analysis of the last note and I can develop this in the next note. The kinetic energy of the gyro is given by MT in their equation (10.149) and the potential energy is V = MGhcos theta where h is the distance from the fixed point of the gyro to its centre of mass. In the Laithwaite configuration, V = 0 because theta is pi / 2 radians (horizontal axis of the spinning wheel). So the total energy is (1/2) I3 omega3 squared, and is conserved. There is a torque in the Laithwaite experiment, defined in the moving frame by Tq3 = I3 domega3 / dt. If axis 3 of the moving frame is constant then this is also the lab frame torque. The dynamics of the system are described completely in terms of the Euler angles (MT Section 10.10). There is never a net upwards force on the point of a gyro in classical dynamics. This is because

F1 squared + F2 squared + F3 squared = FX squared + FY squared + FZ sqaured

and the downward force due to gravitation is always equal and opposite to the net upward force in the moving frame. The same is true for any two gyroscopes, or in general n gyroscopes. So the upwards force must be described by something outside the standard model. I propose ECE2 fluid dynamics. I will develop this in the next note. If there is a net upward force, then the fixed point plus gyroscope must move upwards. It would tend to pull Laithwaite upwards. In the standard theory of gyorscopes the point does not move upwards.

To: EMyrone@aol.com

CC: dwlindstrom@gmail.com, mail@horst-eckardt.de

Sent: 11/01/2017 17:11:51 GMT Standard Time

Subj: Re: 347(5a): Force Analysis of the GyroscopeDear Myron,

I’ve attached the experiment where the increase of Z momentum happens by the scissoring motion of the gyros access to a central point. The movement has to by synchronized for maximum effect.

Regards,

Franklin

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