## Discussion of Note 367(8)

Many thanks! I would say that Eqs. (17) and (20) demonstrate the special case when the k and e sub r axes are parallel. Agree with the second point. This is what is done in Eqs. (15) and (19). Eqs. (8) to (10) were used in UFT270 and are simply the expressions for acceleration components in spherical polars, equivalent to a = d2 X / dt 2 i + ……. in Cartesians. The spherical polar expression for force is equivalent to F = m(dv /dt + omega x v). Finally, Eq. (27) means r is aligned with k. It is meant to be just an illustration. Rigorously, theta dot is zero. As you can see from UFT270, angular momentum terms appear in the spherical polar expressions, and it is also possible to adapt your incisive graphics of UFT270 to the classical gyroscope. It would also be interesting of course to work out the extra F sub Z term from fluid dynamics using gnuplot for example.

To: EMyrone@aol.com
Sent: 16/01/2017 17:24:31 GMT Standard Time
Subj: Re: 367(8): Complete Theory of the Gyroscope

This is now a well understandable note, thanks. I only have two remarks:

Eq.(17) is at variance with eq.(20). I think eq.(14) has to be transformed to spherical polarcoordinates with eqs.(19). This gives eq.(20) correctly. One cannot use eqs.(8-10) because these are equations of dynamics, not transformation equations. Setting theta dot=0 and phi dot=0 makes less sense since then the gyro would fall to the ground, enforcing theta dot not equal to 0.

Eq.(27) should contain phi dot instead of theta dot because phi is the azimut. We have to set theta=pi/2. This gives the corresponding result for the 2D orbit.

Horst

Am 14.01.2017 um 11:30 schrieb EMyrone:

After some discussions with co author Horst Eckardt I produced this new and original theory of the gyroscope, or spinning top, giving the force equation (25) in terms of spherical polar coordinates. It shows very clearly that the force due to gravitation is counterbalanced by three dimensional centrifugal and Coriolis forces when the gyro is spun. Eq. (25) reduces to the Leibnitz equation of planar orbits under the conditions (26). So the gyro is a type of three dimensional orbit. The analysis of three dimensional orbits by Horst and myself in previous UFT papers can be used to see if there are constants of motion in Eq. (25). It could then be solved. Eq. (13) is always true so the gyro cannot lift itself off the ground without an extra force or torque, which may be mechanical or may come from ECE2 fluid dynamics. So I will now write up my part of UFT367, Sections 1 and 2 as usual, around the finished theory of Note 367(8). The use of spherical polar coordinates gives great insight into the motion of the gyro. I congratulate the group on an excellent discussion.