A geometric theory of rapidly spinning tops, tippe tops, and footballs

American Journal of Physics

Volumn 70, Issue 8, 2002, Pages 815-828

  Soodak, Harry ^a,b  

^a University of New York ^*  (United States)

^b CITY COLLEGE OF NEW YORK   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33646377371     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1479741     Document Type: Article

References (22)

1. S. Ebenfeld and F. Scheck, "A new analysis of the tippe top: Asymptotic states and Liapunov stability," Ann. Phys. (Leipzig) 243, 195-217 (1995). This paper uses Eq. (I) for energy considerations.
2. Such a resolution was introduced in a more detailed form in Ref. 3, where it is called the "Hooke-Newton Resolution Principle."
3. H. Soodak and M. S. Tiersten, "Resolution analysis of gyroscopic motion," Am. J. Phys. 62, 687-694 (1994).
4. The motion considered here is one of a general class of guiding center motions, and the class of multiple time-scale motions. Reference 5 discusses the example of charged particles in a magnetic field. Reference 6 touches on multiple time-scale problems and discusses averaging methods used to obtain the variation on the long time scales.
5. Theodore G. Northrop, The Adiabatic Motion of Charged Particles (Inter-science, New York, 1963), Chap. I.
6. A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Motion (Springer-Verlag, Berlin, 1992), 2nd ed., p. 102.
7. p), in the direction of p × I. The value of ψ* is equal to its initial value. For ψ*>λ ψ varies between the limits,ψ* + λ, and ψ*-λ, with ψ̄ very close to ψ* when λ is small compared to unity; the path of s is a looped cycloid, with |δψ|≈λ. For ψ *<λ, the limits are λ + ψ* and λ-ψ*, with ψ̄ very close to λ the path is a wavy cycloid without loops, and with |Δψ|≈ψ*, which is less than λ For ψ* = 0, s parallels I, moving uniformly along the path of I*, displaced from I by the constant distance λ in this case ψ is constant, and Δψ vanishes.
8. S for tops supported on frictionless surfaces is known to follow exactly from the constants of motion, as described in Ref. 9. A similar analysis applies to the fixed point top. It is further noted that a remark regarding the average torque concept appears on p. 513 of Ref. 17.
9. S is demonstrated in Appendix B of the paper, and in footnote 33, it is pointed out that the cycling is related to the absence of dissipation, so that dissipation is required for inversion of a tippe top.
10. The values used here assume a gyroscope consisting of a ring of mass m and radius D, attached by massless spokes to the center of a massless perpendicular axle of length 2D. The fixed point is one end of the axle.
11. J. H. Jellett, A Treatise on the Theory of Friction (MacMillan, London, 1872), p. 185. Reference 9 lists references in which the Jellett constant was rediscovered.
12. h in the direction opposite to u in Eq. (28), thus promoting the possibility of a transition to rolling.
13. 1/2 used in simulations.
14. T. R. Kane and D. A. Levinson, "A realistic solution of the symmetric top problem," J. Appl. Mech. 45, 903-909 (1978). The possibility of early transitions from slipping to rolling was recognized in this paper by using computer simulations.
15. s through the Jellett constant, Eq. (26). The equations are solved using the NDSOLVE program of Mathematica.
16. N. M. Hugenholtz, "On tops rising by friction," Physica (Amsterdam) 18, 515-527 (1952).
17. C. M. Braams, "On the influence of friction on the motion of a top," Physica (Amsterdam) 18, 503-514 (1952).
18. D. G. Parkyn, "The inverting top," Math Gazette 40, 260-265 (1956).
19. W. A. Pliskin, "The tippe top (topsy-turvy top)," Am. J. Phys. 22, 28-32 (1954).
20. R. J. Cohen, "The tippe top revisited," Am. J. Phys. 45, 12-17 (1977).
21. S≈0.05.
22. h, to be initially zero, and approximate its evolution due to the friction force.