Problem 7: Bead Dynamics

A circular hoop rotates about a vertical diameter. A small bead is allowed to roll in a groove on the inside of the hoop. Investigate the relevant parameters affecting the dynamics of the bead.

I. Phenomenon Demonstration

II. Books, Encyclopedia, Discussion and Forum Posts

III. Research Papers

  • Raviola, L. A., Véliz, M. E., Salomone, H. D., Olivieri, N. A., & Rodríguez, E. E. (2016). The bead on a rotating hoop revisited: an unexpected resonance. European Journal of Physics, 38(1), 015005. https://doi.org/10.1088/0143-0807/38/1/015005
  • Dutta, S., & Ray, S. (2011). Bead on a rotating circular hoop: a simple yet feature-rich dynamical system. arXiv: Classical Physics.
  • Johnson, A. K., & Rabchuk, J. A. (2009). A bead on a hoop rotating about a horizontal axis: A one-dimensional ponderomotive trap. American Journal of Physics, 77(11), 1039–1048. https://doi.org/10.1119/1.3167216
  • Dutta, S., & Ray, S. (2012). Damped bead on a rotating circular hoop – a bifurcation zoo. arXiv: Classical Physics.
  • Balandin, D. V., & Shalimova, E. S. (2015). Bifurcations of the relative equilibria of a heavy bead on a hoop uniformly rotating about an inclined axis with dry friction. Journal of Applied Mathematics and Mechanics, 79(5), 440–445. doi:10.1016/j.jappmathmech.2016.03.004

  • Burov, A. A., & Yakushev, I. A. (2014). Bifurcations of the relative equilibria of a heavy bead on a rotating hoop with dry friction. Journal of Applied Mathematics and Mechanics, 78(5), 460–467. doi:10.1016/j.jappmathmech.2015.03.004

  • Basquerotto, C. H. C. C., Righetto, E., & Silva, S. da. (2018). Applications of the Lie symmetries to complete solution of a bead on a rotating wire hoop. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40(2). doi:10.1007/s40430-018-0995-x 

  • Baker, T. E., & Bill, A. (2012). Jacobi elliptic functions and the complete solution to the bead on the hoop problem. American Journal of Physics, 80(6), 506–514. doi:10.1119/1.3682321 

  • Basquerotto, C. H. C. C., & Ruiz, A. (2020). On the reduction of nonlinear mechanical systems via moving frames: a bead on a rotating wire hoop and a spinning top. Acta Mechanica. doi:10.1007/s00707-020-02798-1