Suspend a metal ball from a fixed support using a rubber band and twist it many times around its vertical axis. When the ball is released, standing waves are formed on the rubber band. Investigate this phenomenon and study how the wave depends on relevant parameters.

Background reading from IYPT reference kit

Wikipedia: Rubber band, https://en.wikipedia.org/wiki/Rubber_band Wikipedia: Rubber band motor, https://en.wikipedia.org/wiki/Rubber_band_motor Wikipedia: Rubber Elasticity, https://en.wikipedia.org/wiki/Rubber_elasticity Wikipedia: Hyperelastic Material, https://en.wikipedia.org/wiki/Hyperelastic_material Wikipedia: Mullins Effect, https://en.wikipedia.org/wiki/Mullins_effect Wikipedia: Elastometer, https://en.wikipedia.org/wiki/Elastomer M. Leembruggen, J. Andrejevic, A. Kudrolli, and C. H. Rycroft. Computational model of twisted elastic ribbons. Phys. Rev. E 108, 1, 015003 (2023), https://doi.org/10.1103/PhysRevE.108.015003 L. Slepyan and A. B. Movchan. An overview of elastic waveguides with dynamic sub-structures. Phil. Trans. R. Soc. A 380, 2237, 20210381(2022), https://doi.org/10.1098/rsta.2021.0381 X. Li, B. Sun, Y. Zhang, and Y. Dai. Dynamics of rubber band stretch ejection. Preprints 2021030294 (2021), https://doi.org/10.20944/preprints202103.0294.v1 D. Featonby, D. Keenahan, and M. Fernandez. Standing waves in strings—the answer. Phys. Educ. 6, 55, 067001 (2020), https://doi.org/10.1088/1361-6552/ab7445 *G. Zurlo, J. Blackwell, N. Colgan, and M. Destrade. The Poynting effect. Am. J. Phys. 88, 12,1036-1040 (2020), arXiv:2004.09653 [condmat.soft], https://doi.org/10.1119/10.0001997 A. Majumdar and A. Raisch. Stability of twisted rods, helices and buckling solutions in three dimensions. Nonlinearity 12, 27, 2841-2867 (2014), https://doi.org/10.1088/0951-7715/27/12/2841 J. Liu, J. Huang, T. Su, K. Bertoldi, and D. R. Clarke. Structural transition from helices to hemihelices. PLoS ONE 9, 4, e93183 (2014), https://doi.org/10.1371/journal.pone.0093183 *P. Ciarletta and M. Destrade. Torsion instability of soft solid cylinders. IMA J. Appl. Math. 79, 804-819 (2014), arXiv:2009.09790 [condmat.soft], https://doi.org/10.1093/imamat/hxt052 D. Roundy and M. Rogers. Exploring the thermodynamics of a rubber band. Am. J. Phys. 1, 81,20-23 (2013), https://doi.org/10.1119/1.4757908 T. Shearer, I. D. Abrahams, W. J. Parnell, and C. H. Daros. Torsional wave propagation in a prestressed hyperelastic annular circular cylinder. Q. J. Mech. Appl. Math. 66, 4, 465-487 (2013), https://doi.org/10.1093/qjmam/hbt014 C. A. Triana and F. Fajardo. Dependence of some mechanical properties of elastic bands on the length and load time. Eur. J. Phys. 33, 4, 771-784 (2012), https://doi.org/10.1088/0143-0807/33/4/771 H. J. Schlichting and W. Suhr. The buzzer—a novel physical perspective on a classical toy. Eur. J. Phys. 31, 3, 501–510 (2010), https://doi.org/10.1088/0143-0807/31/3/007 *J. Diani, B. Fayolle, and P. Gilormini. A review on the Mullins effect. Eur. Polymer J. 45, 3, 601-612 (2009), https://doi.org/10.1016/j.eurpolymj.2008.11.017

R. Vermorel, N. Vandenberghe, and E. Villermaux. Rubber band recoil. Proc. R. Soc. A 463, 2079, 641-658 (2006), https://doi.org/10.1098/rspa.2006.1781 V. G. A. Goss, G. H. M. van der Heijden, J. M. T. Thompson, and S. Neukirch. Experiments on snap buckling, hysteresis and loop formation in twisted rods. Exp. Mech. 45, 101-111 (2005), https://doi.org/10.1177/0014485105052318 P. Kozić and R. Pavlović. Stochastic stability of torsion oscillations in moving thin elastic bands. J. Sound Vib. 3-5, 274, 1103-1109 (2004), https://doi.org/10.1016/j.jsv.2003.09.041 N. Pan and D. Brookstein. Physical properties of twisted structures. II. Industrial yarns, cords, and ropes. J. Appl. Polymer Sci. 83, 610-630 (2002), https://doi.org/10.1002/app.2261 J. Pellicer, J. A. Manzanares, J. Zúñiga, and P. Utrillas. Thermodynamics of rubber elasticity. J. Chem. Educ. 78, 2, 263 (2001), https://doi.org/10.1021/ed078p263 S. Przybyl and P. Pieranski. Helical close packings of ideal ropes. Eur. Phys. J. E 4, 4, 445-449 (2001), https://doi.org/10.1007/s101890170099, arXiv:physics/0101080 [physics.comp-ph] R. W. Ogden and D. G. Roxburgh. A pseudo–elastic model for the Mullins effect in filled rubber. Proc. R. Soc. Lond. A 455, 1988, 2861-2877 (1999), https://doi.org/10.1098/rspa.1999.0431

A. Goriely and M. Tabor. Nonlinear dynamics of filaments. IV Spontaneous looping of twisted elastic rods. Proc. R. Soc. Lond. A 454, 3183-3202 (1998), https://doi.org/10.1098/rspa.1998.0297 P. G. Santangelo and C. M. Roland. Chain ends and the Mullins effect in rubber. Rubber Chem. Technol. 65, 5, 965-972 (1992), https://www.doi.org/10.5254/1.3538654 G. Savarino and M. R. Fisch. A general physics laboratory investigation of the thermodynamics of a rubber band. Am. J. Phys. 59, 2, 141-145 (1991), https://doi.org/10.1119/1.16594 T. D. Rossing and D. A. Russell. Laboratory observation of elastic waves in solids. Am. J. Phys. 12, 58, 1153-1162 (1990), https://doi.org/10.1119/1.16245 C. M. Roland. The Mullins effect in crosslinked rubber. J. Rheology 33, 4, 659-670 (1989), https://doi.org/10.1122/1.550032 R. T. Deam and S. F. Edwards. The theory of rubber elasticity. Phil. Trans. R. Soc. Lon. A 280, 1296, 317-353 (1976), https://doi.org/10.1098/rsta.1976.0001 L. Mullins. Effect of stretching on the properties of rubber. Rubber Chem. Technol. 21, 2, 281-300 (1948), https://www.doi.org/10.5254/1.3546914 R. S. Rivlin. Torsion of a rubber cylinder. J. Appl. Phys. 18, 444-449 (1947), https://doi.org/10.1063/1.1697674 W. B. Wiegand and J. W. Snyder. The rubber pendulum, the Joule effect, and the dynamic stressstrain curve. Rubber Chem. Tech. 8, 2, 151-173 (1935), https://www.doi.org/10.5254/1.3539424

Wave Speed on a Rubber Band (Seth Stein, Earth and Planetary Sciences Dept., Northwestern Univ.), https://sites.northwestern.edu/sethstein/a-small-is-beautiful-approach-to-upgrading-abeginning-geophysics-course/wave-speed-on-a-rubber-band/ How Much Energy Can You Store in a Rubber Band? (Rhett Allain, wired.com, MAR 23, 2018), https://www.wired.com/story/how-much-energy-can-you-store-in-a-rubber-band/ Do Rubber Bands Act Like Springs? (Rhett Allain, wired.com, AUG 8, 2012), https://www.wired.com/2012/08/do-rubber-bands-act-like-springs/ Stretching a Rubber Band (schoolphysics.co.uk), http://www.schoolphysics.co.uk/age14-16/Matter/text/Rubber_band/index.html R. E. Hobbs, M. S. Overington, J. W. S. Hearle, and S. J. Banfield. Buckling of fibres and yarns within ropes and other fibre assemblies (tensiontech.com), https://web.archive.org/web/20240219050523/https://www.tensiontech.com/_app_/resources/documents/www.tensiontech.com/buckling_fibres_yarns.pdf L. R. G. Treloar. The physics of rubber elasticity (Oxford University Press, 1975) A. N. Gent. 1 - Rubber Elasticity: Basic Concepts and Behavior. In: Science and Technology of Rubber (eds J. E. Mark, B. Erman, F. R. Eirich, second edition, Academic Press, 1994), pp. 1-22 Hysteresis and Rubber Bands (madphysics.com), https://web.archive.org/web/20060524013139/http://www.madphysics.com/exp/hysteresis_and_rubber_bands.htm

TYPT Supplementary References

University Physics Vol. 1 Ch. 15 Oscillations, Ch. 16 Waves. https://openstax.org/details/books/university-physics-volume-1

Nayfeh, A. H., & Mook, D. T. Nonlinear Oscillations. Wiley, 1979. https://archive.org/details/NonlinOscNayfeh/mode/2up Graff, K. F. Wave Motion in Elastic Solids. Dover, 1991. https://archive.org/details/wavemotioninelas00graf/page/n3/mode/2up

OBSERVATION

We have:

This creates standing waves that are visible as patterns along the rubber band.