The motion of a coin falling to the bottom of a tank filled with liquid can be remarkably similar to the fluttering and tumbling of a falling autumn leaf. Investigate how the motion of the coin depends on relevant parameters.
Autumn coin (youtube, Никита Черников, 06.07.2025), https://youtu.be/EW6tA00SZL8 Autumn coin - top (YouTube, Никита Черников, 06.07.2025), https://youtube.com/shorts/ItA7Z667tYc Coins Dropped Into Crystal Clear Water: A View From The Bottom As The Coins Sink. Ogemaw Springs, MI (youtube, Chosen Won, 01.02.2021), https://youtu.be/jx0acGK7NC8 Wikipedia: Angle of attack, https://en.wikipedia.org/wiki/Angle_of_attack Wikipedia: Froude number, https://en.wikipedia.org/wiki/Froude_number Wikipedia: Free fall, https://en.wikipedia.org/wiki/Free_fall Wikipedia: Drag, https://en.wikipedia.org/wiki/Drag_(physics) *A. Tinklenberg, M. Guala, and F. Coletti. Thin disks falling in air. J. Fluid Mech. 962, A3 (2023), https://doi.org/10.1017/jfm.2023.209 Y. Kubota and O. Mochizuki. Numerical investigation of aerodynamic characteristics by a rotating thin plate. World J. Mech. 5, 3, 42-47 (2015), https://doi.org/10.4236/wjm.2015.53005 S. P. Kuznetsov. Plate falling in a fluid: Regular and chaotic dynamics of finite-dimensional models. Regul. Chaot. Dyn. 20, 345-382, (2015), https://doi.org/10.1134/S1560354715030090 *L. Heisinger, P. Newton, and E. Kanso. Coins falling in water. J. Fluid Mech. 742, 243-253 (2014), https://doi.org/10.1017/jfm.2014.6 *D. M. Hargreaves, B. Kakimpa, and J. S. Owen. The computational fluid dynamics modelling of the autorotation of square, flat plates. J. Fluids Struct. 46, 111-133 (2014), https://doi.org/10.1016/j.jfluidstructs.2013.12.006
S. Michelin and S. G. Llewellyn Smith. Falling cards and flapping flags: understanding fluid–solid interactions using an unsteady point vortex model. Theor. Comput. Fluid Dyn. 24, 1-4, 195-200 (2010), https://doi.org/10.1007/s00162-009-0117-6 C. Eloy, R. Lagrange, C. Soutilliez, and L. Schouveiler. Aeroelastic instability of cantilevered flexible plates in uniform flow. J. Fluid Mech. 611, 97-106 (2008), https://doi.org/10.1017/S002211200800284X, https://www.irphe.fr/~eloy/assets/pdf/FastTrack2008.pdf, arXiv:0804.0774 [physics.flu-dyn] A. V. Borisov, V. V. Kozlov, and I. S. Mamaev. Asymptotic stability and associated problems of dynamics of falling rigid body. Regul. Chaot. Dyn. 12, 5, 531-565 (2007), https://doi.org/10.1134/S1560354707050061 D. Vella, D.-G. Lee and H.-Y. Kim. Sinking of a horizontal cylinder. Langmuir 22, 7, 2972-2974 (2006), https://doi.org/10.1021/la0533260 *A. Andersen, U. Pesavento, and Z. J. Wang. Unsteady aerodynamics of fluttering and tumbling plates. J. Fluid Mech. 541, 1, 65-90 (2005), https://doi.org/10.1017/S002211200500594X, https://dragonfly.tam.cornell.edu/publications/S002211200500594Xa.pdf A. Andersen, U. Pesavento, and Z. J. Wang. Analysis of transitions between fluttering, tumbling and steady descent of falling cards. J. Fluid Mech. 541, 1, 91-104 (2005), https://doi.org/10.1017/S0022112005005847, https://dragonfly.tam.cornell.edu/publications/S0022112005005847a.pdf M. A. Jones and M. J. Shelley. Falling cards. J. Fluid Mech. 540, 393-425 (2005), https://doi.org/10.1017/S0022112005005859
*U. Pesavento and Z. J. Wang. Falling paper: Navier-Stokes solutions, model of fluid forces, and center of mass elevation. Phys. Rev. Lett. 93, 14, 144501 (2004), https://doi.org/10.1103/PhysRevLett.93.144501 L. Mahadevan, W. S. Ryu, and A. D. T. Samuel. Tumbling cards. Phys. Fluids 11, 1, 1-3 (1999), https://doi.org/10.1063/1.869919 A. Belmonte, H. Eisenberg, and E. Moses. From flutter to tumble: Inertial drag and Froude similarity in falling paper. Phys. Rev. Lett. 81, 2, 345-348 (1998), https://doi.org/10.1103/PhysRevLett.81.345 *S. B. Field, M. Klaus, M. G. Moore, and F. Nori. Chaotic dynamics of falling disks. Nature 388, 252-254 (1997), https://doi.org/10.1038/40817, https://public.websites.umich.edu/~nori/disks/long.pdf Y. Tanabe and K. Kaneko. Behavior of a falling paper. Phys. Rev. Lett. 73, 10, 1372-1375 (1994), https://doi.org/10.1103/PhysRevLett.73.1372 W. W. Willmarth, N. E. Hawk, and R. L. Harvey. Steady and unsteady motions and wakes of freely falling disks. Phys. Fluids 7, 2, 197-208 (1964), https://doi.org/10.1063/1.1711133 N. Kamaruddin, J. Potts, and W. Crowther. Aerodynamic performance of flying discs. Aircr. Eng. Aerosp. Technol. 90, 2, 390-397 (2018), https://shura.shu.ac.uk/14521/1/Potts -%20Aerodynamic%20Performance%20of%20Flying%20Discs%20-%20%28AM%29.pdf D. L. Flynn. Falling paper and flying business cards. SIAM News 40, 4 (2007), https://web.archive.org/web/20140602214301/https://www.siam.org/pdf/news/1123.pdf L. Mahadevan. Tumbling of a falling card. C. R. Acad. Sci. Paris IIb, 323, 729-736 (1996)
J. C. Maxwell. On a particular case of the descent of a heavy body in a resisting medium. Camb. Dublin Math. J. IX, 145-148 (1854) J. R. Chasnov. Flow Around a Cylinder (The Hong Kong Univ. of Sci. and Techn., Feb 2022), https://www.math.hkust.edu.hk/~machas/flow-around-a-cylinder.pdf P. Martin. Falling leaves simulation (dspace.cvut.cz, 2022), https://dspace.cvut.cz/bitstream/handle/10467/101330/F3-DP-2022-Pazout-Martin-Master_Thesis.pdf N. M. Kamaruddin. Dynamics and performance of flying discs (PhD thesis, Univ. of Manchester, 2011), https://research.manchester.ac.uk/en/studentTheses/dynamics-and-performance-of-flying-discs/ U. Pesavento. Unsteady aerodynamics of falling plates (PhD thesis, Cornell Univ., 2006), https://ecommons.cornell.edu/items/06483654-e7b8-4900-8922-42bc3b6c7bfa Strange turn of a rotating playing card when I throw it (JakubKivi, physicsforums.com, Feb 23, 2020), https://www.physicsforums.com/threads/strange-turn-of-a-rotating-playing-card-whenithrow-it.984704/ The Physics of Falling Leaves (S. Schreier, October 19, 2019), https://colgatephys111.blogspot.com/2019/10/v-behaviorurldefaultvmlo.html Is a falling leaf an example of a chaotic system? (physics.stackexchange.com, May 7, 2016), https://physics.stackexchange.com/questions/254545/is-a-falling-leaf-an-example-of-a-chaoticsystem
Anthony Scodary. The Aerodynamics and Stability of Flying Discs (large.stanford.edu, October 30, 2007), http://large.stanford.edu/courses/2007/ph210/scodary1/ Aerodynamics: Mathematically Modeling the Flight of an Object (spartan711, physicsforums.com, Sep 1, 2007), https://www.physicsforums.com/threads/aerodynamicsmathematically-modeling-the-flight-ofan-object.182550/ Flutter and tumble in fluids (Elisha Moses, Andrew Behnonte, physicsworld.com, 01 Apr 1999), https://physicsworld.com/a/flutter-and-tumble-in-fluids/ A. C. Bustamante. Free-fall rotation and aerodynamic motion of rectangular plates (Sandia Laboratories, 1968), https://apps.dtic.mil/sti/tr/pdf/ADA395124.pdf Flow around a disk (or an infinite cylinder) (Luc Jaouen, apmr.matelys.com), https://apmr.matelys.com/BasicsMechanics/Fluid/FlowAroundADisk.html