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5. Loaded hoop

5. Loaded hoop. “F asten a small weight to the inside of a hoop and set the hoop in motion by giving it an initial push. Investigate the hoop’s motion. ”. Example of motion - video. Outline. Experimental approach Hoop properties Giving initial push Parameters. Theoretical modeling

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5. Loaded hoop

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  1. 5. Loaded hoop “Fasten a small weight to the inside of a hoop and set the hoop in motion by giving it an initial push. Investigate the hoop’s motion.”

  2. Example of motion - video

  3. Outline • Experimental approach • Hoop properties • Giving initial push • Parameters • Theoretical modeling • Kinematics and dynamics • Modes of motion • Comparison of experimental data and theory • Modes of motion • Dependence of angular frequency on angle for different weight mass • Hop analysis • Energy analysis

  4. Hoop properties 124 mm 29,5 cm 99 mm

  5. Hoop properties

  6. Experimental aproach • Giving initial push • Incline • Colision • Method of motion recording • 3 small areas marked with reflective stickers • Motion recorded in 120 fps • Stickers tracked in program for video analysis ImageJ Frame for centering and releasing the hoop

  7. hoops mass mass of the weight – moment of inertia around center friction force normal force coordinates of the hoops center Theoretical modeling weight Mg N F Kinematics of the hoop: Dynamics of the hoop: r ∆R R

  8. Modes of motion

  9. Rolling • Properties of rolling • Solving equations for dynamics with assumption of rolling we obtain:

  10. Rolling with slipping • Properties of rolling with slipping • whereby is relative velocity between hoop on the contact of the surface and the surface • Two cases: • The hoop is rotating more than it’s translating • The hoop is translating more than it’s rotating

  11. Hop • The center of mass moving along a parabola withconstant speed in the x direction • Conservation of angular momentum • Most probable hop when weight is on highest position

  12. Determing hoop motion - 1 • Rolling condition • Hop Hop

  13. Determing hoop motion - 1 • Rolling condition: • Rolling with sliping condition: • Hop:

  14. Hop analysis - 2

  15. Analysis of motion without hop - 3 • Rolling condition

  16. Analysis of motion without hop - 3 • The motion of the hoop can be considerated as rolling

  17. Conservation of energy Energy of a hoop:

  18. Dependence of angular velocity on angle for different weight mass ,

  19. Conclusion • Three modes of motion were observed • Rolling • Rolling with sliping • Hop • We analyticaly solved equation of motion for rolling case • , and the solution fits the experimental data • When the hoop hop its center of mass will move on parabolic trajectory and it will have constant angular momentum

  20. Reference • M. F. Maritz, W.F.D. Theron; Experimental verification of the motion of a loaded hoop; 2012 American journal of physics • M. F. Maritz, W.F.D. Theron; The amazing variety of motions of a loaded hoop; Mathematical and Computer Modelling 47 (2008) 1077-1088 • Lisandro A. Raviola, O. Zarate, E. E. Rodriguez; Modeling and experimentation with asymmetric rigid bodies: a variation on disks and inclines; March 31, 2014 • W. F. D. Theron, Analysis of the Rolling Motion of Loaded Hoops, dissertation, March 2008

  21. IYPT 2014 CroatiaN team Thank you Reporter: Domagoj Pluščec

  22. Hoop1

  23. Determing hoops motion – another example

  24. Rotational velocity in time – another example

  25. Energy – another example

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