1414 Loaded hoop Mário Lipovský 5

1414 Task 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

1414 Types of hoop motion 1)Rolling 2)Spinning Skidding 3)Jump 3 ω N v

1414 1) Rolling 4 m weight = 22g m hoop = 28g r hoop = 41 mm

1414 1) Rolling Conservation of mechanical energy Rolling condition can be calculated from 5 θ G[x, y] Friction force is strong enough to prevent slipping ω N F Potential energy of the system Translational kinetic energy of the system Rotational kinetic energy of the system

1414 2) Spinning/Skidding 6 m weight = 322g m hoop = 28g r hoop = 41 mm

1414 2) Spinning/Skidding Newton’s laws of motion ω can be calculated by numerical integration of ε 7 θ G[x, y] N F Friction force reaches it’s maximum Torque equation my : mx : mg

1414 3) Jump 8 m weight = 220g m hoop = 28g r hoop = 41 mm

1414 my : 3) Conditions needed to jump Forces –gravity –centrifugal Condition of leaving the ground 9 ω mx : mg N

1414 3) When does hoop jump? 10 Jump occurs θ G[x, y] ω

1414 3) When does hoop jump? 11 Jump occurs θ G[x, y] ω

1414 3) Jump Hoop – Constant ω around centre of gravity Centre of gravity – projectile motion 12

1414 Types of hoop motion 1)Rolling 2)Spinning Skidding 3)Jump 13 ω N v F How to determine changes of motion types?

1414 What has been done 14 Analysis of Rolling Motion of Loaded Hoops [ W.F.D. Theron ] 4 types of motion Combined into 36 patterns Numerical simulation Observed only 1 x 360° rotation Stopped right after the hoop leaves ground Our own simulation Compare with our experiments Simulate jump

1414 Simulation 15 3) Jump 1) Rolling 2) Skidding / Spinning Numerical solution, using method of small increments (t) Init. conditions (x, θ, ω)

1414 Simulation 16

1414 Experiments Constant parameters –r – hoop radius –hoop material (rigid) –f – coefficient of friction (ground) Changing parameters –ω – initial velocity –m – mass of weight 17

1414 1) Rolling 18 m weight = 22g m hoop = 28g r hoop = 41 mm

1414 Angular displacement in timeθ(t) 19

1414 Position of centre of mass y(t) 20

1414 2) Spinning/skidding 21 m weight = 322g m hoop = 28g r hoop = 41 mm

1414 Position of centre of mass y(t) 22

1414 3) Jump 23 m weight = 220g m hoop = 28g r hoop = 41 mm

1414 4) Oscillation 24 m weight = 3,6g m hoop = 3,9g r hoop = 20,4 mm

1414 m weight = 22g 25 rolling skidding spinning jump Phase diagram of movement = small eccentricity mostly boring rolling What happens in time? ω 0 = 10 rad/s - rolling ω 0 = 15 rad/s - still rolling … ω 0 = 20 rad/s - spinning/skidding occur ω 0 = 25 rad/s – finally some jump

1414 m weight = 214g 26 Irregularities – spinning/skidding occur at lower velocities Immediate jump rolling skidding spinning jump Phase diagram of movement = higher eccentricity

1414 m weight = 322g 27 Increasing eccentricity – shift of patterns to lower velocities rolling skidding spinning jump Phase diagram of movement

1414 1) Rolling 2) Spinning/Skidding 3) Jump 1) Rolling 2) Spinning/Skidding 3) Jump Theoretical prediction of motions Numerical simulation Conclusion Description of types of motion 28 Thank you for your attention Theory & simulation proved by experiments

1414 THANK YOU FOR YOUR ATTENTION 29

1414 APPENDICES 30

1414 Comparison to original paper [W.F.D. Theron] 31 γ = 3/4 f = 0,4

γ = 3/4 f = 0,09 Comparison to original paper [W.F.D. Theron]

1414 my : Description of the system 33 θ G[x, y] S[X,Y] ω N F Centre of hoop S [X, Y] Angular displacement θ Centre of gravity G [x, y] Normal force Friction force mx : mg

1414 my : Description of the system 34 θ G[x, y] S[X,Y] ω N F mx :

1414 Elastic hoop – obvious difference in motion 35