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4. Spring thread

4. Spring thread. Task. Pull a thread through the button holes as shown in the picture. The button can be put into rotating motion by pulling the thread. One can feel some elasticity of the thread. Explain the elastic properties of such a system. Outline. Theory What is elasticity?

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4. Spring thread

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  1. 4. Spring thread

  2. Task Pull a thread through the button holes as shown in the picture. The button can be put into rotating motion by pulling the thread. One can feel some elasticity of the thread. Explain the elastic properties of such a system.

  3. Outline • Theory • What is elasticity? • Deriving the motion equation • Experiments • Conlusion

  4. What is elasticity? • Elastic material acts according to Hooke‘s law • In other notation • Force is directly proportional to relative extension. • Hooke‘s law can be used also for compressing an elastic material. Then e is relative contraction.

  5. Our system • Two extremal views of the situation: 1.) Fixed distance of ends of the thread. - When rotating the button, the thread has to extend, so it acts due to Hooke‘s law 2.) The distance changes, we assume no extension in real lenght of the thread – It changes only due to entangling of the thread.

  6. Where is the reality? • Somewhere between the extremes. • The distance changes significantly. • There are some small changes in real lenght of the thread.

  7. Used threads • We used three different types of thread • Thin thread (green, orange, yellow); r = 0.12 mm • Thick thread (white); r = 0.29 mm • Silon; r = 0.65 mm

  8. Measuring the Young modulus

  9. Young modulus of used threads • Thin thread

  10. Young modulus of used threads • Thick thread

  11. Young modulus of used threads • Silon

  12. Our model • Extensibilities due to Hooke’s law are small – we will neglect them. • The system is only being shortened due to convolution

  13. Is this system elastic? • Force F can be arbitrary, whatever the relative contraction of the system is. • Against Hooke’s law • Therefore THE SYSTEM IS NOT ELASTIC. • However, we can feel some “elasticity”.

  14. Parameters • Half-lenght of the system in steady state (not shortened) – L0 • Actual half-lenght – L • Radius of the thread – r • Angle of rotation of the button (in comparison with steady state) – j • Angle of convolution of the thread - a

  15. Parameters shown in a picture

  16. L0 a L rolled thread Basic equtions • The thread is homogeneously rolled on a cyllindrical surface with radius r.

  17. Length of the system • According to Phytagorean theorem: • For shortening of the system: • N is number of windings. • For small number of windings:

  18. Measuring the shortening fixed end free end button

  19. Shortening of used threads • Thin thread

  20. Shortening of used threads • Thick thread

  21. Shortening of used threads • Silon

  22. Further equations • In the thread there will be a strain T:

  23. Motion equation • For torque we can obtain: • For small number of windings • Direct proportion:

  24. changeable wieght weight of mass F/g Prooving the direct proportion

  25. Prooving the direct proportion

  26. Measuring the direct proportion • In equilibrum: • Theoretically: • d = radius of a button • M = mass suspended at free end of the system • m = mass of changeable weight

  27. Results • Thin thread M = 1kg ; d = 14 mm ; L0= 0.2 m ; m = 0.5 g

  28. Results • Thick thread M = 1kg ; d = 14 mm ; L0= 0.6 m ; m = 0.5 g

  29. Results • Silon M = 1kg ; d = 14 mm ; L0= 0.6 m ; m = 0.5 g

  30. Motion equations • We obtained • Therefore • Linear harmonic oscillator with period

  31. Motion equations • If we assume some damping b: (where ) • Well-known solution of this equation (for w0=0) is:

  32. Damped oscillations

  33. Spring thread as a toy  • When playing with the toy, we act • with larger force when the system is expanding • with smaller force when the system is shortening • In our model, we suppose the forces to be F and F/2.

  34. Simulation • We can see, that the system begins to rezonate

  35. k m Conclusion • Elasticity of the system is a dynamic property. • We can feel it only when the button is rotating. • Spring thread is rezonating torsional oscillator - THIS IS THE ELASTICITY we can feel.

  36. Thank you for your attention

  37. 18. Appendix The motion equation (1) We suppose a solution . When substitued into the differential equation, we obtain: We suppose that

  38. 19. Appendix The motion equation (2) Substitution: We have: For j0: Since j0 is real, and

  39. 20. Appendix The motion equation (3) For angular velocity: In zero time:

  40. 21. Appendix The motion equation (4) We have and Therefore we can simplify: Finally we have: We will use:

  41. 22. Appendix The motion equation (5) By putting together and simplifying we obtain: Under special condition :

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