1. Team of Slovakia International Young Physicists’ Tournament, Seoul 2007 4. Spring thread

2. No. 4. Spring Thread Presented by Tomas Bzdusek 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. No. 4. Spring Thread Presented by Tomas Bzdusek Outline Theory –What is elasticity? –Deriving the motion equation Experiments Conlusion

4. No. 4. Spring Thread Presented by Tomas Bzdusek 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  is relative contraction.

5. No. 4. Spring Thread Presented by Tomas Bzdusek 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. No. 4. Spring Thread Presented by Tomas Bzdusek Where is the reality? Somewhere between the extremes. –The distance changes significantly. –There are some small changes in real lenght of the thread.

7. No. 4. Spring Thread Presented by Tomas Bzdusek 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. No. 4. Spring Thread Presented by Tomas Bzdusek Measuring the Young modulus

9. No. 4. Spring Thread Presented by Tomas Bzdusek Young modulus of used threads Thin thread

10. No. 4. Spring Thread Presented by Tomas Bzdusek Young modulus of used threads Thick thread

11. No. 4. Spring Thread Presented by Tomas Bzdusek Young modulus of used threads Silon

12. No. 4. Spring Thread Presented by Tomas Bzdusek Our model Extensibilities due to Hooke’s law are small – we will neglect them. The system is only being shortened due to convolution

13. No. 4. Spring Thread Presented by Tomas Bzdusek 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. No. 4. Spring Thread Presented by Tomas Bzdusek Parameters Half-lenght of the system in steady state (not shortened) – L 0 Actual half-lenght – L Radius of the thread – r Angle of rotation of the button (in comparison with steady state) –  Angle of convolution of the thread - 

15. No. 4. Spring Thread Presented by Tomas Bzdusek Parameters shown in a picture

16. No. 4. Spring Thread Presented by Tomas Bzdusek Basic equtions The thread is homogeneously rolled on a cyllindrical surface with radius r.  L L0L0 rolled thread

17. No. 4. Spring Thread Presented by Tomas Bzdusek Length of the system According to Phytagorean theorem: For shortening of the system: –N is number of windings. For small number of windings:

18. No. 4. Spring Thread Presented by Tomas Bzdusek Measuring the shortening button fixed endfree end

19. No. 4. Spring Thread Presented by Tomas Bzdusek Shortening of used threads Thin thread

20. No. 4. Spring Thread Presented by Tomas Bzdusek Shortening of used threads Thick thread

21. No. 4. Spring Thread Presented by Tomas Bzdusek Shortening of used threads Silon

22. No. 4. Spring Thread Presented by Tomas Bzdusek Further equations In the thread there will be a strain T:

23. No. 4. Spring Thread Presented by Tomas Bzdusek Motion equation For torque we can obtain: For small number of windings –Direct proportion:

24. No. 4. Spring Thread Presented by Tomas Bzdusek Prooving the direct proportion weight of mass F/g changeable wieght

25. No. 4. Spring Thread Presented by Tomas Bzdusek Prooving the direct proportion

26. No. 4. Spring Thread Presented by Tomas Bzdusek 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. No. 4. Spring Thread Presented by Tomas Bzdusek Results Thin thread M = 1kg ; d = 14 mm ; L 0 = 0.2 m ; m = 0.5 g

28. No. 4. Spring Thread Presented by Tomas Bzdusek Results Thick thread M = 1kg ; d = 14 mm ; L 0 = 0.6 m ; m = 0.5 g

29. No. 4. Spring Thread Presented by Tomas Bzdusek Results Silon M = 1kg ; d = 14 mm ; L 0 = 0.6 m ; m = 0.5 g

30. No. 4. Spring Thread Presented by Tomas Bzdusek Motion equations We obtained Therefore Linear harmonic oscillator with period

31. No. 4. Spring Thread Presented by Tomas Bzdusek Motion equations If we assume some damping b: (where ) Well-known solution of this equation (for  0 =0) is:

32. No. 4. Spring Thread Presented by Tomas Bzdusek Damped oscillations

33. No. 4. Spring Thread Presented by Tomas Bzdusek 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. No. 4. Spring Thread Presented by Tomas Bzdusek Simulation We can see, that the system begins to rezonate

35. No. 4. Spring Thread Presented by Tomas Bzdusek 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. k m

36. No. 4. Spring Thread Presented by Tomas Bzdusek Thank you for your attention

37. No. 4. Spring Thread Presented by Tomas Bzdusek 18. Appendix The motion equation (1) We suppose a solution. When substitued into the differential equation, we obtain: We suppose that

38. No. 4. Spring Thread Presented by Tomas Bzdusek 19. Appendix The motion equation (2) Substitution: We have: For  0 : Since  0 is real, and

39. No. 4. Spring Thread Presented by Tomas Bzdusek 20. Appendix The motion equation (3) For angular velocity: In zero time:

40. No. 4. Spring Thread Presented by Tomas Bzdusek 21. Appendix The motion equation (4) We have and Therefore we can simplify: Finally we have: We will use:

41. No. 4. Spring Thread Presented by Tomas Bzdusek 22. Appendix The motion equation (5) By putting together and simplifying we obtain: Under special condition :