1. Simple Harmonic Motion Harmonic Motion is any motion that repeats itself. Examples of Harmonic Motion.

6. Period Displacement Time for one oscillation Frequency Number of oscillations in one second AmplitudeMaximum displacement Distance from equilibrium

7. Simple harmonic motion is a special type of harmonic motion. Consider a mass on a spring. The cart is in equilibrium, because the total force is The acceleration is also (this doesn’t mean its stationary) zero.

8. Lets look at the forces force dispt = -A

9. force dispt = -A/2

10. Force = 0 dispt = 0

11. force dispt = A/2

12. force dispt = A

13. Notice that as the displacement increases, the restoring force increases. Notice that the restoring force is always in the opposite direction to the displacement force dispt = A

14. Now we’ll look at the acceleration force dispt = -A acceleration

15. force dispt = -A/2 acceleration

16. Force = 0 dispt = 0 Acceleration =0

17. force dispt = A/2 acceleration

18. force dispt = A acceleration

19. Notice that as the displacement increases, the acceleration increases. Notice that the acceleration is always in the opposite direction to the displacement acceleration dispt = A

20. The relation between acceleration and displacement is ….. Acceleration is proportional to displacement Acceleration is in opposite direction to displacement.

21. Acceleration/position graph acceleration position

22. Acceleration/position graph acceleration position

23. Force/position graph force position

24. Graphs of SHM We have looked at simple harmonic motion as a function of position. Now we’ll look at it as a function of time

25. link to graphslink to g

26. graphical treatment

27. phasor diagram and graph

28. http://www.edumedia- sciences.com/a266_l2-shm.htmlhttp://www.edumedia- sciences.com/a266_l2-shm.html

29. Reference Circle

30. mass on a spring (start with “graph”)mass on a spring

31. Reference Circle Red ball moves in SHM horizontally Blue ball moves in a circle Amplitude of SHM equals radius of circle Both have same period Both have same horizontal displacement

32. To find the position of a swing at a certain time. The period is 4.0s The amplitude is 2.0m Where is the swing 2.0s after release?

33. The period is 4.0s The amplitude is 2.0m Where is the swing 1.0s after release?

34. Where is the swing 0.5s after release? Convert time to angle (1period = 360 o ) 45 0 2.0m x

35. Where is the swing 2.5s after release? Convert time to angle (1period = 360 o ) 45 0 2.0m x

36. How long does it take to go 1.4m from the start? (1) Calculate angle (2) Convert angle to time (1period = 360 o ) 60 0 2.0m 1.41m 0.59m

37. The top of the sky tower is oscillating with an amplitude of 2.0 m and a period of 14 s. How long is it more than 0.80m from equilibrium each cycle? What is the horizontal acceleration when the displacement is maximum?

38. Equations 1

39. Equations 2

40. Equations 3

42. Anisha is on a swing. Kate pulls her back 2.0m and lets her go. Her period is 4.0s. (a) Calculate her maximum speed. (where is it?) (b) Calculate her maximum acceleration. (where is it?)

43. Anisha is on a swing. Kate pulls her back 2.0m and lets her go. Her period is 4.0s. (a) Calculate her speed 0.50s after being released (b) Calculate her acceleration 0.50s after being released

44. Nik is bungee jumping. In one oscillation he travels 12 m and it takes 8.0s. Tahi starts videoing him as he passes through the mid position moving UP. (a)Calculate his velocity 1.0s after the video starts (b) Calculate his acceleration 2.0s after the video starts.

45. Mass on a Spring As the mass increases, the period… As the spring stiffness increases the period … increases

46. Effect of mass: As the mass increases, the acceleration… As the acceleration decreases the period … A larger mass means a longer period. decreases increases (assuming constant force)

47. Effect of spring stiffness: As the stiffness increases, the restoring force… As the restoring force increases the acceleration … As the acceleration increases the period … A stiffer spring means a shorter period. increases (assuming same displacement) increases decreases

48. Summary mass ↑ acceln↓ period ↑ stiffness ↑ force ↑ acceln ↑ period ↓ equation

49. Extension …..derivation of the equation: consider a mass on a spring.

50. energy of motion

51. Simple Pendulum This is where all the mass is concentrated in one point.

52. What provides the restoring force? the restoring force is the Tension plus Gravity

53. Why is the motion SHM? As the displacement increases, the restoring force. increases. the restoring force is always towards equilibrium

55. This next bit is very important

56. Why does length affect period? For the same amplitude, if the pendulum is shorter, the angle of the string to the vertical is greater. The restoring force is greater. The acceleration is greater So the period is shorter

57. period of a pendulum How is length measured?

58. As the pendulum expands down, The mercury expands up This keeps the center of mass in the same place Same length same period.

59. Energy of SHM

60. energy of motion

61. a sprung system

62. shock absorbers? - dampers

63. energy dissipation plunger hydraulic oil dividing piston high pressure nitrogen gas

64. bridge dampers

65. Resonance Any elastic system has a natural period of oscillation. If bursts of energy (pushes) are supplied at the natural period, the amplitude will increase. This is called resonance

66. Examples of resonance

68. Examples of resonance tacoma narrows. tacoma narrows.

69. The glass has a natural frequency of vibration. If you tap the glass, it vibrates at the natural frequency causing sound. If you put energy in at the natural frequency, the amplitude increases. This is resonance. If the amplitude gets high enough, the glass can break.

70. Bay of Fundy

71. The period of the tide is 12 hours. The time for a wave to move up the bay and back is 12 hours