1. Chapter 10 Waves and Vibrations Simple Harmonic Motion SHM

2. Observing Oscillating Systems  Questions to observe: Does the rate of oscillation depend on the amplitude of the motion? Does the rate of oscillation depend on the mass being oscillated?

3. Periodic Motion  Periodic motion- a motion that is repeated with some set frequency.  Two common types: pendulum Spring and mass Source: Wikipedia

4. Describing Periodic Motion

5. Describing Simple Harmonic Motion  Frequency is the reciprocal of period, so: Source: Wikipedia

6. Spring & mass oscillator - Review From chapter 6 (Hooke’s Law), we know that the force exerted by the spring is opposite the displacement, and can be expressed as

7.  SHM mass and ball SHM mass and ball  SHM – Waves SHM – Waves  SHM - Waves 2 SHM - Waves 2  SHM - Waves 3 (2:50-3:17) SHM - Waves 3 (2:50-3:17) Spring & mass oscillator – Review Wave motion

8. DISPLACEMENT

9. 10.2 Simple Harmonic Motion and the Reference Circle DISPLACEMENT

10. 10.2 Simple Harmonic Motion and the Reference Circle

11. DISPLACEMENT

12. 10.2 Simple Harmonic Motion and the Reference Circle period T: the time required to complete one cycle frequency f: the number of cycles per second (measured in Hz) amplitude A: the maximum displacement

13. Velocity (ω in rad/s)

14. Oscillating Spring/mass Systems © 2002 HRW

15.  The velocity is at a maximum when the displacement is zero. Maxima and Minima for SHM

16. Maxima and minima for SHM The restoring force, and thus the acceleration, are at a maximum when displacement is maximum.

18. 10.2 Simple Harmonic Motion and the Reference Circle VELOCITY

19. Example 1 The Maximum Speed of a Loudspeaker Diaphragm 10.2 Simple Harmonic Motion and the Reference Circle (b)The maximum speed occurs midway between the ends of its motion.

20. Acceleration (ω in rad/s)

21. 10.2 Simple Harmonic Motion and the Reference Circle ACCELERATION

22. Hooke’s Law  F elastic = -kx where F elastic is the force of the spring. k is the spring constant, Units: N/m x is the displacement from equilibrium. F elastic ©2008 by W.H. Freeman and Company

23. Frequency and Period of a Spring and Mass System ( ꙍ=angular frequency)  The frequency and period for a spring and mass are

24.  What determines the period of a torsional pendulum? Simple Harmonic Motion Example: The Torsion Pendulum

25. 10.2 Simple Harmonic Motion and the Reference Circle FREQUENCY OF VIBRATION

26. CONCEPTUAL REVIEW (AP – TYPE MC QUESTIONS) Simple Harmonic Motion

27. Graphs of SHM  Displacement:  Velocity:  Acceleration

28. CLASS ASSIGNMENT Section 1 Assignment

29. Waves and Vibrations Measuring Simple Harmonic Motion

30. Oscillating Spring/Mass Systems  Springs can vibrate horizontally (on a frictionless surface) or vertically. HorizontalSpring and Mass Animation source: Wikipedia

31. Simple Harmonic Motion  Any periodic motion that is the result of a restoring force which is proportional to the displacement can be described as simple harmonic motion. F elastic ©2008 by W.H. Freeman and Company

32. SHM and Hooke’s Law  Since a spring exhibits a restoring force which is proportional to the displacement, a spring/mass system will exhibit simple harmonic motion. Spring Mass System and SHM F elastic ©2008 by W.H. Freeman and Company

33. Period of a Mass-Spring System

34. Example Problem: 3

35. Pendulum Motion  The forces on the bob are the weight, mg, and the tension, T.  The restoring force is the tangential component of the weight, -mg sin θ.

36. Pendulum Motion  For small angles,  The restoring force is  The outcome is SHM

37. Pendulum Motion  The mass cancels out when Newton’s 2 nd Law is applied:

38. Pendulum Motion  The frequency and period are

39. Period of a Simple Pendulum

40. Example Problem: 4

41. From Holt Physics © Holt, Rinehart

42. Damping  In an ideal system, the mass-spring system would oscillate indefinitely.  Damping occurs when friction slows down motion of the object. Damping causes the system to come to rest after a period of time. If we observe the system over a short period of time, damping is minimal, and we can treat the system like as ideal.

43. Comparison of a Pendulum and an Oscillating Spring Equilibrium position Max. PE Min. KE Min. PE Max. KE

44. 10.8 Stress, Strain, and Hooke’s Law

45. CLASS ASSIGNMENT Section 2 Assignment

46. ENERGY AND SHM Section 3

47. 10.3 Energy and Simple Harmonic Motion  A stretched or compressed spring has elastic potential energy and therefore it can do work.

48. Energy of a Spring and Mass System  If there is no friction, mechanical energy is conserved. A0 -A

49. x

50. 10.3 Energy and Simple Harmonic Motion Conceptual Example 3 Changing the Mass of a Simple Harmonic Oscillator The box rests on a horizontal, frictionless surface. The spring is stretched to x=A and released. When the box is passing through x=0, a second box of the same mass is attached to it at the same speed. Discuss what happens to the (a) maximum speed, (b) amplitude, and (c) angular frequency.

51. Example 5: Adding a Mass to a Simple Harmonic Oscillator 10.3 Energy and Simple Harmonic Motion

52. Example 5: Adding a Mass to a Simple Harmonic Oscillator

53. ____ ____ ____ ____ ____ ____ _______________ U S K ME Energy Bar Chart The potential, kinetic and mechanical energy can be displayed on an energy bar chart.

54. ____ ____ ____ ____ ____ ____ _______________ U S K ME____ ____ ____ ____ ____ ____ _______________ U S K ME ____ ____ ____ ____ ____ ____ _______________ U S K ME ____ ____ ____ ____ ____ ____ _______________ U S K ME____ ____ ____ ____ ____ ____ _______________ U S K ME____ ____ ____ ____ ____ ____ _______________ U S K ME

56. H/W – DUE FIRST THING TOMORROW!! Section 3 Assignment

57. Graphs for SHM (Prepare to discuss your graphs)  Displacement:  Velocity:  Acceleration:

59. Uniform Circular motion - Recap

60. Oscillating Spring/mass Systems – what they have in common  SHM produces wave graphs

61. Example 5: Adding a Mass to a Simple Harmonic Oscillator 10.3 Energy and Simple Harmonic Motion