1. Physical Manifestations of Periodic Functions Matthew Koss College of the Holy Cross July 12, 2012 IQR Workshop: Foundational Mathematics Concepts for the High School to College Transition

2. Simple Block and Spring

4. Data Studio 500

5. Simple Harmonic Motion

6. Simple Harmonic Oscillations A Amplitude  t +  Phase (radians)/Angle (radians)  Phase Constant (radians)  Angular Frequency (rad/s) TPeriod (s) fFrequency (Hz)

7. Simple Harmonic Motion for Block and Spring

8. Another Representation

9. or

10. Review A Periodic Function (sine or cosine) is the Recorded History of the Oscillations of an object attached to a spring.

11. Position, velocity, and acceleration If you know calculus

12. Calculus Approach

13. If Not, then …

14. Zero Offset Oscillations do not always occur about the zero point. To account for this, there is one additional term called the zero offset which is middle value in the oscillations. So, more completely:

15. iPads and Video Physics

16. Physics Toolkit

17. Atom Can Execute Simple Periodic Motions

18. States of Matter Simulation

19. SHM is the Projection of Circular Motion

20. Illustration y(t)y(t) y2(t)y2(t) y1(t)y1(t) y 2 y 1 A A y(t)y(t)

21. PhET Rotation Simulation

22. Simple Pendulum

23. PhET Pendulum Simulation

24. Same as a simple pendulum, but… Physical Pendulum axis cm

25. Oscillations on a String

26. Tangent on Traveling Waves A wave is a disturbance in position propagating in time. Many traveling waves are periodic in both position and time, e.g.

27. Mathematical Relationships A Amplitude kx  t+  Phase (radians)  Angular Frequency (rad/s) TPeriod (s) fFrequency (Hz) k (Angular) Wave number Wavelength In general: Specifically: Periodic Sine Waves

28. Waves and Oscillations Compared An oscillation in time is a “history” of a wave at a particular place. An oscillation in space is a “snapshot” of a wave at a particular time,

29. Sum of Two Traveling Waves Makes Standing Waves Last Slide of Digression

30. Standing Waves on a String, or Oscillations on a String

31. String Vibrates the Air

32. Guitar Strings The strings on a guitar can be effectively shortened by fingering, raising the fundamental pitch. The pitch of a string of a given length can also be altered by using a string of different density.

33. Sound is a Periodic Oscillation of the Air

34. Tuning Forks

35. Data Studio 500 Redux

36. Beats If the two interfering oscillations have different frequencies they will superimpose, but the resulting oscillation is more complex. This is still a superposition effect. Under these conditions, the resultant oscillation is referred to as a beat.

38. Beat Frequency Mathematics f Beat = f 1 -f 2

39. Amplitude (I) of Sound Oscillations I 0 is taken to be the threshold of hearing: The loudness of a sound is much more closely related to the logarithm of the intensity. Sound level is measured in decibels (dB) and is defined as:

40. MacScope II

41. Audacity

42. iPads & I Phones

43. More Complex Sounds

44. Fundamental/Normal Modes

46. Time and Frequency Domains

47. Sample Musical Instrument Sounds in the Frequency Domain

48. Web References/Resources PhET Simulations http://phet.colorado.edu/en/simulations/category/new Springs http://phet.colorado.edu/en/simulation/mass-spring-lab Rotation http://phet.colorado.edu/en/simulation/rotation Atomic Oscillation http://phet.colorado.edu/en/simulation/states-of-matter Pendulum http://phet.colorado.edu/en/simulation/pendulum-lab Normal Modes http://phet.colorado.edu/en/simulation/normal-modes Making Waves http://phet.colorado.edu/en/simulation/fourier Video Physics http://itunes.apple.com/us/app/vernier-video-physics/id389784247?mt=8 Physics Toolkit http://physicstoolkit.com/ MacScope & Physics2000 http://www.physics2000.com/Pages/Downloads.html Audacity http://audacity.sourceforge.net/download/