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Physical Manifestations of Periodic Functions

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. Simple Block and Spring. Data Studio 500. Simple Harmonic Motion.

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Physical Manifestations of Periodic Functions

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  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

  3. Data Studio 500

  4. Simple Harmonic Motion

  5. Simple Harmonic Oscillations A Amplitude w t + f Phase (radians)/Angle (radians) f Phase Constant (radians) w Angular Frequency (rad/s) T Period (s) f Frequency (Hz)

  6. Simple Harmonic Motion for Block and Spring

  7. Another Representation

  8. or

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

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

  11. Calculus Approach

  12. If Not, then …

  13. 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:

  14. iPadsand Video Physics

  15. Physics Toolkit

  16. Atom Can Execute Simple Periodic Motions

  17. States of Matter Simulation

  18. SHM is the Projection of Circular Motion

  19. y(t) A A y(t) y2 y1 y2(t) y1(t) Illustration

  20. PhET Rotation Simulation

  21. Simple Pendulum

  22. PhET Pendulum Simulation

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

  24. Oscillations on a String

  25. 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.

  26. Mathematical Relationships In general: Specifically: Periodic Sine Waves A Amplitude kx-wt+f Phase (radians) w Angular Frequency (rad/s) T Period (s) f Frequency (Hz) k (Angular) Wave number  Wavelength

  27. 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,

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

  29. Standing Waves on a String, orOscillations on a String

  30. String Vibrates the Air

  31. 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.

  32. Sound is a Periodic Oscillation of the Air

  33. Tuning Forks

  34. Data Studio 500 Redux

  35. 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.

  36. Beat Frequency Mathematics fBeat= f1 -f2

  37. Amplitude (I) of Sound Oscillations 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: I0 is taken to be the threshold of hearing:

  38. MacScope II

  39. Audacity

  40. iPads & I Phones

  41. More Complex Sounds

  42. Fundamental/Normal Modes

  43. Time and Frequency Domains

  44. Sample Musical Instrument Sounds in the Frequency Domain

  45. 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/

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