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Oscillations: Understanding Periodic Motion and Simple Harmonic Motion

This article explains the concepts of periodic motion, simple harmonic motion, and the properties of waves. It covers topics such as the period and frequency of oscillations, the behavior of masses on springs and pendulums, and the phenomenon of resonance. It also discusses the different types of waves, their reflection and transmission, and the characteristics of sound.

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Oscillations: Understanding Periodic Motion and Simple Harmonic Motion

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  1. Oscillations About Equilibrium

  2. 7.1 Periodic Motion

  3. Periodic Motion – repeat, same time, same path Period (T) – time required for one complete cycle (seconds) or seconds/cycle Frequency (f) – the number of oscillations per second (s-1 or hertz) 7.2 Simple Harmonic Motion

  4. 7.2 Simple Harmonic Motion

  5. A form of Periodic Motion Simple Harmonic Motion A restoring force is applied proportional to the distance from equilibrium So Hooke’s Law 7.2 Simple Harmonic Motion

  6. If a graph of simple harmonic motion is created And spread out over time We get a wave pattern Amplitude – maximum displacement 7.2 Simple Harmonic Motion

  7. 7.3 The Period of a Mass on a Spring

  8. The period of a spring is given by the equation A larger mass would have greater inertia – longer period A larger spring constant would produce more acceleration, so a shorter period The period is independent of amplitude 7.3 The Period of a Mass on a Spring

  9. 7.5 The Pendulum

  10. q Lcosq L L-Lcosq A simple Pendulum The potential energy is So potential energy is zero at equilibrium (like SHM) 7.5 The Pendulum

  11. The period of a pendulum is given as Independent of the mass of the bob 7.5 The Pendulum

  12. T mgcosq mgsinq W Restoring Force Forces Components A pendulum does not act as a Simple Harmonic Oscillator, but at small angles (<30o) it approximates SHM 7.5 The Pendulum

  13. 7.7 Driven Oscillations and Resonance

  14. Natural Frequency – depends on the variables (m,k or L,g) of the object Forced Vibrations – caused by an external force 7.7 Driven Oscillations and Resonance

  15. Resonant Frequency – the natural vibrating frequency of a system Resonance – when the external frequency is near the natural frequency and damping is small Tacoma Narrow Bridge 7.7 Driven Oscillations and Resonance

  16. 7.8 Types of Waves

  17. Mechanical Waves – travels through a medium The wave travels through the medium, but the medium undergoes simple harmonic motion Wave motion Particle motion 7.8 Types of Waves

  18. Waves transfer energy, not particles A single bump of a wave is called a pulse A wave is formed when a force is applied to one end Each successive particle is moved by the one next to it 7.8 Types of Waves

  19. Parts of a wave Transverse wave – particle motion perpenduclar to wave motion Wavelength (l) measured in meters Frequency (f) measured in Hertz (Hz) Wave Velocity (v) meters/second 7.8 Types of Waves

  20. Longitudinal (Compressional) Wave Particles move parallel to the direction of wave motion Rarefaction – where particles are spread out Compression – particles are close 7.8 Types of Waves

  21. Earthquakes S wave – Transverse P wave – Longitudinal Surface Waves – can travel along the boundary Notice the circular motion of the particles 7.8 Types of Waves

  22. 7.9 Reflection and Transmission of Waves

  23. When a wave comes to a boundary (change in medium) at least some of the wave is reflected The type of reflection depends on if the boundary is fixed (hard) - inverted 7.9 Reflection and Transmission of Waves

  24. When a wave comes to a boundary (change in medium) at least some of the wave is reflected Or movable (soft) – in phase 7.9 Reflection and Transmission of Waves

  25. For two or three dimensional we think in terms of wave fronts A line drawn perpendicular to the wave front is called a ray When the waves get far from their source and are nearly straight, they are called plane waves 7.9 Reflection and Transmission of Waves

  26. Law of Reflection – the angle of reflection equals the angle of incidence Angles are always measured from the normal 7.9 Reflection and Transmission of Waves

  27. 7.10 Characteristics of Sound

  28. Sound is a longitudinal wave Caused by the vibration of a medium The speed of sound depends on the medium it is in, and the temperature For air, it is calculated as 7.10 Characteristics of Sound

  29. Loudness – sensation of intensity Pitch – sensation of frequency Range of human hearing – 20Hz to 20,000 Hz ultrasonic – higher than human hearing dogs hear to 50,000 Hz, bats to 100,000 Hz infrasonic – lower than human hearing 7.10 Characteristics of Sound

  30. Often called pressure waves Vibration produces areas of higher pressure These changes in pressure are recorded by the ear drum 7.10 Characteristics of Sound

  31. 7.11 Intensity of Sound

  32. Loudness – sensation Relative to surrounding and intensity Intensity – power per unit area Humans can detect intensities as low as 10-12 W/m2 The threshold of pain is 1 W/m2 7.11 Intensity of Sound

  33. Sound intensity is usually measured in decibels (dB) • Sound level is given as • I – intensity of the sound • I0 – threshold of hearing (10-12 W/m2) • – sound level in dB Some common relative intensities 7.11 Intensity of Sound

  34. 7.12 The Ear

  35. Steps in sound transmission 7.12 The Ear

  36. 7.13 Sources of Sound: Strings and Air Columns

  37. Vibrations in strings Fundamental frequency Next Harmonic 7.13 Sources of Sound

  38. Vibrations in strings Next Harmonic Strings produce all harmonics – all whole number multiples of the fundamental frequency 7.13 Sources of Sound

  39. Vibrations in an open ended tube (both ends) Fundamental frequency Next Harmonic 7.13 Sources of Sound

  40. Vibrations in open ended tubes Next Harmonic Open ended tubes produce all harmonics – all whole number multiples of the fundamental frequency Examples include organ pipes and flutes. 7.13 Sources of Sound

  41. Vibrations in an closed end tube (one end) Fundamental frequency Next Harmonic 7.13 Sources of Sound

  42. Vibrations in open ended tubes Next Harmonic Closed end tubes produce only odd harmonics Examples include reeded wind instruments and brass instruments 7.13 Sources of Sound

  43. 7.14 Interference of Sound Waves: Beats

  44. If waves are produced by two identical sources A pattern of constructive and destructive interference forms Applet 7.14 Interference of Sound Waves: Beats

  45. 7.15 The Doppler Effect

  46. Doppler Effect – the change in pitch due to the relative motion between a source of sound and the receiver Applies to all wave phenomena Objects moving toward you have a higher apparent frequency Objects moving away have a lower apparent frequency Doppler Effect Light Doppler 7.15 The Doppler Effect

  47. If an object is stationary the equation for the wave velocity is Sound waves travel outward evenly in all directions If the object moves toward the observed, the waves travel at the same velocity, but each new vibration is created closer to the observer Doppler Applet 7.15 The Doppler Effect

  48. The general equation is The values of Vo (speed of observer) and Vs (speed of source) is positive when they approach each other 7.15 The Doppler Effect Radar Gun

  49. 7.16 Interference

  50. Interference – two waves pass through the same region of space at the same time The waves pass through each other Principle of Superposition – at the point where the waves meet the displacement of the medium is the algebraic sum of their separate displacements 7.16 Interference

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