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Chapter 11

Chapter 11. Section 1 Simple Harmonic Motion. Chapter 11. Periodic (vibrating) motion is repeated motion. swing pendulum mass on a spring metronome pendulum clocks oscillations waves. V ibrations and WAVES. Chapter 11. Chapter 11. Vibrations and Waves. Table of Contents.

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Chapter 11

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  1. Chapter 11

  2. Section 1 Simple Harmonic Motion Chapter 11 • Periodic (vibrating) motion is repeated motion. • swing • pendulum • mass on a spring • metronome • pendulum clocks • oscillations • waves

  3. Vibrations and WAVES Chapter 11

  4. Chapter 11 Vibrations and Waves Table of Contents Section 1 Simple Harmonic Motion Section 2 Measuring Simple Harmonic Motion Section 3 Properties of Waves Section 4 Wave Interactions

  5. Section 1 Simple Harmonic Motion Chapter 11 The Vibrating Spring • One example ofperiodic motionis the motion of a mass attached to a spring. • The direction of the force acting on the mass (Felastic) is always opposite the direction of the mass’s displacement from equilibrium (x = 0). • Felis a restoring force, because it acts to restore the system to equilibrium

  6. Section 1 Simple Harmonic Motion Chapter 11 Hooke’s Law • Measurements show that thespringforce, or restoring force,isdirectly proportionalto thedisplacementof the mass. • This relationship is known asHooke’s Law: Felastic = –kx spring force = –(spring constant  displacement) • The quantitykis a positive constant called thespring constant with SI unit N/m. PE elastic = 1/2kx2 PE max = 1/2kx2max • The stiffer the spring the greater the spring constant.

  7. Section 1 Simple Harmonic Motion Chapter 11 Sample Problem Hooke’s Law If a mass of 0.55 kg attached to a vertical spring stretches the spring 2.0 cm from its original equilibrium position, what is the spring constant?

  8. Section 1 Simple Harmonic Motion Chapter 11 • Determine the spring constant. • Unknown: • k = ? • Diagram: • Given: • m = 0.55 kg • x = – 2.0 cm = – 0.02 m • g = 9.81 m/s2 • Fg = mg = 0.55 kg x 9.81 m/s2 = 5.4 N • F = -kx

  9. Section 1 Simple Harmonic Motion Chapter 11 When the mass is attached to the spring, the equilibrium position changes. At the new equilibrium position, the net force acting on the mass is zero. So the spring force (given by Hooke’s law) must be equal and opposite to the weight of the mass. Fnet = 0 = Felastic + Fg (Fel is up and Fg is down) Felastic + Fg= 0 Felastic= –kx Fg = – 5.4 N –kx– 5.4 N = 0 –kx= 5.4 N k = Felastic /–x k = 5.4 N /–(-.02 m) k = 270 N/m

  10. Section 1 Simple Harmonic Motion Chapter 11 Evaluate your answer. The value of k implies that 270 N of force is required to displace the spring 1 m. A 270 N force is the weight of a 27.5 kg mass.

  11. Practice A p. 371 • a. 15 N/m b. less stiff • 3.2 x 102 N/m • 2.7 x 103 N/m • 81 N

  12. Chapter 11 Section 1 Simple Harmonic Motion • All vibrating systems that obey Hooke’s law demonstrate simple harmonic motion. • Simple harmonic motion describes any periodic motion that is the result of a restoring force that is proportional to displacement. (Hooke’s Law) • Because simple harmonic motion involves a restoring force, every simple harmonic motion is a back-and-forth motion over the same path.

  13. Harmonic Oscillators A harmonic oscillator is an oscillator with a restoring force proportional to its displacement from equilibrium. Its period of oscillation depends only on the stiffness of that restoring force and on its mass, not on its amplitude of oscillation. Period (T) is the time for 1 complete oscillation and in measured in seconds. Any harmonic oscillator can be thought of as having a restoring force component that drives the motion and an inertial component that resists the motion. http://upload.wikimedia.org/wikipedia/commons/2/21/Pendulum_animation.gif http://upload.wikimedia.org/wikipedia/commons/e/ea/Simple_Harmonic_Motion_Orbit.gif

  14. Clocks are based on Simple Harmonic Oscillators Pendulum and Balance Clocks The accuracy of pendulum and balance clocks is limited by friction, air resistance, and thermal expansion to about 10 s/year. Electronic Clocks Quartz crystals are piezoelectric materials that respond mechanically to electrical stress and electrically to mechanical stress. Loses or gains less than a 0.1 s/year Atomic Clocks (Gain / lose 1 s/ 10 million years) The second is the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. The meter is the length of the path travelled by light emitted by krypton-86 in vacuum during a time interval of 1/299 792 458 of a second.

  15. Section 2 Measuring Simple Harmonic Motion Chapter 11 Amplitude, Period, and Frequency in SHM • In SHM, the maximum displacement from equilibrium is defined as theamplitude of the vibration. • Apendulum’s amplitude is measured by the angle between the pendulum’s equilibrium position and its maximum displacement. • For amass-spring system,the amplitude is the maximum amount the spring is stretched or compressed from its equilibrium position. • TheSI unitsof amplitude are theradian (rad)and themeter (m).

  16. Section 2 Measuring Simple Harmonic Motion Chapter 11 Amplitude, Period, and Frequency in SHM • Theperiod(T)is the time that it takes a complete cycle to occur. • The SI unit of period is seconds (s). • The frequency(f)is the number of cycles or vibrations per unit of time. (cycles/s) • The SI unit of frequency is hertz (Hz). • Hz = s–1

  17. Section 2 Measuring Simple Harmonic Motion Chapter 11 Amplitude, Period, and Frequency in SHM • Period and frequency areinversely related: • Any time you know the period or the frequency, you can calculate the other value.

  18. Section 2 Measuring Simple Harmonic Motion Chapter 11 Measures of Simple Harmonic Motion

  19. Simple Harmonic Motion http://webphysics.davidson.edu/physlet_resources/bu_semester1/c18_SHM_graphs.html y = A sinwt y = A sin 2pf

  20. Simple Harmonic Motion

  21. Section 1 Simple Harmonic Motion Chapter 11 Skip to Next Simple Harmonic Motion

  22. Section 1 Simple Harmonic Motion Chapter 11 Force and Energy in Simple Harmonic Motion

  23. Section 1 Simple Harmonic Motion Chapter 11 Simple Harmonic Motion

  24. Section 2 Measuring Simple Harmonic Motion Chapter 11 Measures of Simple Harmonic Motion

  25. Section 1 Simple Harmonic Motion Chapter 11 Simple Harmonic Motion • Why must the angle of displacement be small for a pendulum to exhibit simple harmonic motion? • For SHM condition the restoring force must be proportional to the displacement. • F must be proportional to q and not sin q. 360o = 2P radians 15o x 2P radians/ 360o

  26. Chapter 11

  27. Section 3 Properties of Waves Chapter 11 WAVES

  28. Section 3 Properties of Waves Chapter 11 Objectives • Distinguishlocal particle vibrations from overall wave motion. • Differentiatebetween pulse waves and periodic waves. • Interpret waveforms of transverse and longitudinal waves. • Applythe relationship among wave speed, frequency, and wavelength to solve problems. • Relateenergy and amplitude.

  29. Section 3 Properties of Waves Chapter 11 Wave Motion • Awaveis the motion of a disturbance and is produced by a vibration. A wave transfers energy. • Amediumis a physical environment through which a disturbance can travel. For example, water is the medium for ripple waves in a pond. • Waves that require a medium through which to travel are calledmechanical waves.Water waves and sound waves are mechanical waves. • Electromagnetic wavessuch as visible light do not require a medium.

  30. Section 3 Properties of Waves Chapter 11 Wave Types • A wave that consists of a single traveling pulse is called apulse wave. • Whenever a wave’s source is periodic motion, such as a vibration or the motion of your hand moving up and down repeatedly, aperiodic waveis produced. • A wave generated by a simple harmonic oscillator is called asine wave.

  31. Section 3 Properties of Waves Chapter 11 Relationship Between SHM and Wave Motion As the sine wave created by this vibrating blade travels to the right, a single point on the string vibrates up and down with simple harmonic motion.

  32. Section 3 Properties of Waves Chapter 11 Wave Types • Atransverse waveis a wave whose particles vibrateperpendicularlyto the direction of the wave motion. • Thecrestis the highestpoint above the equilibrium position, and thetroughis thelowestpoint below the equilibrium position. • Thewavelength (l)is the distance between two adjacent similar points of a wave. • Amplitude is maximum displacement from rest or equilibrium.

  33. Section 3 Properties of Waves Chapter 11 Transverse Waves

  34. Section 3 Properties of Waves Chapter 11 Wave Types • Alongitudinal wave is a wave whose particles vibrate parallel to the direction the wave is traveling. • A longitudinal wave on a spring at some instant in time, t, can be represented by a graph. Thecrestscorrespond to compressed regions, and thetroughscorrespond to stretched regions. • The crests are regions of high density and pressure (relative to the equilibrium density or pressure of the medium), and the troughsare regions of low density and pressure.

  35. Section 3 Properties of Waves Chapter 11 Longitudinal Waves

  36. Section 3 Properties of Waves Chapter 11 Period, Frequency, and Wave Speed • The frequency of a wavedescribes the number of waves that pass a given point in a unit of time. • Theperiod of a wavedescribes the time it takes for a complete wavelength to pass a given point. • The relationship between period and frequency in SHM holds true for waves as well; the period of a wave isinversely relatedto its frequency.

  37. Section 3 Properties of Waves Chapter 11 Amplitude, Period, and Frequency • Theperiod(T)is the time that it takes a complete cycle of a wave to occur. • The SI unit of period is seconds (s). • The frequency(f)is the number of waves or vibrations per unit of time. (waves/s) • The SI unit of frequency is hertz (Hz). • Hz = s–1

  38. Section 3 Properties of Waves Chapter 11 Amplitude, Period, and Frequency • Period and frequency areinversely related: • Any time you know the period or the frequency, you can calculate the other value.

  39. Section 3 Properties of Waves Chapter 11 Characteristics of a Wave

  40. Section 3 Properties of Waves Chapter 11 Period, Frequency, and Wave Speed, continued • The speed of a mechanicalwave is constant for any given medium. • Thespeed of a waveis given by the following equation: v = fl wave speed = frequency  wavelength • This equation applies to both mechanical and electromagnetic waves.

  41. Section 3 Properties of Waves Chapter 11 Waves and Energy Transfer • Waves transfer energy by the vibration of matter. • Waves are often able to transport energy efficiently. • The rate at which a wave transfers energy depends on theamplitude. • The greater the amplitude, the more energy a wave carries in a given time interval. • For a mechanical wave, the energy transferred is proportional to the square of the wave’s amplitude. • E a amp2 • The amplitude of a wave gradually diminishes over time as its energy is dissipated.

  42. Practice D p. 387 • 0.081 m <l< 12 m • a. 3.41 m b. 5.0 x 10-7 m c. 1.0 x 10-10 m • 4.74 x 1014 Hz • a. 346 m/s b. 5.86 m

  43. Section 4 Wave Interactions Chapter 11 Objectives • Apply the superposition principle. • Differentiatebetween constructive and destructive interference. • Predictwhen a reflected wave will be inverted. • Predictwhether specific traveling waves will produce a standing wave. • Identifynodes and antinodes of a standing wave.

  44. Section 4 Wave Interactions Chapter 11 Wave Interference • Two different material objects can never occupy the same space at the same time. • Because mechanical waves are not matter but rather are displacements of matter, two waves can occupy the same space at the same time. • The combination of two overlapping waves is calledsuperposition.

  45. Section 4 Wave Interactions Chapter 11 Wave Interference, continued In constructive interference,individual displacements on the same side of the equilibrium position are added together to form the resultant wave.

  46. Section 4 Wave Interactions Chapter 11 Wave Interference In destructive interference,individual displacements on opposite sides of the equilibrium position are added together to form the resultant wave.

  47. Section 4 Wave Interactions Chapter 11 Comparing Constructive and Destructive Interference

  48. Section 4 Wave Interactions Chapter 11 Reflection • What happens to the motion of a wave when it reaches a boundary? • At afree boundary, waves arereflected. • At afixedboundary, waves arereflectedand inverted. Free boundary Fixed boundary PhET Wave on a String - Interference, Harmonic Motion, Frequency , Amplitude

  49. Section 4 Wave Interactions Chapter 11 Standing Waves

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