1. Chapter 16 Vibrations and Waves

2. Vibrations/Oscillations Object at the end of a spring Object at the end of a spring Tuning fork Tuning fork Pendulum Pendulum String of a violin String of a violin Atoms in a crystal Atoms in a crystal Source of a wave Source of a wave

3. Hooke’s Law F s = - k x F s = - k x F s is the spring forceF s is the spring force k is the spring constantk is the spring constant It is a measure of the stiffness of the spring It is a measure of the stiffness of the spring A large k indicates a stiff spring and a small k indicates a soft springA large k indicates a stiff spring and a small k indicates a soft spring x is the displacement of the object from its equilibrium positionx is the displacement of the object from its equilibrium position x = 0 at the equilibrium position x = 0 at the equilibrium position The negative sign indicates that the force is always directed opposite to the displacementThe negative sign indicates that the force is always directed opposite to the displacement

4. Hooke’s Law Force The force always acts toward the equilibrium position The force always acts toward the equilibrium position It is called the restoring forceIt is called the restoring force The direction of the restoring force is such that the object is being either pushed or pulled toward the equilibrium position The direction of the restoring force is such that the object is being either pushed or pulled toward the equilibrium position

5. Hooke’s Law Applied to a Spring – Mass System When x is positive (to the right), F is negative (to the left) When x is positive (to the right), F is negative (to the left) When x = 0 (at equilibrium), F is 0 When x = 0 (at equilibrium), F is 0 When x is negative (to the left), F is positive (to the right) When x is negative (to the left), F is positive (to the right)

6. Motion of the Spring-Mass System Assume the object is initially pulled to a distance A and released from rest Assume the object is initially pulled to a distance A and released from rest As the object moves toward the equilibrium position, F and a decrease, but v increases As the object moves toward the equilibrium position, F and a decrease, but v increases At x = 0, F and a are zero, but v is a maximum At x = 0, F and a are zero, but v is a maximum The object’s momentum causes it to overshoot the equilibrium position The object’s momentum causes it to overshoot the equilibrium position

7. Motion of the Spring-Mass System, cont The force and acceleration start to increase in the opposite direction and velocity decreases The force and acceleration start to increase in the opposite direction and velocity decreases The motion momentarily comes to a stop at x = - A The motion momentarily comes to a stop at x = - A It then accelerates back toward the equilibrium position It then accelerates back toward the equilibrium position The motion continues indefinitely The motion continues indefinitely

8. Simple Harmonic Motion Motion that occurs when the net force along the direction of motion obeys Hooke’s Law Motion that occurs when the net force along the direction of motion obeys Hooke’s Law The force is proportional to the displacement and always directed toward the equilibrium positionThe force is proportional to the displacement and always directed toward the equilibrium position The motion of a spring mass system is an example of Simple Harmonic Motion The motion of a spring mass system is an example of Simple Harmonic Motion

9. Simple Harmonic Motion, cont. Not all periodic motion over the same path can be considered Simple Harmonic motion Not all periodic motion over the same path can be considered Simple Harmonic motion To be Simple Harmonic motion, the force needs to obey Hooke’s Law To be Simple Harmonic motion, the force needs to obey Hooke’s Law

10. Amplitude Amplitude, A Amplitude, A The amplitude is the maximum position of the object relative to the equilibrium positionThe amplitude is the maximum position of the object relative to the equilibrium position In the absence of friction, an object in simple harmonic motion will oscillate between the positions x = ±AIn the absence of friction, an object in simple harmonic motion will oscillate between the positions x = ±A

11. Period and Frequency The period, T, is the time that it takes for the object to complete one complete cycle of motion The period, T, is the time that it takes for the object to complete one complete cycle of motion From x = A to x = - A and back to x = AFrom x = A to x = - A and back to x = A The frequency, ƒ, is the number of complete cycles or vibrations per unit time The frequency, ƒ, is the number of complete cycles or vibrations per unit time ƒ = 1 / Tƒ = 1 / T Frequency is the reciprocal of the periodFrequency is the reciprocal of the period Unit: Hertz, 1 Hz = 1 cycle/sUnit: Hertz, 1 Hz = 1 cycle/s

12. Acceleration of an Object in Simple Harmonic Motion Newton’s second law will relate force and acceleration Newton’s second law will relate force and acceleration The force is given by Hooke’s Law The force is given by Hooke’s Law F = - k x = m a F = - k x = m a a = -kx / ma = -kx / m The acceleration is a function of position The acceleration is a function of position Acceleration is not constant and therefore the uniformly accelerated motion equation cannot be appliedAcceleration is not constant and therefore the uniformly accelerated motion equation cannot be applied

13. Elastic Potential Energy A compressed spring has potential energy A compressed spring has potential energy The compressed spring, when allowed to expand, can apply a force to an objectThe compressed spring, when allowed to expand, can apply a force to an object The potential energy of the spring can be transformed into kinetic energy of the objectThe potential energy of the spring can be transformed into kinetic energy of the object

14. Elastic Potential Energy, cont The energy stored in a stretched or compressed spring or other elastic material is called elastic potential energy The energy stored in a stretched or compressed spring or other elastic material is called elastic potential energy PE s = ½kx 2PE s = ½kx 2 The energy is stored only when the spring is stretched or compressed The energy is stored only when the spring is stretched or compressed Elastic potential energy can be added to the statements of Conservation of Energy and Work-Energy Elastic potential energy can be added to the statements of Conservation of Energy and Work-Energy

15. Energy in a Spring Mass System A block sliding on a frictionless system collides with a light spring A block sliding on a frictionless system collides with a light spring The block attaches to the spring The block attaches to the spring The system oscillates in Simple Harmonic Motion The system oscillates in Simple Harmonic Motion

16. Energy Transformations The block is moving on a frictionless surface The block is moving on a frictionless surface The total mechanical energy of the system is the kinetic energy of the block The total mechanical energy of the system is the kinetic energy of the block

17. Energy Transformations, 2 The spring is partially compressed The spring is partially compressed The energy is shared between kinetic energy and elastic potential energy The energy is shared between kinetic energy and elastic potential energy The total mechanical energy is the sum of the kinetic energy and the elastic potential energy The total mechanical energy is the sum of the kinetic energy and the elastic potential energy

18. Energy Transformations, 3 The spring is now fully compressed The spring is now fully compressed The block momentarily stops The block momentarily stops The total mechanical energy is stored as elastic potential energy of the spring The total mechanical energy is stored as elastic potential energy of the spring

19. Energy Transformations, 4 When the block leaves the spring, the total mechanical energy is in the kinetic energy of the block When the block leaves the spring, the total mechanical energy is in the kinetic energy of the block The spring force is conservative and the total energy of the system remains constant The spring force is conservative and the total energy of the system remains constant

20. Graphical Representation of Motion When x is a maximum or minimum, velocity is zero When x is a maximum or minimum, velocity is zero When x is zero, the velocity is a maximum When x is zero, the velocity is a maximum When x is a maximum in the positive direction, a is a maximum in the negative direction When x is a maximum in the positive direction, a is a maximum in the negative direction

21. Period of a SHM It can be shown that It can be shown that

22. Simple Pendulum The simple pendulum is another example of simple harmonic motion The simple pendulum is another example of simple harmonic motion The force is the component of the weight tangent to the path of motion The force is the component of the weight tangent to the path of motion F t = - m g sin θF t = - m g sin θ

23. Simple Pendulum, cont In general, the motion of a pendulum is not simple harmonic In general, the motion of a pendulum is not simple harmonic However, for small angles, it becomes simple harmonic However, for small angles, it becomes simple harmonic In general, angles < 15° are small enoughIn general, angles < 15° are small enough

24. Period of Simple Pendulum This shows that the period is independent of the amplitude This shows that the period is independent of the amplitude The period depends on the length of the pendulum and the acceleration of gravity at the location of the pendulum The period depends on the length of the pendulum and the acceleration of gravity at the location of the pendulum Measure g Measure g

25. Example A body with a mass of 5.0 kg is suspended by a spring, which stretches 10 cm when the body is attached. The body is then pulled downward an additional 5 cm and released. Find k, T, f, E, v max and a max.

26. Example Period of a simple pendulum is 2s on Earth. What would be the period of the same pendulum on the moon? (g moon =1.67 m/s 2 )

27. Forced Vibrations and Resonance Shaking (vibration) as specific frequency Shaking (vibration) as specific frequency Pushing child on swing Pushing child on swing glass glass Tacoma Narrows Bridge Tacoma Narrows Bridge ! Every object has its natural frequencies ! Every object has its natural frequencies

30. Wave Motion A wave is the motion of a disturbance A wave is the motion of a disturbance Mechanical waves require Mechanical waves require Some source of disturbanceSome source of disturbance A medium that can be disturbedA medium that can be disturbed Some physical connection between or mechanism though which adjacent portions of the medium influence each otherSome physical connection between or mechanism though which adjacent portions of the medium influence each other All waves carry energy and momentum All waves carry energy and momentum

31. Types of Waves – Traveling Waves Flip one end of a long rope that is under tension and fixed at one end Flip one end of a long rope that is under tension and fixed at one end The pulse travels to the right with a definite speed The pulse travels to the right with a definite speed A disturbance of this type is called a traveling wave A disturbance of this type is called a traveling wave

32. Types of Waves – Transverse In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion

33. Types of Waves – Longitudinal In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave A longitudinal wave is also called a compression wave A longitudinal wave is also called a compression wave

34. Other Types of Waves Waves may be a combination of transverse and longitudinal Waves may be a combination of transverse and longitudinal Mainly consider periodic sinusoidal waves Mainly consider periodic sinusoidal waves

35. Waveform – A Picture of a Wave The brown curve is a “snapshot” of the wave at some instant in time The brown curve is a “snapshot” of the wave at some instant in time The blue curve is later in time The blue curve is later in time The high points are crests of the wave The high points are crests of the wave The low points are troughs of the wave The low points are troughs of the wave

36. Longitudinal Wave Represented as a Sine Curve A longitudinal wave can also be represented as a sine curve A longitudinal wave can also be represented as a sine curve Compressions correspond to crests and stretches correspond to troughs Compressions correspond to crests and stretches correspond to troughs Also called density waves or pressure waves Also called density waves or pressure waves

37. Amplitude and Wavelength Amplitude is the maximum displacement of string above the equilibrium position Amplitude is the maximum displacement of string above the equilibrium position Wavelength, λ, is the distance between two successive points that behave identically Wavelength, λ, is the distance between two successive points that behave identically

38. Speed of a Wave v = ƒ λ v = ƒ λ Is derived from the basic speed equation of distance/timeIs derived from the basic speed equation of distance/time This is a general equation that can be applied to many types of waves This is a general equation that can be applied to many types of waves

39. Speed of a Wave on a String The speed of wave on a stretched rope under some tension, F The speed of wave on a stretched rope under some tension, F  is called the linear density The speed depends only upon the properties of the medium through which the disturbance travels The speed depends only upon the properties of the medium through which the disturbance travels

40. Example Mass and length of the string are 0.9 kg and 8 m. What is the speed of wave on the string?

41. Superposition Principle Two traveling waves can meet and pass through each other without being destroyed or even altered Two traveling waves can meet and pass through each other without being destroyed or even altered Waves obey the Superposition Principle Waves obey the Superposition Principle If two or more traveling waves are moving through a medium, the resulting wave is found by adding together the displacements of the individual waves point by pointIf two or more traveling waves are moving through a medium, the resulting wave is found by adding together the displacements of the individual waves point by point Actually only true for waves with small amplitudesActually only true for waves with small amplitudes

42. Constructive Interference Two waves, a and b, have the same frequency and amplitude Two waves, a and b, have the same frequency and amplitude Are in phaseAre in phase The combined wave, c, has the same frequency and a greater amplitude The combined wave, c, has the same frequency and a greater amplitude

43. Constructive Interference in a String Two pulses are traveling in opposite directions Two pulses are traveling in opposite directions The net displacement when they overlap is the sum of the displacements of the pulses The net displacement when they overlap is the sum of the displacements of the pulses Note that the pulses are unchanged after the interference Note that the pulses are unchanged after the interference

44. Destructive Interference Two waves, a and b, have the same amplitude and frequency Two waves, a and b, have the same amplitude and frequency They are 180° out of phase They are 180° out of phase When they combine, the waveforms cancel When they combine, the waveforms cancel

45. Destructive Interference in a String Two pulses are traveling in opposite directions Two pulses are traveling in opposite directions The net displacement when they overlap is decreased since the displacements of the pulses subtract The net displacement when they overlap is decreased since the displacements of the pulses subtract Note that the pulses are unchanged after the interference Note that the pulses are unchanged after the interference

46. Chapter 17 Sound

47. Producing a Sound Wave Sound waves are longitudinal waves traveling through a medium Sound waves are longitudinal waves traveling through a medium A tuning fork can be used as an example of producing a sound wave A tuning fork can be used as an example of producing a sound wave

48. Using a Tuning Fork to Produce a Sound Wave A tuning fork will produce a pure musical note A tuning fork will produce a pure musical note As the tines vibrate, they disturb the air near them As the tines vibrate, they disturb the air near them As the tine swings to the right, it forces the air molecules near it closer together As the tine swings to the right, it forces the air molecules near it closer together This produces a high density area in the air This produces a high density area in the air This is an area of compressionThis is an area of compression

49. Using a Tuning Fork, cont. As the tine moves toward the left, the air molecules to the right of the tine spread out As the tine moves toward the left, the air molecules to the right of the tine spread out This produces an area of low density This produces an area of low density This area is called a rarefactionThis area is called a rarefaction

50. Using a Tuning Fork, final As the tuning fork continues to vibrate, a succession of compressions and rarefactions spread out from the fork As the tuning fork continues to vibrate, a succession of compressions and rarefactions spread out from the fork A sinusoidal curve can be used to represent the longitudinal wave A sinusoidal curve can be used to represent the longitudinal wave Crests correspond to compressions and troughs to rarefactions

51. Categories of Sound Waves Audible waves Audible waves Lay within the normal range of hearing of the human earLay within the normal range of hearing of the human ear Normally between 20 Hz to 20,000 HzNormally between 20 Hz to 20,000 Hz Infrasonic waves Infrasonic waves Frequencies are below the audible rangeFrequencies are below the audible range Earthquakes are an exampleEarthquakes are an example Ultrasonic waves Ultrasonic waves Frequencies are above the audible rangeFrequencies are above the audible range Dog whistles are an exampleDog whistles are an example

52. Applications of Ultrasound Can be used to produce images of small objects Can be used to produce images of small objects Widely used as a diagnostic and treatment tool in medicine Widely used as a diagnostic and treatment tool in medicine Ultrasounds to observe babies in the wombUltrasounds to observe babies in the womb Cavitron Ultrasonic Surgical Aspirator (CUSA) used to surgically remove brain tumorsCavitron Ultrasonic Surgical Aspirator (CUSA) used to surgically remove brain tumors Ultrasonic ranging unit for cameras Ultrasonic ranging unit for cameras

53. Speed of Sound, General The speed of sound is higher in solids than in gases The speed of sound is higher in solids than in gases The speed is slower in liquids than in solids The speed is slower in liquids than in solids

54. Speed of Sound in Air 331 m/s is the speed of sound at 0°C and 1 atm 331 m/s is the speed of sound at 0°C and 1 atm Changes with temperature Changes with temperature T in °C T in °C At 20 °C, 343 m/s At 20 °C, 343 m/s In other substances In other substances in He: 1000 m/s in Water: 1500 m/s in Al: 5000 m/s

55. Intensity of Sound Waves The average intensity of a wave is the rate at which the energy flows through a unit area, A, oriented perpendicular to the direction of travel of the wave The average intensity of a wave is the rate at which the energy flows through a unit area, A, oriented perpendicular to the direction of travel of the wave The rate of energy transfer is the power The rate of energy transfer is the power Units are W/m 2 Units are W/m 2

56. Various Intensities of Sound Threshold of hearing Threshold of hearing Faintest sound most humans can hearFaintest sound most humans can hear About 1 x 10 -12 W/m 2About 1 x 10 -12 W/m 2 Threshold of pain Threshold of pain Loudest sound most humans can tolerateLoudest sound most humans can tolerate About 1 W/m 2About 1 W/m 2 The ear is a very sensitive detector of sound waves The ear is a very sensitive detector of sound waves It can detect pressure fluctuations as small as about 3 parts in 10 10It can detect pressure fluctuations as small as about 3 parts in 10 10

57. Intensity Level of Sound Waves The sensation of loudness is logarithmic in the human hear The sensation of loudness is logarithmic in the human hear β is the intensity level or the decibel level of the sound β is the intensity level or the decibel level of the sound I o = 1 x 10 -12 W/m 2 is the threshold of hearing I o = 1 x 10 -12 W/m 2 is the threshold of hearing

58. Various Intensity Levels Threshold of hearing is 0 dB Threshold of hearing is 0 dB Threshold of pain is 120 dB Threshold of pain is 120 dB Jet airplanes are about 150 dB Jet airplanes are about 150 dB Table of next slide lists intensity levels of various sounds Table of next slide lists intensity levels of various sounds Multiplying a given intensity by 10 adds 10 dB to the intensity levelMultiplying a given intensity by 10 adds 10 dB to the intensity level

60. Example Quite automobile: 10 -7 W/m 2 1000 automobiles:?

61. Standing Waves When a traveling wave reflects back on itself, it creates traveling waves in both directions When a traveling wave reflects back on itself, it creates traveling waves in both directions The wave and its reflection interfere according to the superposition principle The wave and its reflection interfere according to the superposition principle With exactly the right frequency, the wave will appear to stand still With exactly the right frequency, the wave will appear to stand still This is called a standing waveThis is called a standing wave

62. Standing Waves, cont A node occurs where the two traveling waves have the same magnitude of displacement, but the displacements are in opposite directions A node occurs where the two traveling waves have the same magnitude of displacement, but the displacements are in opposite directions Net displacement is zero at that pointNet displacement is zero at that point The distance between two nodes is ½λThe distance between two nodes is ½λ An antinode occurs where the standing wave vibrates at maximum amplitude An antinode occurs where the standing wave vibrates at maximum amplitude

63. Standing Waves on a String Nodes must occur at the ends of the string because these points are fixed Nodes must occur at the ends of the string because these points are fixed

64. Standing Waves, cont. The pink arrows indicate the direction of motion of the parts of the string The pink arrows indicate the direction of motion of the parts of the string All points on the string oscillate together vertically with the same frequency, but different points have different amplitudes of motion All points on the string oscillate together vertically with the same frequency, but different points have different amplitudes of motion

65. Standing Waves on a String, final The lowest frequency of vibration (b) is called the fundamental frequency The lowest frequency of vibration (b) is called the fundamental frequency

66. Standing Waves on a String – Frequencies ƒ 1, ƒ 2, ƒ 3 form a harmonic series ƒ 1, ƒ 2, ƒ 3 form a harmonic series ƒ 1 is the fundamental and also the first harmonicƒ 1 is the fundamental and also the first harmonic ƒ 2 is the second harmonic (1 st overtone)ƒ 2 is the second harmonic (1 st overtone) Waves in the string that are not in the harmonic series are quickly damped out Waves in the string that are not in the harmonic series are quickly damped out In effect, when the string is disturbed, it “selects” the standing wave frequenciesIn effect, when the string is disturbed, it “selects” the standing wave frequencies

67. Example A guitar has 0.6 m long string. Wave speed on the string is 420 m/s. What are the frequencies of the first few harmonics?

68. Standing Waves in Air Columns If one end of the air column is closed, a node must exist at this end since the movement of the air is restricted If one end of the air column is closed, a node must exist at this end since the movement of the air is restricted If the end is open, the elements of the air have complete freedom of movement and an antinode exists If the end is open, the elements of the air have complete freedom of movement and an antinode exists

69. Tube Open at Both Ends

70. Resonance in Air Column Open at Both Ends In a pipe open at both ends, the natural frequency of vibration forms a series whose harmonics are equal to integral multiples of the fundamental frequency In a pipe open at both ends, the natural frequency of vibration forms a series whose harmonics are equal to integral multiples of the fundamental frequency

71. Tube Closed at One End Closed pipe Closed pipe

72. Resonance in an Air Column Closed at One End The closed end must be a node The closed end must be a node The open end is an antinode The open end is an antinode There are no even multiples of the fundamental harmonic There are no even multiples of the fundamental harmonic

73. Example An open organ pipe has a fundamental frequency of 660 Hz at 0 C and 1 atm. a. Frequency of 2 nd overtone? b. Fundamental at 20 C? c. Replacing air with He?

74. Beats Beats are alternations in loudness, due to interference Beats are alternations in loudness, due to interference Waves have slightly different frequencies and the time between constructive and destructive interference alternates Waves have slightly different frequencies and the time between constructive and destructive interference alternates The beat frequency equals the difference in frequency between the two sources: The beat frequency equals the difference in frequency between the two sources:

75. Doppler Effect A Doppler effect is experienced whenever there is relative motion between a source of waves and an observer. A Doppler effect is experienced whenever there is relative motion between a source of waves and an observer. When the source and the observer are moving toward each other, the observer hears a higher frequencyWhen the source and the observer are moving toward each other, the observer hears a higher frequency When the source and the observer are moving away from each other, the observer hears a lower frequencyWhen the source and the observer are moving away from each other, the observer hears a lower frequency

76. Doppler Effect, General Case Both the source and the observer could be moving Both the source and the observer could be moving Use positive values of v o if observer moving toward source Use positive values of v o if observer moving toward source Frequency appears higherFrequency appears higher Use positive values of v s if source moving away from observer Use positive values of v s if source moving away from observer Frequency appears lowerFrequency appears lower

77. Example F s = 300 Hz and v=300m/s a. Observer moves 30m/s away from source. b. Source moves 30 m/s toward observer. c. Both moving Police radar Police radar “Redshift” of distant galaxies “Redshift” of distant galaxies

78. Shock Waves A shock wave results when the source velocity exceeds the speed of the wave itself A shock wave results when the source velocity exceeds the speed of the wave itself The circles represent the wave fronts emitted by the source The circles represent the wave fronts emitted by the source

79. Shock Waves, final Shock waves carry energy concentrated on the surface of the cone, with correspondingly great pressure variations Shock waves carry energy concentrated on the surface of the cone, with correspondingly great pressure variations A jet produces a shock wave seen as a fog A jet produces a shock wave seen as a fog