Unit 10 Simple Harmonic Waves and Sound Holt Chapter 12 Student Outline

Unit 10 Simple Harmonic Waves and Sound Holt Chapter 12 Student Outline Variables introduced or used in chapter: Quantity Symbol Units Vector or Scalar? Spring Force Spring Constant Displacement Period Frequency Length of Pendulum Mass Wavelength Wave Speed Formula Chart Hooke s Law Relationship between f and T Period of Pendulum in SHM Period of a Mass-Spring System in SHM Speed of a Wave Vocabulary: Define using complete sentences: 1. Simple Harmonic Motion 2. Simple Pendulum 3. Amplitude 4. Period 1

5. Frequency 6. Medium 7. Mechanical Wave 8. Pulse Wave 9. Periodic Wave 10. Transverse Wave 11. Crest 12. Trough 13. Wavelength 14. Longitudinal wave 15. Superposition Principle 16. Constructive Interference 17. Destructive Interference 18. Standing Wave 19. Node 20. Antinodes 2

Holt Chapter 13 Student Outline Variables introduced or used in chapter: Quantity Symbol Units Vector or Scalar? Distance from source Power Intensity Decibel Level Frequency of nth Harmonic Harmonic Number Speed Wavelength Length of string or pipe Formula Chart Intensity of a Spherical Wave Fundamental Frequency (Wave on String) Harmonic Series (Wave on String) Harmonic Series (Open Pipe) Harmonic Series (Closed Pipe) Beat Frequency Vocabulary: Define using complete sentences: 1. Compression 2. Rarefaction 3. Pitch 4. Doppler Effect 3

5. Intensity 6. Decibel Level 7. Forced Vibrations 8. Sympathetic Vibrations 9. Natural Frequency 10. Resonance 11. Fundamental Frequency 12. Harmonic Series 13. Timbre 14. Beat Questions: Answer using complete sentences: 1. Why are sound waves in air characterized as longitudinal? 2. What is the difference between frequency and pitch? 3. What are the differences between infrasonic, audible and ultrasonic waves? 4. How does temperature affect the speed of sound in a medium? 4

Simple Harmonic Motion and Waves 1. 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? [269.5 N/m] 2. Suppose the spring in problem #1 is replaced with a spring that stretches 36cm from its equilibrium position. What is the new spring s spring constant? Is the spring stiffer or less stiff than the spring in problem #1? [14.972 N/m] 3. A load of 45 N is attached to a spring that is hanging vertically stretches the spring 0.14 m. What is the spring constant? [321.429 N/m] 4. A slingshot consists of a light leather cup attached between two rubber bands. If it takes a force of 32 N to stretch the bands 1.2 cm, what is the equivalent spring constant of the rubber bands? [2667 N/s] b. How much force is required to pull the cup of the slingshot 3.0 cm from its equilibrium position? [80.01 N] 5. Which of these periodic motions are simple harmonic? a. A child swinging on a playground swing at a small angle. b. A record rotating on a turntable c. An oscillating clock pendulum 6. A pinball machine uses a spring that is compressed 4.0 cm to launch a ball. If the spring constant is 13 N/m, what is the force on the ball at the moment the spring is released? [0.52 N] 7. How does the restoring force acting on a pendulum bob change as the bob swings toward the equilibrium position? How do the bob s acceleration (along the direction of motion) and velocity change? 8. When an acrobat reaches the equilibrium position, the net force acting along the direction of motion is zero. Why does the acrobat swing past the equilibrium position? 9. You need to know the height of a tower, but darkness obscures the ceiling. You note that a pendulum extending from the ceiling almost touches the floor and its period is 12 s. How tall is the tower? [35.746 m] b. If the period of the pendulum were 24 s, how tall would the tower be? [142.984 m] 10. You are designing a pendulum clock to have a period of 1.0 s. How long should the pendulum be? [24.8 cm] 11. Calculate the period and frequency of a 3.5 m long pendulum at the following locations: a. the North Pole, where g = 9.832 m/s 2 [3.749 s, 0.2667 Hz] b. Chicago, where g = 9.803 m/s 2 [3.754 s, 0.2664 Hz] c. Jakarta, Indonesia, where g = 9.782 m/s 2 [3.758 s, 0.2661 Hz] 12. The free-fall acceleration on the surface of the moon is approximately one-sixth of the free-fall acceleration on the surface of the earth. Compare the period of a pendulum on Earth with that of an identical pendulum set in motion on the moon. 5

13. Why is a pendulum a reliable time-keeping device, even if its oscillations gradually decrease in amplitude over time? 14. The body of a 1275 kg car is supported on a frame by four springs. Two people riding in the car have a combined mass of 153 kg. When driven over a pothole in the road, the frame vibrates with a period of 0.840 s. For the first few seconds, the vibration approximates SHM. Find the spring constant for a single spring. [19974.2 N/m] b. If 2 more people get in the car so that the total mass is 255 kg, what is the period of vibration of the car when it drives over the pothole? [0.86 s] 15. A spring of spring constant 30.0 N/m is attached to different masses and the system is set in motion. Find the period and frequency of vibration for masses of the following magnitudes: a. 2.3 kg [1.74 s, 0.57 Hz] b. 15 g [0.14 s, 7.14 Hz] 16. Two mass-spring systems vibrate with simple harmonic motion. If the spring constants of each system are equal and the mass of one is twice that of the other, which system has a greater period? 17. A child swings on a playground swing with a 2.5 m long chain. What is the period of the child s motion? [3.17 s] b. What is the frequency of vibration? [0.32 Hz] 18. A 0.75 kg mass attached to a vertical spring stretches the spring 0.30 m. What is the spring constant? [24.5 N/m] b. The mass-spring system is now placed on a horizontal surface and set vibrating. What is the period of the vibration? [1.1 s] 19. The reading on a metronome indicates the number of oscillations per minute. What are the period and frequency of the metronome s vibration when the metronome is set at 180? [0.33 s, 3.0 Hz] 20. The piano string tuned to middle C vibrates with a frequency of 264 Hz. Assuming the speed of sound in air is 343 m/s, find the wavelength of the sound waves produced by the string. [1.30 m] 21. A piano emits frequencies that range from a low of about 28 Hz to a high of about 4200 Hz. Find the range of wavelengths in air attained by this instrument when the speed of sound in air is 340 m/s. [0.081 m 12.14 m] 22. The red light emitted by a He-Ne laser has a wavelength of 633 nm in air and travels at 3.0 x 10 8 m/s. Find the frequency of the laser light. [4.74 x 10 14 Hz] 23. A tuning fork produces a sound with a frequency of 256 Hz and a wavelength in air of 1.35 m. What value does this give for the speed of sound in air? [345.6 m/s] b. What would be the wavelength of the wave produced by this tuning fork in water in which sound travels at 1500 m/s? [5.86 m] 24. As waves pass by a duck floating on a lake, the duck bobs up and down but remains in essentially one place. Explain why the duck is not carried along by the wave motion. 25. Sketch each of the following waves on a spring that is attached to a wall at one end: 6

a. a pulse wave that is longitudinal b. a periodic wave that is longitudinal c. a pulse wave that is transverse d. a periodic wave that is transverse 26. Draw a graph for waves (b) and (d) in the problem above, and label the y-axis of each graph with the appropriate variable. Label the following on each graph, where appropriate: crest, trough, wavelength, amplitude, compression, rarefaction. 27. If the amplitude of a sound wave is increased by a factor of four, how does the energy carried by the sound wave in a given time interval change? Why? [16 times greater] 28. A wave of amplitude 0.30 m interferes with a second wave of amplitude 0.20 m. What is the largest resultant displacement that may occur? [0.50 m] 29. A string is rigidly attached to a post at one end. Several pulses of amplitude 0.15 m sent down the string are reflected at the post and travel back down the string without a loss of amplitude. What is the amplitude at a point on the string where the maximum displacement points of two pulses cross? What type of interference is this? b. What if the same pulses were sent down a string whose end was free? What is the amplitude at a point on the string where the maximum displacement points of two pulses cross? What type of interference is this? 30. A stretched string fixed at both ends is 2.0 m long. What are three wavelengths that will produce standing waves on this string? b. name at least one wavelength that would not produce a standing wave pattern and explain your answer. Sound 31. If you hear a higher pitch from a trumpet than from a saxophone, how do the frequencies of the sound waves from the trumpet compare with those from the saxophone? 32. Dolphins can produce sound waves with frequencies ranging from 0.25 khz 220 khz, but only those at the upper end of the spectrum are used in echolocation. Explain why high-frequency waves work better than low-frequency waves. 33. Sound pulses emitted by a dolphin travel through 20 o C ocean water at a rate of 1450 m/s. In 20 o C air, these pulses would travel 342.9 m/s. how can you account for this difference in speed? 34. As a dolphin swims toward a fish, it sends out sound waves to determine the direction the fish is moving. If the frequency of the reflected waves is increased, is the dolphin catching up to the fish or falling behind? 35. What is the intensity of the sound wave produced by a trumpet at a distance of 3.2 m when the power output of the trumpet is 0.20 W? Assume that the sound waves are spherical. [1.55 x10-3 W/m 2 ] 36. At a maximum level of loudness, the power output of a 75-piece orchestra radiated as sound is 70 W. What is the intensity of these sound waves to a listener who is sitting 25 m from the orchestra? [8.91x10-3 W/m 2 ] 7

37. If the intensity of a person s voice is 4.6 x 10-7 W/m 2 at a distance of 2.0 m, how much sound power does that person generate? [2.3 x10-5 W] 38. If a 15 person musical ensemble gains 15 new members, so that its size doubles, will a listener perceive the music created by the ensemble to be twice as loud? Why or why not? 39. Federal regulations require that no office or factory worker be exposed to noise levels that average above 90 db over an 8 hour day. Thus, a factory that currently averages 100 db must reduce its noise level by 10 db. Assuming that each piece of machinery produces the same amount of noise, what percentage of equipment must be removed? Explain your answer. 40. Opera singers have been known to set crystal goblets in vibration with their powerful voices. In fact, an amplified human voice can shatter the glass, but only at certain fundamental frequencies. Speculate about why only certain fundamental frequencies will break the glass. 41. Electric guitars, which use electric amplifiers to magnify their sound, can have a variety of shapes, but acoustic guitars must have an hourglass shape. Explain why. 42. Which of the following change when a sound gets louder? Which change when a pitch gets higher? a. Intensity b. Speed of sound waves d. Decibel level e. Wavelength c. Frequency f. Amplitude 43. A certain microphone placed in the ocean is sensitive to sounds emitted by dolphins. To produce a usable signal, sound waves striking the microphone must have a decibel level of 10 db. If dolphins emit sound waves with a power of 0.050 W, how far can a dolphin be from the microphone and still be heard? Assume the sound wave propagate spherically and disregard absorption of the sound waves. [1.99 x 10 4 m] 44. When the decibel level of traffic in the street goes from 40 to 60 db, how much louder does the traffic noise seem? How much greater is the intensity? 45. What are the first three harmonics in a 2.45 m long pipe that is open at both ends? [70.41 Hz, 140.82 Hz, 211.22 Hz] What are the first three harmonics of this pipe when one end of the pipe is closed? [35.20 Hz, 105.61 Hz, 176.02 Hz] Assume that the speed of sound in air is 345 m/s for both of these situations. 46. A violin string that is 50 cm long has a fundamental frequency of 440 Hz. What is the speed of the waves on this string? [440 m/s] 47. What is the fundamental frequency of a 0.20 m long organ pipe that is closed at one end, when the speed of sound in the pipe is 352 m/s? [440 Hz] 48. On a piano, the note middle C has a fundamental frequency of 264 Hz. What is the second harmonic of this note? [528 Hz] If the piano wire is 66 cm long, what is the speed of waves on the wire? [348.5 m/s] 49. A piano tuner using a 392 Hz tuning fork to tune the wire for G-natural hears four beats per second. What are the two possible frequencies of vibration of this piano wire? [388 Hz & 396 Hz] 8

Standing Waves Lab Theory: Each harmonic is the number of antinodes in the string. One complete wavelength is the length of two antinodes. If the string is 1 meter long and you have 3 antinodes then the wavelength is 2/3 meter, since 3 antinodes equal 1 meter and a whole wave is 2 of the 3 antinodes. The larger the amplitude of a wave, the more energy the wave has, because it takes more force to stretch the string a greater distance. The wiggler applies electrical energy to vibrate the string. The wiggler supplies about the same amount of energy to each harmonic. Purpose: To observe how frequency relates to standing waves. Materials: CPO Wave Generator CPO Wiggler CPO Timer Ruler Procedure: 1. Turn on timer and set the wave generator to WAVES. 2. Set the timer to measure FREQUENCY. You should get a reading of about 10 Hz, which means the wiggler is oscillating back and forth 10 times per second. 3. Adjust the frequency of the wiggler with the frequency control on the Wave Generator. 4. Adjust the frequency to obtain the 2 nd harmonic of the string 5. Fine-tune the frequency to obtain the largest amplitude before recording the data. 6. Using a ruler, measure the amplitude. include the 6 th harmonic or one higher. Remember the amplitude is ½ the width of the wave at the widest point. 7. Record the frequency, wavelength and amplitude in the data table below. 8. Find and record the 3 rd and 4 th harmonics. 9. Find the 1 st harmonic. Use the data you collected to help find it. 10. Calculate the velocity of the waves. Summarize the Purpose of the Lab: Observations: Harmonic # Frequency (Hz) Wavelength (m) Speed of wave (m/s) Amplitude (cm) Sample speed of wave calculation: Analysis: 1. Describe how the frequencies of the different harmonic patterns are related to each other. 2. If the frequency increases, what happens to the wavelength? 3. Make a amplitude (Y-axis) vs frequency (X-axis) graph on a separate piece of graph paper and attach to the lab. Label axes (with units), include a title, make as large as is reasonable, and use best fit line. 4. Suppose you had a wave maker which allowed you to adjust the input energy so all the waves could have the same amplitude. You then used this wave maker to create two waves with equal amplitude, but one had a higher frequency than the other. If the 9

amplitude is the same, which wave has more energy, the higher frequency wave or the lower frequency wave? Use your results/graph to explain your answer. Conclusion: a. Write in paragraph form using proper English b. Restate purpose and if it was met. Back up your answer with data. c. Relate your results to the equations and the concepts of the physics that apply. Don t fail to see the forest for the trees there are often general conclusions to be drawn as well as specific. d. Discuss error list SPECIFIC sources of error and their size and impact on the results. Include how you could reduce the amount of error. Always calculate percent error or percent difference when possible. 10

Hooke s Law Lab Theory: Force When a force is applied to a spring, the spring stretches. A greater force stretches the spring further. Will twice the force stretch the spring twice as far, or four times as far? The relationship between force and the spring is linear. Both the stretching and the unstretching can be defined by the same equation. F=-kx The k is called the spring constant. It measures the stiffness of the spring and depends on the material of the spring. The negative sign is present in the equation because the force of the spring is a restoring force and is constantly in the opposite direction of the displacement of the spring. Work When the force vs. displacement is graphed, the slope of the line is the spring constant and the area under the graph is the work done on the spring. Since this line forms a triangle with the x axis, the area is: W = ½ kx 2 Since the spring returns this same work, the elastic potential energy stored in the spring is given by: PE = ½ kx 2 Purpose: Determine the relationship between the force on a spring and its stretch, both stretching and unstretching and investigate the work done on a spring. Materials: Hooke s Law Apparatus Assorted Masses Mass Hanger Graph paper Procedure: 1. Hang the spring from the hook of the Hooke s Law Apparatus and the mass hanger from the spring. The mass hanger will be viewed as being a part of the spring. 2. Add even increments of mass to the mass hanger and record the stretch of the spring after each addition. Choose the mass increment so that it is small enough for at least ten increments to be added without over stretching the spring. Make sure the increments are large enough that there is a noticeable stretch of the spring with each addition. 3. Record the measurements in the data table. 4. After adding 10 increments of mass, begin removing the mass in the same intervals it was added. Take care during the addition and subtraction of the masses not to let the spring oscillate. 5. Record the measurements on the data table. 6. Create a single graph illustrating the force versus the displacement of the stretching and unstretching of the spring. Use a symbol for the stretch data (trials 1-10) and superimpose the unstretched data (trials 11-20) with a different symbol. This will create two lines on one graph. 7. Draw a trend line and compute the slope to find k. 8. Use the graph to predict what the extension of the spring would be if 600 g were supported from the spring. 9. Calculate how much stored potential energy is in the spring under each mass load and record it on the data table. 11

10. Use the graph to determine the area under the line, and compare this to the calculated potential energy on the data table. Summarize the Purpose of the Lab: Observations: Trial Mass (kg) Force (N) Stretch (m) Elastic Potential Energy (J) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Calculations: 2. Attach graph. Include title and label axes (include units) 3. Show work for calculating k. k = 4. Stretch of spring if it was supporting 600 g. x = 12

5. Show a sample of work for calculating elastic potential energy from each mass load. 6. Use graph to calculate the area under the line. Area under line =. Analysis: 1. What does the area under the graph equal? Use your calculations to justify your answer. 2. What would happen to the results if the spring is over stretched? 3. What would happen to the graph if the spring was denser? Conclusion: a. Write in paragraph form using proper English b. Restate purpose and if it was met. Back up your answer with data. c. Relate your results to the equations and the concepts of the physics that apply. Don t fail to see the forest for the trees there are often general conclusions to be drawn as well as specific. d. Discuss error list SPECIFIC sources of error and their size and impact on the results. Include how you could reduce the amount of error. Always calculate percent error or percent difference when possible. 13

Speed of Sound in Air Demo Materials: Large container with water (e.g. bucket or 1L graduated cylinder) PVC pipe (approx. 1 diameter, at least 70cm long) Tuning fork, >320Hz (higher frequency is better) Rubber Mallet Meterstick Note: You want to have a pipe that will give you at least 2 resonance points with your tuning fork. Procedure: 1. Fill the cylinder with water, put the PVC pipe in the water. 2. Strike the tuning fork with a rubber mallet. 3. Hold the tuning fork about an inch above the PVC pipe (not touching it) while slowly raising the PVC pipe (and tuning fork) as if you were taking it out of the water. 4. At some point you will hear the sound of the tuning fork resonating in the PVC pipe. Measure the length of PVC pipe that is out of the water. 5. Repeat 3 and 4 while continuing to raise the PVC pipe out of the water. See if you can find any more resonance points. 6. If the distance between the resonance points is ½ λ, calculate the speed of sound in air. Observations & Calculations: L 1 = L 2 = ΔL = (show work) Use v = f λ λ = (show work) v = (show work) 14

Simple Harmonic Motion and Waves Make sure to study all of your notes, problems worked in class and on previous homeworks and be familiar with what was done on lab days. 1. You want to know the height of the ceiling at the Museum of Natural Science. A pendulum is attached to the ceiling which almost touches the floor and its period is 15 s. How tall is the ceiling? [55.85 m] b. What is the frequency of the pendulum? [0.067 Hz] 2. The body of a 1400 kg car is supported on a frame by four springs. Two people riding in the car have a combined mass of 160 kg. When driven over a pothole in the road, the frame vibrates with a period of 0.960 s. For the first few seconds, the vibration approximates SHM. Find the spring constant for a single spring. [16706.36N/m] b. If 2 more people get in the car so that the total mass of people is 315 kg, what is the period of vibration of the car when it drives over the pothole? [1.01 s] 3. A pinball machine uses a spring that is compressed 5.8 cm to launch a ball. If the spring constant is 14.2 N/m, what is the force on the ball at the moment the spring is released? [0.824 N] 4. If a mass of 0.82 kg attached to a vertical spring stretches the spring 6.7 cm from its original equilibrium position, what is the spring constant? [119.94 N/m] 5. A 0.78 kg mass attached to a vertical spring stretches the spring 0.15 m. What is the spring constant? [50.96 N/m] b. The mass-spring system is now placed on a horizontal surface and set vibrating. What is the period of the vibration? [0.777 s] 6. The piano string tuned to middle C vibrates with a frequency of 264 Hz. Assuming the speed of sound in air is 349 m/s, find the wavelength of the sound waves produced by the string. [1.322 m] 7. A tuning fork produces a sound with a frequency of 256 Hz and a wavelength in air of 1.22 m. What value does this give for the speed of sound in air? [312.32 m/s] b. What would be the wavelength of the wave produced by this tuning fork in water in which sound travels at 1500 m/s? [5.86 m] 8. If the amplitude of a sound wave is increased by a factor of three, how does the energy carried by the sound wave in a given time interval change? Why? [9 times] 9. A wave of amplitude 0.40 m interferes with a second wave of amplitude 0.22 m. What is the largest resultant displacement that may occur? [0.62 m] 10. A stretched string fixed at both ends is 3.0 m long. What are three wavelengths that will produce standing waves on this string? b. Name at least one wavelength that would not produce a standing wave pattern and explain your answer. 15

Sound 1. Sound waves are waves. a. longitudinal b. transverse c. compression d. a & b e. a & c 2. When a sound wave is graphed as a sine wave, the compressions correspond to the on the sine graph and the rarefactions correspond to the. Use the following words for #3-4. high, low, spread apart, close together 3. Compressions are regions of density, pressure and the molecules are. 4. Rarefactions are regions of density, pressure and the molecules are. 5. The frequency of a sound wave can be defined as 6. The range frequencies a human can hear is. 7. Sound waves lower than 20 Hz are called. List some examples of when this type of wave is used. 8. Sound waves higher than 20,000 Hz are called. List some examples of when this type of wave is used. 9. The higher the frequency, the the wavelength. 10. How are pitch and frequency related? Which one is subjective (perceived differently by individuals)? Which one is objective (can be measured precisely)? 11. List in order of increasing speed of sound the 3 states of matter. Why does sound travel fastest through the medium that it does? 12. How does the temperature of a gas medium affect the speed of sound? 13. Draw a representation of a spherical sound wave. Label the wave front, wavelength, source, and ray. Draw the sine curve that corresponds to your diagram. 14. Explain how the Doppler Effect works including diagrams. How does the pitch change as an object moves towards and away from a stationary observer? What are some examples of things that use the Doppler Effect? 15. How is intensity related to distance from the source of sound? If the distance from the source were to double, what would happen to the sound intensity? 16. At a maximum level of loudness, the power output of a 150-piece orchestra radiated as sound is 95 W. What is the intensity of these sound waves to a listener who is sitting 20 m from the orchestra? 17. If the intensity of a person s voice is 5.2 x 10-7 W/m 2 at a distance of 1.2 m, how much sound power does that person generate? 16

18. The softest sound human can hear is known as the, while the loudest sound a human can tolerate is known as the. 19. An increase of 10 db corresponds to a of the volume. 20. If the intensity of a sound is multiplied by 10, there is an increase of db. 21. What is relative intensity? It is also referred to as the level because relative intensity is measured in. 22. What is an example of forced vibration? 23. What is natural frequency? 24. Explain resonance. Give at least one example. 25. Why is our outer ear shaped the way it is? 26. What types of waves are produced on strings and in air columns of instruments? 27. The frequency is the lowest possible frequency of a standing wave. 28. In what situation can you hear ALL of the harmonics? In what situation can you only hear the ODD harmonics? 29. Explain what is happening when you can hear beats. 30. When the decibel level goes from 30 db to 60 db, how much louder does the noise seem? How much greater is the intensity? 31. If the speed of sound is 344 m/s, what are the first three harmonics in a 0.75 m long pipe with both ends open? 32. If the speed of sound is 340 m/s, what are the first three harmonics in a 1.5 m long pipe with one end closed? 33. On a piano, the note E has a fundamental frequency of 320 Hz. What is the second harmonic of this note? If the piano wire is 60 cm long, what is the speed of waves on the wire? 34. You have two tuning forks, one with a frequency of 345 Hz and a second with an unknown frequency. If you can hear 6 beats per second, what frequency could the unknown tuning fork be? 17

35. Label the following graph of 2 tuning forks with slightly different frequencies. Include destructive interference, constructive interference, loud, soft, in-phase, out of phase. What does this graph represent? 18