PHYSICS LAB 10 SP211. Standing Waves

Transcription

1 PHYSICS LAB 10 SP211 Standing Waves I. Introduction The phenomenon of standing waves is responsible for most of the sounds of musical instruments. Unwanted standing waves can be responsible for uneven cooking in a microwave oven and are what marching groups are trying to avoid when they break cadence when crossing a bridge. In this lab, we will study standing waves on guitar strings and aluminum rods. Even though standing waves are responsible for the sounds that a guitar makes, it is not easy to see them on a guitar. The first part of this lab is designed so that we can see the standing waves. The phenomenon that makes this possible is resonance. Resonance occurs when a "driving frequency" matches (is the same as) the "natural frequency" of an oscillating system. Furthermore a standing wave in one dimension is actually composed of two traveling waves moving in opposite directions, due to reflections from the ends. Knowing this, we will be able to make measurements to calculate the speed of traveling waves on strings. In the last part of the lab, we will measure frequencies associated with standing waves on an aluminum rod. That information, along with assumed wavelengths, will enable us to calculate the speed of sound in aluminum. In this lab we will study the standing waves on a string and a rod with setups similar to those shown below. Page 1 of 6

2 II. Objectives At the end of this activity, you should: 1. Be able to calculate and measure the frequencies for the lowest four modes of oscillation for a vibrating string. 2. Be able to calculate and observe the two lowest harmonic frequencies in a vibrating bar. 3. Become vaguely familiar with a Fast Fourier Transform (FFT) and see why it is usual for signal analysis. 4. Observe the effects of Harmonics. III. Needed Equipment Your instructor will show you the experimental setup, which consists of: For experiment 1: A signal generator to create the sound wave, a speak to produce the vibrations, a piece of Guitar string, a pulley, and a hanging mass as shown below. For experiment 2: A microphone plugged into Lab Pro, an aluminum rod, and set up your laptop with Logger Pro so that we can graph the cosine wave produced in the rod. Page 2 of 6

3 IV. Turn in your Pre lab/homework problem if assigned. V. Discussion In order to support wave motion, a string must be fixed at both ends, so the ends of the string must be nodes of any wave that stands on the string. This means that the only wavelengths that can stand on the string are given by n = (2L)/n, where n = 1, 2, 3... but not zero. Since the wave speed is the same for all wavelengths, the harmonic frequencies can be found. A rod is rigid, so it need not have its ends fixed. However, even a rod has to be held somewhere, and any place it is held will be a node. If you hold a rod at its midpoint, that will be a node, and if you strike one end of the rod to make it vibrate, both ends will be anti nodes, and the length of the rod will be half a wavelength. If you hold the rod L/4 from one end, that will be a node. The ends will be anti nodes, so L/4 = /4, and the second harmonic has wavelength same as the length of the rod. Can you see that for a rod, the harmonic wavelengths will be given by the same pattern as for a string, that is, by n = (2L)/n? VI. Procedure A. Experiment I: Standing Waves on Guitar Strings: A.1. We will predict and measure the frequencies for the lowest four modes of oscillation for a vibrating string. We will do this four times, changing exactly one parameter each time. Connect the Pasco signal generator to the speaker. Connect the black terminal (GND) to one terminal on the speaker, and the red terminal (Lo ) on the generator to the other terminal on the speaker. Make sure the function is set for SINE waves. A.2. Preliminary Measurements: The value of g, and the diameters and linear mass densities for two guitar strings are given in the spreadsheet template. A.3. Start with the E string. Adjust the length so that it is between 0.8 and 0.9 meters long, and measure that length carefully. Hang a 200 gram weight on the end of the string. Calculate the tension in the string and the wave speed, and then predict the lowest four resonant frequencies. Page 3 of 6

4 Turn on the signal generator, and turn the frequency down to less than 20 Hz. Slowly increase the frequency until you reach the first resonant frequency. Record that frequency and its uncertainty. Continue increasing the frequency until you have found and recorded the first four resonant frequencies and their uncertainties. Do the measured frequencies agree with the predicted values? A.4. Replace the 200 gram weight with a 500 gram weight. Calculate the new wave speed and predict the new frequencies, and then measure the new frequencies. Is there agreement? A.5. Replace the E string with a G string, which is heavier. Calculate the new wave speed and predict the new frequencies, and then measure the new frequencies. Is there agreement? A.6. Finally, shorten the length of the string by moving the speaker. Measure the new length carefully, and record it. Predict and measure the new frequencies. Is there agreement? A.7. Conclusion: In a short paragraph, discuss agreement between your predictions and your measurements. What is the effect of changing (a) the tension, (b) the linear density, and, (c) the length of the string? B. Experiment II: Standing Waves on an Aluminum Rod B.1. Preliminary Data: We don't know what specific aluminum alloy our rods are made of. We can measure its density, but we have no way to measure its bulk modulus in our lab. The density and bulk modulus for a common aluminum alloy named 6061 T6 are given in the spreadsheet template, but we have no assurance this is the alloy used for our rods. You'll have to measure the length of your own rod. Also measure its diameter and mass, and calculate its density. Is this density consistent with alloy 6061 T6? Whatever the answer, let's assume that 6061 T6 specifications apply to our rods. B.2. Predictions: Calculate the wave speed for sound in 6061 T6, and then predict the two lowest harmonic frequencies. B.3. Measurement of Harmonic Frequencies: Connect microphone to LabPro, and start Logger Pro. Make sure the microphone shows up as a sensor, and set the data collection to 0.02 s and 50,000 samples/second. It is not necessary to zero the microphone. Page 4 of 6

5 We will use equipment to carry out a Fast Fourier Transform (FFT) on the data. That resolves a time varying signal into its constituent frequencies. To understand this, recall that mathematical functions can be approximated by a sum of terms called a series. A periodic (repeating) function can be represented by a sum of sine and cosine waves. This is called a Fourier series. FFTs use some linear algebra to calculate the amplitude and frequency of the sine and cosine terms that make up this series. These are called the constituent frequencies. When the time varying signal is almost exactly sinusoidal, the FFT simply calculates the dominant frequency for you. From the menu bar at the top of your Logger Pro screen, choose "Insert" > "Additional Graphs" > "FFT". Resize Sound Pressure graph so both graphs fit. B.3.a. First Harmonic: Hold the rod by its midpoint, and strike it on the floor. Hold the end of the rod near the mike, and, after the higher harmonics die out, press "collect" and acquire Graph #1. Annotate your graph, and find the frequency both by finding the time for a large number of periods, and from the FFT. B.3.b. Second Harmonic: Try the experiment again, this time, holding the rod 1/4 of its length form the end. Call this data Graph #2, and analyze it the same way you analyzed Graph #1. B.4. Conclusion: By now, you should know what to write in your conclusion. Page 5 of 6

6 VII. Lab Report to Hand In: A. Spreadsheet from part A; don t forget uncertainties and conclusion. B. Spreadsheet from part B; don t forget uncertainties and conclusion. C. Graph #1 and #2 from parts B.3.a and B.3.b, in support of your calculations. VIII. Clean Up A. End of Lab Checkout: Before leaving the laboratory, please tidy up the equipment at the workstation and quit all running software. B. The lab station should be in better condition than when you arrived and more importantly, should be of an appearance that you would be PROUD to show to your legal guardians during a Parents Weekend. C. Have your instructor inspect your lab station and receive their permission to leave the Lab Room. D. You SHALL follow this procedure doing every lab for BOTH SP211 and SP212! Many thanks to Dr. Huddle for his assistance in producing this Laboratory procedure; specific references can be supplied on request. LCDR Timothy Shivok Page 6 of 6