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Standing Waves 11-28-2005 8:54 AM SP 211 Lab - Standing Waves 11-28-2005 8:54 AM Use Internet Explorer for this laboratory. Save your work often. NADN ID: guest49 Section Number: guest All Team Members: Your Name: SP 211 Lab - Standing Waves Version: November 17, 2003 Standing Waves 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 the guitar. However, 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. Further, as you know, a standing wave in one dimension is actually composed of two traveling waves. Consequently, we will also make measurements that will enable us to calculate the speed of the traveling waves on the 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 a Play Video Clips of Bridge: luminum. Oscillations and Collapse (Long Version) Oscillations Only (Short Version) Equipment for Experiments 1-4 A schematic of the equipment is shown below. The important feature is the guitar string that is attached to a board at the left end. Close to that end, the string is in contact with a block of wood that is glued to the center of a speaker. Since the amplitude of the motion of the block of wood is very small, the string has a node very close to the block of wood. At the right, the string is passed over a pulley and the tension is controlled by a mass hanging from the right end of the string. Since the pulley is at rest, there is a node in the str ing where it is in contact with http://intranet.usna.edu/cgi-bin/labform_sp211 Page 1 of 11 SP 211 Lab - Standing Waves 11-28-2005 8:54 AM the pulley. The Pasco PI 9587C Digital Function Generator-Amplifier controls the speaker that makes the wood undergo simple harmonic motion. Since the wood is in contact with the guitar string, the wood drives the guitar string sinusoidally. At resonance (when the frequency of the speaker matches a natural frequency of the string), the amplitude of the standing wave on the string should be large enough for us to see and make measurements on it. Theory (Standing Waves) The general theory of standing waves is given in section 15-9 of the textbook and is given for a string anchored at both ends. (There are nodes at both ends.) As you know, the string has natural patterns of vibration (normal modes) that have natural frequencies given by (n = 1, 2, 3, ...) (1) L is the total length of the string associated with the standing wave i.e. L is the distance between the nodes near the ends of the string. This also applies to the aluminum rod. However, for the aluminum rod, there are antinodes at the ends since the ends are “free” to vibrate. Also, for the aluminum rod, there will be a node wherever we hold the rod. While eq. (1) looks a bit cluttered, it is just an application of the basic equation for traveling sinusoidal waves, v = fλ. (This may be obvious from the first equality in eq. (1)). Of course, v is neither the "speed of standing waves" nor the speed of sound in air. Rather, v is the speed of the constituent traveling waves i.e. the speed of waves along the string or the aluminum rod. The second equality in eq. (1) specifies the allowed wavelengths: Text eq.(18.6) (n = 1, 2, 3, ...) (2) One way to interpret eq. (2) is that the wavelength associated with a standing wave is twice the distance between adjacent nodes. This may be clear from the sketch at the right which is meant to represent a string vibrating with a value of n = 3. (Note that n is the number of antinodes.) The thicker line shows the associated wavelength. Since we can measure L, we can predict what the wavelengths should be for our standing waves. Further, as is apparent from eq. (1), if we know the speed of traveling waves, we can predict what the frequencies of the standing waves should be. We learned how to do that in section 16.3 of the textbook where it is shown that the speed of traveling waves on a string is: Text eq. (16.18) (3) T is the magnitude of the tension in the string and µ is the mass per length. Let us begin. Experiment 1: Predicting and Measuring the Natural Frequencies for a Guitar String Predicting the Natural Frequencies of Vibration 1. Locate the Black Diamond nylon guitar string 1 (E). It is the smaller diameter (0.711 mm) string. We have added the loops at the ends for convenience of use. For string 1, we have determined that µ = (4.15 ± 0.10)x10-4 kg/m. R1: Describe an experiment that could be carried out to determine the value of µ. Also, suggest a possible reason why you have not been asked to measure the value of µ. (Hint: Consider the mass of a short length of the guitar string.) http://intranet.usna.edu/cgi-bin/labform_sp211 Page 2 of 11 SP 211 Lab - Standing Waves 11-28-2005 8:54 AM 2. Be sure that the speaker is positioned as far from the pulley as possible. If it isn't, move the speaker so that it is. Hook one of the loops in the end of the string over the screw near the speaker, run the string over the pulley and hang a 200 g mass from the other end. Be sure that the string is in contact with the wood at the center of the speaker. If it isn't, tighten the screw or move the string to the back of the screw. 3. Calculate the magnitude of the tension in the string and record it in the space provided. R2: T = ± N R3: v = ± m/s 4. Use eq. (3) to calculate the speed of traveling waves on the string. Record the value in the space provided. 5. We need one other quantity before we can predict what the natural frequencies (frequencies of the standing waves) of the string should be. is needed is the length, L, which is the distance between the nodes at the ends of the string. (Note: We will let L be the distance from the top of the pulley to the wood at the middle of the speaker. This is based on the assumption that that there are nodes at the wood at the middle of the speaker and at the top of the pulley. Since the wood is moving slightly, it may be apparent that there is not actually a node there. However, it turns out that the node is very, very close to that point.) Measure L and record the value in the space provided. R4: L = ± m R5: Make sketches of the expected shape of the string for the various standing waves. According to instructions from your instructor, either enter the sketches into an Excel spreadsheet or draw them on a sheet of paper. Save your Responses 6. Use eq. (2) to predict wavelengths for the n = 1 to n = 3 normal modes. R6: Record the predicted wavelengths in the next table. Show a typical calculation. Typical Calculation: 7. Use eq. (1) or v = fλ to predict the values of the natural frequencies. R7: Enter the predicted frequencies in the appropriate column of the table. Measuring the Natural Frequencies of Vibration 1. On the frequency generator, turn the AMPLITUDE knob counterclockwise until it stops in order to start with the amplitude at a minimum. 2. Using the switch on the back, turn on the frequency generator. Adjust the frequency using the ADJUST knob and adjust the amplitude using the AMPLITUDE knob until you observe a standing wave. (Please do not turn the AMPLITUDE knob fully clockwise. There should be no need to increase the AMPLITUDE knob beyond about 1/3 of the way to maximum.) Once one standing wave is observed, it should be straightforward to find the others. 3. For each standing wave, adjust the frequency until the amplitude of the standing wave is a maximum. However, for a very interesting reason, it is important that the amplitude of the standing wave be small (less than about 1 cm). What happens if the amplitude of the standing wave is large is that the system becomes "nonlinear" and new phenomena, including chaos, become observable. However, all that is beyond the scope of this course. Consequently, keep the amplitude of the standing wave small via the AMPLITUDE knob and find the frequencies for which the amplitude of the standing wave is a maximum. R8: Record the measured natural frequencies in appropriate column of the table. http://intranet.usna.edu/cgi-bin/labform_sp211 Page 3 of 11 SP 211 Lab - Standing Waves 11-28-2005 8:54 AM n Predicted Predicted Measured Wavelength Frequency Frequency (m) (Hz) (Hz) 1 ± ± ± 2 ± ± ± 3 ± ± ± Save your Responses R9: Do the predicted and measured natural frequencies agree? Yes No Discuss.
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