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Standing Waves on a String PURPOSE San Diego Miramar College Physics 197 NAME: _______________________________________ DATE: _________________________ LABORATORY PARTNERS : ______________________________________________________ Standing Waves on a String PURPOSE: To study the relationship among stretching force (FT), wavelength (흺), vibrational frequency (f), linear mass density (µ), and wave velocity (v) in a vibrating string; to observe standing waves and the harmonic frequencies of a stretched string. THEORY: Standing waves can be produced when two waves of identical wavelength, velocity, and amplitude are traveling in opposite directions through the same medium. Standing waves can be established using a stretched string to create a train of waves, set up by a vibrating body, and reflected at the end of the string. Newly generated waves will interfere with the old reflected waves. If the conditions are right, then a standing wave pattern will be created. A Stretched string has many modes of vibration, i.e. standing waves. It may vibrate as a single segment; its length is then equal to one half of the wavelength of the vibrations produced. It may also vibrate in two segments, with a node at each end in the middle; the wavelength produced is then equal to the length of the string. It may also vibrate in a larger number of segments. In every case, the length of the string is some integer multiple of half wavelengths. So, if a string is stretched between two fixed points, the ends are constrained not to move; hence, these are the nodes. In a standing wave the nodes, or the points that do not vibrate, occur every half the wavelength; thus, the ends of the string must correspond to nodes and the whole length of the string must accommodate an integer nymber of half wavelengths. Roughly midway between the nodes are antinodes, which is where the standing wave has maximum amplitude. Remember that a wave traveling along a string will have a velocity given by Where FT is the tension in the string and µ is the linesar density of the string. If the wave velocity is “right”, then the standing wave will have a wave velocity of Thus the fundamental frequency for a particular string (where FT and µ are constant) would be Notice that the length of the string is L and that a standing wave of the fundamental frequency is ½ 흺, so 흺 = 2L. The other allowable frequencies (that generate standing waves) would be an integer multiples of this fundamental frequency. This can be written as It is fairly easy to measure the vibrating length of the string L and, hence, determine 흺. It is also possible to measure the “taut: length of the string and its mass to determine the linear mass density, µ. Thus we can solve for the experimental value of tension as APPARATUS: PASCO SF -9324 Mechanical vibrator, 2 PASCO universal Table Clamp, support rod, pulley, two types of string, electronic balance, meter stick, 500 gram hooked mass or 200 gram hooked mass, 2 jumper cables, PASCO PI -9587B digital function generator. PROCEDURE: Set up the equipment as illustrated in the figure below ( but do not attach the string to the resonator yet). 1. First take a piece of string about 2.5 meter long, mass it and measure its length when “taut”. Using the measured value, calculate the linear density of the string. Also record the mass and length in the following table. 2. Tie one end of the string to the bar, pass it over the vibrator, then ovr the pulley and attach it to the 500g mass. Be sure the string is paralled to the table top. Measure the length of the string from the resonator to the pulley and record. 3. Turn on the power supply and start the frequency at 10 Hz. Increase the frequency until you establish a 1 segment (2 nodes) standing wave pattern. To adjust the frequency, just turn the knob. If you believe you have one segment but might not have maximum amplitude, you might have to move the pulley/resonator backward or forward from it’s clamp for “tuning” or adjust the frequency until you get maximum amplitude. Record your frequency value in the table. 4. Measure and record the segment’s length between the resonator and the pulley. This can be done by holding a 2-m stick to measure the distance from one node to the next consecutive node. Remember that one segment of a standing wave pattern is 1/2 흺. Calculate and record the wavelength (by doubling the length of one segment). Also calculate the experimental tension, and the percent difference of that tension with the theoretical tesional value. a. To get a standing wave that has 2 segments, should you increase or decrease the frequency? b. By how much should you change the frequency? String1: Mass of the string______________ Total length of the string ______________ Linear density of the string _______________________ Tension (theoretical) ________________________ Segment Frequency(Hz) Wavelength(m) Tension(N) %difference 1 2 3 4 5 6 7 8 Calculation of experimental tension: Calculation of % difference of tension: 5. After finishing this measurement, repeat the process for 2, 3, 4, 5, 6, 7 and 8 segment standing waves by adjusting the frequency. Besides recording the frequency and measuring the length of one segment for each standing wave, calculate and record the experimental tension and % difference of tension. a. Calculate the wave velocity in the string when you have a standing wave pattern of 3 segment. b. For 7 segment. c. What was the frequency value for the fundamental frequency? 6. Now repeat this process for the second string. Is the Linear density larger or smaller? Do you expect the string to have a higher fundamental frequency or less? String2: Mass of the string______________ Total length of the string ______________ Linear density of the string _______________________ Tension (theoretical) ________________________ Segment Frequency(Hz) Wavelength(m) Tension(N) %difference 1 2 3 4 5 6 7 8 NAME: _______________________________________ DATE: _________________________ LABORATORY PARTNERS : ______________________________________________________ Standing Waves Resonance OBJECTIVE: We are going to examine the relationship between wave speed, frequency, and wavelength using standing waves. THEORY: When two waves traveling in opposite directions produce a standing wave. We are studying standing sound waves in an air column. The length of the pipe determines the wavelength and the frequency of the various harmonics. For the first harmonic, speed of sound can be found by v = f흺 and L, the length of the tube is odd multiple of 흺/4. EQUIPMENT: spatial standing tubes, meter stick, tuning forks PROCEDURE: Examine the tube and draw the standing wave for the tube for harmonics up to at least the third harmonic. Describe how you determine the speed of the sound in air using the tube. Follow the procedure you have described above using at least three tuning forks and for each tuning fork repeat the process three times, for at least nine entries. Observer Frequency Harmonic Resonance Wavelength Wave speed Number length Calculate the average wave speed from your data. Calculate percent error from the theoretical speed of sound at 25° C of 346 m/s. .
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