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resonance lab

resonance lab - Labs IIIa and IIIb Physics of Music and...

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Labs IIIa and IIIb - Physics of Music and Color Kelsey Schur October 11, 2007 Introduction In the previous lab, the class investigated resonance by varying the frequency effecting a string with a fixed tension. However, in this lab, the frequency effecting the string or air column is fixed, and the tension or length of the air column will be adjusted to find a resonant setting. The string will be the subject of Lab IIIa, and the air column will be experimented with in Lab IIIb. For Lab IIIa, it is important to know the equations that describe the relationships between the frequency, velocity, tension, string length, and mass density of the string in the given situation. First, we remember that the frequency for a given mode of a string is determined by: f = v/2 l for the fundamental f = v/ l for the first overtone f = 3v/2 l for the second overtone, and so on...: where f is frequency, v is velocity, and l is length Students must also recall the equation for wave velocity: v = √( T/m) where v is velocity, T is tension, and m is mass density For Lab IIIb, students should know the equations which determine the length of a closed pipe that will resonate at a given frequency. l = v/4f T for the fundamental l 2 = 3v/4f T for the first overtone, where l is the length of the pipe, v is the wave velocity, and f is the frequency. It should be noted that closed pipes only have frequencies at odd harmonics. Results Part IIIa Procedure 1.
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Mode Number Tension (grams) Ratios 1 T 1 = 5400 g T 1 /T 1 = 1 2 T 2 = 1750 g T 1 /T 2 = 108/35 = 3.086 3 T 3 = 650 g T 1 /T 3 = 108/13 = 8.308 4 T 4 = 350 g T 1 /T 4 = 108/7 = 15.429 5 T 5 = 200 g T 1 /T 5 = 27 3. See attached: Figure 2. Part IIIb Analysis Frequency of Tuning Fork Length of Air Column Overtone Excited Sound Velocity A 426.7 17.5 cm fundamental 298.69 m/s A 426.7 57.5 cm first 327.1367 m/s C 523.3 14 cm fundamental 293.048 m/s C 523.3 47 cm first 327.92467 m/s C 523.3 72 cm second 301.4208 m/s C 1024 6 cm fundamental 314.0267 m/s C 1024 23 cm first 245.76 m/s -Length vs. frequency and overtone data can be found on the attached bar graph. Discussion Lab IIIa 4 Questions and Problems In this lab, we will be adjusting the tension of a string to cause it to resonate with a tuning fork. Only certain tensions will produce resonances because they must fit into one of the mathematical relations given in the introduction. The harmonic that leads to the resonance will change with the tension because different tensions will fulfill the relations for different harmonics. The equations for frequency and wave velocity given in the introduction can be combined and rearranged to create an
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equation that will allow us to calculate the right tension to match a given frequency at a particular harmonic. The given equations, once again, are: f = v/2 l and v =√( T / m) Because v =√( T / m) , it can be substituted into the equation for frequency: f T = √( T / m)/ 2 l Now, solve the equation for T . Remember to square both sides.
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