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Lab 5A: Mersenne's Laws & Melde's Experiment

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CSUEB Physics 1780 Lab 5a: Mersenne & Melde Page 1 Lab 5a: Mersenne’s Laws & Melde's experiment Introduction Vincenzo Galilei (1520 –1591), the father of Galileo Galilei, was an Italian lutenist, composer, and music theorist. It is reported that he showed that the pitch of a vibrating string was proportional to the square root of its tension. In other words, in order to increase the pitch by an octave (factor of 2 in frequency) it would require the tension to be quadrupled. Marin Mersenne (1588-1648, see picture), often called the “father of acoustics” , around 1630 published a book summarizing the properties of sound, based on the earlier work of Vincenzo and Galileo. He established that: • Frequency is inversely proportional to length of string • Frequency is inversely proportional to the diameter of the string • Frequency is proportional to the root of the tension • Frequency is inversely proportional to the root of the mass (density) of the string. So in summary, if you want a low note, use a fat (massive) long string under low tension. For a high pitch, use a thin short string under high tension. We can summarize all of his results with a single equation showing that the frequency “f” is proportional to: 1 F f ∝ , (1) L µ where “L” is the length of the violin string, “F” is the tension (in Newtons of force) and µ (“mu”) is the mass density of the string (in units of mass per unit length, e.g. kg per meter). In this experiment the string is assumed to be vibrating in its “fundamental” mode, like this, (figure 1). In other words, the wavelength of the vibration is: λ1=2L. It is said that Pythagoras found that if you put your finger midway on the string, the string would sing an octave higher (i.e. double the frequency). In this case we have, (figure 2) and now the wavelength of the vibration is λ2=L. Further, if you put the finger 1/3 of the way along the string it would vibrate, Version 4.5 © 2013July24 all rights reserved Dr. Bill Pezzaglia CSUEB Physics 1780 Lab 5a: Mersenne & Melde Page 2 (figure 3) where the wavelength of the vibration is: λ1=(2/3)L . Detailed study of the modes of vibration of a string and the relationship to frequency was done (I think) by Joseph Sauveur (1653-1713), who around 1730 suggested the term " acoustics " from the greek word for "sound" [see also Jean-Phillipe Rameau]. He coined the term “ node ” for the part of the string that does not seem to move, and “ fundamental ” for the lowest frequency-mode vibration of a string. He also introduced the term “ harmonic ”. Hence, figure 1 (the fundamental) which has two nodes (and one antinode ) is called the “ first harmonic ”. Figure 2 (3 nodes, and 2 antinodes) is the 2nd harmonic . Figure 3 (4 nodes and 3 antinodes) is the 3rd harmonic . Hence the number of antinodes is the “number” of the harmonic. 2L The wavelengths of the “nth” harmonic is hence: λ = , where “n” is the number of n n antinodes (or the number of nodes minus 1). WaveSpeed : From observed data, it was found that the wavespeed: v=f λ. was the same for all the harmonics on a string (assuming we don’t change the tension of the string). Putting Sauveur’s ideas together with Mersenne’s formula one can show that the speed of the wave on a string must be proportional to the root of the ratio of the tension divided by mass density. Indeed this is the case (validated by other mathematicians describing the elastic nature of a string), F v = . (2) µ We are essentially going to check this relation by using Franz Melde’s classic experiment. He used an electric vibrator connected to a string which goes over a pulley (creates a node here) with a mass suspended at the end. Hence the tension in the string is just F=mg (where m is the mass, g is the acceleration of gravity). He was able to get “ standing waves ” (a term he coined) which appear to stand still, if the tension is just right. Version 4.5 © 2013July24 all rights reserved Dr. Bill Pezzaglia CSUEB Physics 1780 Lab 5a: Mersenne & Melde Page 3 EXPERIMENT: We will hold the length fixed (approximately L=0.70 meters), and vary either the frequency or the tension (hence the wavespeed) to create the different standing nodes. By measuring the wavelength (and knowing the frequency) we can determine the wavespeed. Hence we can show if the formula (2) is valid by plotting Tension Force “F” vs the squared wavespeed “v” and seeing if it comes out to be a line. The slope of the line will give the mass density of the string. A. Setup 1. Physical setup : Make the distance from the vibrating reed to the pulley approximately L=0.70 meters. Vibrator Metal Reed Pulley Hanger Extra Masses Meter Stick Interface Box 2. Computer Set-up The vibrator is already connected to the output of the interface box. Call up DataStudio and click on Create Experiment. At the bottom of the Sensors panel you will find Signal Output . Click on Output . A window titled Signal Generator will pop up like the one below. Version 4.5 © 2013July24 all rights reserved Dr. Bill Pezzaglia CSUEB Physics 1780 Lab 5a: Mersenne & Melde Page 4 The signal generator controls the frequency and amplitude of the vibrating metal reed. Click the Auto button. This activates the On/Off buttons. So you can turn the vibrator on and off. Under Frequency the +- buttons will change the frequency in steps of 100 Hz (this is the default). The arrow buttons ◄► will change the step size. Possible step sizes are 1000 Hz, 100 Hz, 10 Hz, 1 Hz, 0.1 Hz. The Amplitude of the reed can be adjusted in a similar manner. B. Variable Mode Number Here we keep the length "L" fixed, and keep tension fixed (use approximately 200 grams? More?), but vary the frequency to excite the various modes of oscillation. 1. Hang 0.200 kg on the end of the string, including the 0.050 kg mass hanger. 2. Determine the frequencies of modes 1 through at least 5. 3. For each mode, measure the wavelength λ 4. Calculate the wavespeed: v=f λ for each mode Question 1. (a) What is your fundamental frequency? (b) We expect that the frequency of the “n-th” mode is “n” times the fundamental. Is this true? (c) The wavespeeds should all be the same. Are they? Question 2. (a) What is the tension in your string? [Hint: F=mg] (b) From your tension and average wavespeed calculate your measured mass density from equation (2). Question 3. Using a sample of string, measure its mass and length (be sure to record your values!), and determine the mass density. Compare to your result in question 2. Question 4. [extra credit] Plot frequency vs mode number (Plot #1). The slope will give you a better determination of the fundamental frequency. Plot 1: Determination of Fudamental Frequency (Variable Modes) 300 y = 56.6x - 0.2 250 R2 = 0.9999 200 150 100 Frequency, Hz 50 0 0 1 2 3 4 5 6 Mode Number Version 4.5 © 2013July24 all rights reserved Dr. Bill Pezzaglia CSUEB Physics 1780 Lab 5a: Mersenne & Melde Page 5 C. Variable Tension [if time permits] Hold length "L" fixed. For this experiment you will change the tension on the string, and find the resonant frequency for the n=2 mode. Take at least 5 data points for the mass ranging from 50 grams (0.05 kg) to 1 kg. 1. Data table should include: • Mass • Force=mg • Measured resonant frequency "f" (for mode n=2) • Wavelength (should stay constant if you don't vary "L") • Wavespeed: v=(frequency)x(wavelength) • squared wavespeed 2. Reduction : PLOT tension force vs squared wavespeed (Plot #2). Graph should be a line, with slope giving the mass. • Calculate tension and wave velocity • Plot tension F vs square speed v2 • Draw the best line (if line looks weird, ask instructor to look at it) • Get the slope of the line. • A value of R 2 of 0.99 means you did well. If R 2<0.9 probably you have bad measurements. SUMMARY (essay questions based on Plot #2) Question 5. We expect that the square of the velocity will be proportional to the tension. Does your graph #2 validate this? [Hint: do the points fall on a line?] Question 6. From the formulas, the expected slope of the line will be the mass density µ (in kg per meter). What is the slope of your graph? Question 7. Compare your “measured” mass density with the value the lab tech told − me: µ= 3.0 × 10 4 kg/m. Are you close? Version 4.5 © 2013July24 all rights reserved Dr. Bill Pezzaglia CSUEB Physics 1780 Lab 5a: Mersenne & Melde Page 6 Sample Plot #2 Note the slope here implies the mass density is 0.0032 kg/meter The title shoud include the frequency used and the string length. Version 4.5 © 2013July24 all rights reserved Dr. Bill Pezzaglia .
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