Tensions of Guitar Strings

Transcription

1 1 ensions of Guitar Strings Darl Achilles 1/1/00 Phsics 398 EMI

2 Introduction he object of this eperiment was to determine the tensions of various tpes of guitar strings when tuned to the proper pitch. It was not necessar to actuall place the strings on the guitar to make measurements. he tensions could be found from simple measurements of the mass and the length. he tension of a string could prove to be a ver hand thing to know for future eperiments. heor* raveling Wave Analsis Guitar strings vibrate as standing waves, which are a result of a harmonic wave reflecting off of fied ends. hese standing waves can be epressed as two traveling waves, traveling in opposite directions:, Asin π /? t / τ + Asin π /? + t / τ where is the wavelength, τ is the period, is the longitudinal displacement, t is the time, and is the transverse displacement. We will just look at the term that moves to the right, i.e. the first term, since the are the same wave moving in opposite directions. We can rewrite the traveling wave as π, Asin τ We also know that a wave that moves to the right can be epressed as π, Asin V herefore, we see that V f Eqn. 1 τ Wave Equation On the other hand, we can analze the motion of a string using the wave equation. Look at a segment of string that spans the region from to + Fig.1.

3 3 Fig.1 he force in the direction is ' sin sin F + θ θ Using the definition of a double derivative + it is easil shown that F Now, using Newton s nd law, we equate this force to the mass times the acceleration in the direction t ma F µ where µ is the mass per unit length. he wave equation states 1 t V herefore µ V Eqn. Solving for ension he velocit in the direction has now been solved via two different methods: a traveling wave analsis and the wave equation. If we equate the two epressions we have for the velocit µ f V and solve for the tension f V µ µ

4 4 he tension is now in terms of the mass per unit length, the frequenc and the wavelength. All three of these quantities are easil solved for. he mass per unit length is the onl variable that needs to be determined eperimentall. It is a straightforward measurement of the mass of a certain length of string divided b the length. he frequenc is determined b which of the si strings on the guitar is being analzed. he frequenc for each guitar string is shown in figure. String Low E A D G B High E Frequenc Hz Fig. he wavelength of the fundamental lowest mode of vibration of a standing wave on a guitar string is determined b the so-called scale-length of the guitar, L S. he wavelength of the fundamental is twice the length of the guitar, i.e. L S as shown in Fig. 3. For our eperiment a guitar with a scale length of L S 5 ½ inches was used. herefore, L S 51.0 inches meters. Eperimental Procedure Four different tpes of strings were chosen for this eperiment: Fender Nickel Plated Super 50 Regulars, Fender Nickel Plated Super 50 Lights, D Addario Nickel Wound Regular Light Gauge EXL110 s, and D Addario Nickel Wound Super Light EXL10 s. Both ends of all the strings used were cut off to ensure the strings homogeneit. For each set of si strings, the length was measured five times with a standard meter stick, the diameter was measured ten times with a digital caliper accurate to a tenth of a mil or an analog caliper that was less precise, and the mass was measured with a scale accurate to a thousandth of a gram. Vinl gloves were worn the entire time to ensure no bod oils came in contact with the nickel guitar strings. his not onl aided in accurate mass measurements, but also will etend the lifetime of the strings. Bod oils will make the metal strings deteriorate more quickl than uncontaminated strings.

5 5 Eperimental Results he results of the measurements are as follows: Fender Super 50L s String Hi E ~.009 B ~.011 G ~.016 D ~.04 A ~.03 Low E ~.04 MASS Kg 3.3E E E E E E-03 DENSIYKg/m 3.09E E E E E E-03 DENSIYKg/Vol 7.97E E E E E E+03 FREQUENCYHz ENSION N Fender Super 50R s String Hi E ~.010 B ~.013 G ~.017 D ~.06 A ~.036 Low E ~.046 MASS 3.3E E E E E E-03 DENSIYKg/m 3.09E E E E E E-03 DENSIYVol 7.97E E E E E E+03 FREQUENCY ENSION D Addario EXL 10 s String Hi E ~.009 B ~.011 G ~.016 D ~.04 A ~.03 Low E ~.04 MASS 3.6E E E E-03 4.E E-03 DENSIYKg/m 3.9E E E E E E-03 DENSIYVol 7.94E E E E E E+03 FREQUENCY ENSION D Addario EXL 110 s String Hi E ~.010 B ~.013 G ~.017 D ~.06 A ~.036 Low E ~.046 MASS 3.85E E E-03.19E E E-03 DENSIYKg/m 3.89E E E-03.3E E E-03 DENSIYVol 7.69E E+03 8.E E E E+03 FREQUENCY ENSION he following is a graph showing the si tensions for each set of guitar strings. It makes it easier to see the correlations that eist.

6 6 ensions of All Strings ension D'Addario 110's D'Addario 10's Fender 50L's Fendre 50R's 0.00 StringHigh to Low he average value for each string was: Fig. 4 String High E B G D A Low E AVE Average ension 100 ension N String High to Low Fig. 5

7 7 Conclusions As is seen in Figure 5, the average tension for each string is essentiall between 60 and 80 Newtons. However, more specificall, one can see a correlation between two sets of strings in Figure 4. he D Addario 110 s seem to match up well with the Fender 50R s and the D Addario 10 s match up with the Fender 50L s. his makes sense because each of those pairs has the same diameter as its partner. An interesting side note is that the A string of the D Addario EXL 10 s was said to have a diameter of 3 mils. his was quite wrong, and the diameter of the A string was actuall ~36 mils. Knowing the tensions of the strings on a guitar is ver useful. hese figures can be used for man other different eperiments. One ke problem that would be interesting to investigate would be the detuning of the guitar as the plaer frets higher and higher notes. It is a well known fact that as the strings are fretted more, the become out of tune with each other, while when plaed open, the are in tune. Guitar manufacturers attempted to fi this ailment b having the length of the strings be adjustable near the pickup. However, this onl partiall solves the problem. Perhaps a solution to the problem would be to find a set of strings that have equal tension. his ma improve the intonation between strings. he tension of the strings could also be used in eperiments studing the damping of strings or in eperiments studing how the string deforms when it has been stretched across the guitar for some period of time. he tension is a quantit that is in general a ver useful thing to know, and can be applied to man other topics. his eperiment is just the foundation for what could be man ver interesting studies. *Acknowledgements I would like to thank Prof. Steve Errede for providing the instruction and class notes that lead to a thorough understanding of the theor behind the vibration of strings.