We considered whether constructing an unknown metal into a tuning fork of known dimensions and identifying its fundamental frequency is accurate enough to find what the metal is

Hopkinson, Alfie

Logan, Matthew

Page, Archie

Peacock, Billy

Weston, Jack

English

University of Leicester Open Journals

Journal of Physics Special TopicsAn undergraduate physics journalP5 8 Listening to MetalB. Peacock, A. Hopkins, J. Weston, M. Logan and A. PageDepartment of Physics and Astronomy, University of Leicester, Leicester, LE1 7RHDecember 5, 2019AbstractWe considered whether constructing an unknown metal into a tuning fork of known dimensionsand identifying its fundamental frequency is accurate enough to find what the metal is. Metalshave specific modulus with differences of± 105 m2 s-2, and the tuning for gives the specific modulusto an accuracy of ± 8X104 m2 s-2 so metals can be told apart.IntroductionThe frequency of a tuning fork depends on itsdimensions and the material the tuning fork ismade from. Each type of metal would thereforeproduce its own unique sound which could beused to identify it, which is the reason why dif-ferent materials are used to make guitar strings[4]. We consider how accurately we can deter-mine the frequency of a tuning fork made of anunknown metal with known dimensions and howthis can be used to determine the material.Theory and ResultsA tuning fork is an acoustic resonator used toproduce a fixed tone. Unlike other types of res-onators, such as a plucked wire, the tuning forkhas most of its vibrational energy at the fun-damental frequency, with little energy going tothe overtone frequencies. The tuning fork’s over-tones die out fast, leaving a pure sine wave at thefundamental frequency. By comparison, the firstovertone of a vibrating wire is twice the funda-mental, so when the wire is plucked its vibrationstend to mix the fundamental and overtone fre-quencies.the frequency produced by a tuning fork withFigure 1: A tuning fork’s vibrational modes [2].cylindrical prongs of known material and dimen-sions can be found from the following:f =1.8752r4pi`2√Eρ(1)Where f is the frequency the fork vi-brates, 1.875 is the smallest positive solution ofcos(x)cosh(x) = 1 [3], ` is the length of the prongsin m, E is the Young’s modulus of for material,ρ is the density of the fork material, and r is theradius of the circular cross-section of the cylin-drical prongs. Re-arranging equation 1 for theratio Eρ , we find:Eρ=16pi2`41.8754r2f2 (2)This ratio is known as the specific modulus,and can be seen in the graph below:Figure 2: The density of different materials against itsYoung’s modulus [1].We will take our known dimensions arbitrarily:` = 0.15 ± 5x10-5 m, and r = 0.005 ± 5x10-5 m,where the error is that from a standard caliperin the construction of the fork [7]. Assumingthe frequency measurements are taken at 20 oCtemperature, it can be found that the error inthe frequency will be about 1/10000 th of themeasurement for an average frequency counter[6]. We can Propagate these errors using thefollowing error propagation equations:If: Z = Xn then:∆ZZ= n∆XX(3)If: Z = X/Y or Z = XYthen:∆ZZ=√(∆XX)2 + (∆YY)2(4)Equation 3 can be used to propagate theknown errors to the powers that are on the vari-ables in the equation. Equation 4 is then usedto combine these errors propagated to the pow-ers to find the error of the specific modulus overa given specific modulus to be 8.0069x10-3. thespecific modulus for a given metal will be of themagnitude 107 m2 s-2 [5], giving the error to theorder of ± 8X104 m2 s-2. This is small enoughthat metals can be easily differentiated, as theyare roughly ± 105 m2 s-2 difference between eachmetal [5].DiscussionAt different temperatures the tuning forks willhave slightly different values of frequency, as thematerial will have a different specific modulus fordifferent temperatures, so there will be a devia-tion of 86 ppm per oC from the standard tem-perature of 20oC.ConclusionWe found by making an unknown metal intoa tuning fork of known dimensions we could findwhat type of metal it was from the frequency ofthe tuning fork.References[1] https://ocw.tudelft.nl/wp-content/uploads/Materiaalkunde_1_slides_chapter4.pdf[Accessed 20 October 2019][2] https://physics.stackexchange.com/questions/51838/why-do-tuning-forks-have-two-prongs/51849. [Accessed 20 October 2019][3] http://emweb.unl.edu/Mechanics-Pages/Scott-Whitney/325hweb/Beams.htm. [Accessed 20 Oc-tober 2019][4] https://www.gearank.com/articles/guitar-string-types [Accessed 20 Octo-ber 2019][5] https://www.engineeringtoolbox.com/engineering-materials-properties-d_1225.html [Accessed 20 October 2019][6] https://www.picotech.com/support/viewtopic.php?t=35211 [Accessed 20October 2019][7] http://www.physics.smu.edu/~scalise/apparatus/caliper/ [Accessed 20 October2019]

P5_8 Listening to Metal

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P5_8 Listening to Metal

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