NEW APPROACHES weights of those components, such as the tail itself weight of the tail of the fish will be eliminated. and the linkage members, which are also raised in Through this improvisation it also becomes possible the process and to the force required to to increase the number of elements beyond four and counterbalance the spring loading within the tail explore the eventual limits to the development of the which returns the device to its original state after machine. use. All of these factors give rise in practice to some Note. The Lazyfish is manufactured and distributed loss of efficiency. Further measurements could be by by Bacchanal Limited, Unit 15, Hale Trading made with the spring balance attached at H and K Estate, Lower Church Lane, Tipton, West Midlands in turn to investigate the changes in mechanical DY4 7PQ (tel: 0121 520 4727; fax: 0121 520 7637). advantage which result when the number of elements in the linkage is reduced. Received 22 April 1996, in final form 10 July 1996 If the commercial device is not available then the linkage, which is the essential component for teaching purposes, could be very easily improvised Reference from metal strips such as Meccano. In this case the Sandor B I 1983 Engineering Mechanics: Statics effects of the spring loading in the tail and the and Dynamics (Englewood Cliffs, NJ: Prentice-Hall)

Speed of in tuning fork metal V Anantha Narayanan Savannah State College, GA, USA and Radha Narayanan Student, Windsor Forest High School, Savannah, GA, USA

A procedure to find the in tuning Transverse motions of the prongs cause an up and fork metal is described. The formula needed is down motion in the stem of the tuning fork [1, 2]. extracted from the literature and explained. Since Tuning forks are commonly used in in the equipment needed for this project is readily experiments with air columns to determine the available in most high school and introductory speed of sound in air very accurately. The level college science laboratories, this exercise can of a tuning fork can be determined experimentally be done without any additional cost. by using a sonometer. Both these experiments are frequently performed in introductory level physics The details of the tuning fork construction formula courses in high school and/or college. An are extracted from the literature and presented here. elementary derivation of the tuning fork frequency Calculation procedures to evaluate the speed of formula using dimensional analysis is also given in a sound in the tuning fork material are outlined. book [3]. The mathematical treatment of transverse vibrations Theory in a straight rod or bar is very complicated [1, 4]. The vibrations of the tuning fork are controlled by The tuning fork is an example of a bar clamped at the elasticity and the inertia of the prongs of the one end. The symmetrical modes of a tuning fork are fork. The complex vibrations of the tuning fork treated as being due to two vibrating bars fixed at involve bending deformation of the prongs. their lower ends. For such a bar the vibration nodes

389 NEW APPROACHES for are not evenly spaced and therefore or the overtones are not integral multiples of the 2 2 fundamental tone. The mathematical solution and v ϭ f1L /0.162t ϭ 6.173f1L /t. (7) the essential details of the transverse vibrations of a bar clamped at one end are succinctly summarized in We can rewrite equation (7) as references [1, 4–9]. Such analysis leads to the v ϭ␭f (8) following formula for the frequency: 1 2 ␲ EK 1/2 2 2 2 2 2 where ␭ϭ6.173L2/t is the wavelength in m of the fn ϭᎏᎏ2 ᎏᎏ (1.194 , 2.988 , 5 , 7 ,... (2nϪ1) ). (1) 8L ΂ ␳ ΃ compression waves in the metal for the fundamental This expression is the result of applying the frequency f1. boundary conditions to the vibrations of a finite bar, When a tuning fork is gently struck the amplitude of thereby limiting the allowed modes to a discrete set the overtones is not entirely negligible. Thus, of . Here f ϭ frequency in Hz, n ϭ 1, 2, immediately after being struck, a distinct metallic 3, 4, . . ., E ϭ Young’s modulus of the material of the quality to the sound is produced and lasts for a short Ϫ bar in N m 2, ␳ϭdensity of the bar metal in kg time. The amplitudes of the overtones are much Ϫ m 3, L ϭ length of the bar in m, and K ϭ radius of smaller than the fundamental. The high-frequency gyration of the bar (of rectangular cross section with metallic sound rapidly dies out [1, 4, 7, 8]. The final t ϭ thickness of the bar in m in the direction of sustaining sound is a mellowed pure tone of the vibration) = t/121/2. fundamental mode. This is the frequency number Figure 1 gives a pictorial representation of t and L that is printed on laboratory tuning forks by the for a tuning fork. In equation (1) the numerical manufacturers. 2 2 terms (1.194) for f1 and (2.988) for f2 should be An experiment involving the formula of a tuning fork 2 used as given by detailed derivations in the original as f1 = constant ϫ t/L is given as a laboratory treatises dealing with the theory without rounding exercise in a manual [10]. However, it does not give them off [1, p 73]. They are numerically significant the details about the relation of this constant to v, ␭ and result from the trigonometric solutions involved and the moment of inertia or the connecting in the derivations. If we substitute v ϭ speed of references needed to appreciate a determination of Ϫ sound in the bar in m s 1 ϭ (E/␳)1/2, we can write v by experiment.

2 2 f1 ϭ␲vK ϫ 1.194 /8L (2)

2 2 f2 ϭ␲vK ϫ 2.988 /8L (3)

2 2 f3 ϭ␲vK ϫ 5 /8L (4)

2 2 f4 ϭ␲vK ϫ 7 /8L . (5)

Also f2 /f1 ϭ 6.26, f3 /f1 ϭ 17.54 and f4 /f1 ϭ 34.37. Hence the first four frequencies are f1, 6.26f1, 17.54f1 and 34.37f1. The extreme anharmonicity of the vibrations of the tuning fork is evident from these frequency values, in contrast to the integer multiples of f as frequencies in a vibrating string. We can rearrange equation (2) as

2 1/2 2 f1 ϭ (␲/8) (v/L ) (t/12 ) 1.194

ϭ 0.162vt/L2 ϭϭϭϭϭϭ (6) Figure 1. Tuning fork: representation of L and t.

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Experimental details The manufacturer of the tuning forks specifies the Besides a collection of tuning forks, the items right-hand side of equation (9), which is an error in needed to perform this experiment are a rule f1, to be Ϯ 0.5%. These frequencies as given by the and a vernier caliper. manufacturer seem quite reliable on the basis of our experiments involving these tuning forks in In table 1 are listed the f1 values of nine tuning forks resonance in air columns and the sonometer. We made of an aluminium alloy as printed by the estimate ⌬L to be no more than 0.001 m, and the manufacturer, and the measured values of t and L in term (2⌬L/L) ϫ 100% varies from 1.3% for fork #1 columns 3 and 4. All these tuning forks happen to to 1.9% for fork #9, giving an average of about have the same t values. Column 5 gives the values 1.6% error due to this term. The ⌬t in our of calculated wavelength. Column 6 gives the measurement is about 0.0002 m, so the error term values of v calculated from equation (8). (⌬t/t) ϫ 100% may amount to 1.4%. If we assume that all these errors add cumulatively, the Results percentage error in the extreme case for the The speed of sound in aluminium can be calculated calculated value of v can be 3.5%. 10 Ϫ2 Ϫ3 using E ϭ 7 ϫ 10 N m and ␳ϭ2700 kg m Another contributing factor can be due to the fact 1/2 2 and the formula v ϭ (E/␳) . We take 51 ϫ 10 m that the speed of sound in the aluminium alloy used Ϫ1 s as the table value of v. The average of the to construct the fork may not be exactly 51 ϫ 102 m experimental value of v from the last column in sϪ1. Pure aluminium is soft and lacks strength. It is 2 Ϫ1 table 1 is 49 ϫ 10 m s . The apparent alloyed with very small amounts of other elements percentage error in the value of v is about 4% like copper, magnesium and silicon. Three common from these data. commercial aluminium alloys are listed in the Taking logarithms of equation (3), then Handbook of Physics and Chemistry [11]. Using the differentiating and finally multiplying by 100 values of E and ␳ listed therein the v were calculated 2 Ϫ1 throughout, we get the percentage error to be 50.3, 50.8 and 51.3 ϫ 10 m s . Thus assuming 51 ϫ 102 m sϪ1 as the table value of v in ⌬v ⌬f 2⌬L ᎏᎏ ϫ 100% ϭ ᎏᎏ1 ϫ 100% ϩ ᎏᎏ ϫ 100% our calculations can introduce an additional v f1 L uncertainty in the error estimate in v of a little over Ϫ⌬t 1%. Overall, based on these analyses, our ϩ ᎏᎏ ϫ 100% . (9) t measurements seem to be giving satisfactory values for v. Table 1. List of tuning fork frequencies, t, L, ␭ and calculated values of v. ✝ Column 1 Column 2 Column 3 Column 4 Column 5 Column 6 2 # f1 tL␭ϭ 6.173 L /tvϭ␭f1 (Hz) (m) (m) (m) (m sϪ1) (corrected to a metre) 1 256 0.0072 0.151 19.55 5005 2 288 0.0072 0.140 16.80 4840 3 320 0.0073 0.134 15.18 4859 4 341.3 0.0072 0.128 14.05 4794 5 384 0.0072 0.123 12.97 4981 6 426.6 0.0073 0.117 11.58 4938 7 440 0.0073 0.113 10.80 4751 8 480 0.0071 0.109 10.33 4958 9 512 0.0073 0.106 9.50 4865 ✝ Average value of v ϭ 49.0 ϫ 102 m sϪ1.

391 NEW APPROACHES

engineering details that go into making such a simple piece of equipment as a tuning fork. Received 28 November 1995, in final form 17 June 1996

References [1] Kinsler L E, Frey A R, Coppens A B and Sanders JV 1982 Fundamentals of Acoustics 3rd edn (New York: John Wiley) pp 57–77 [2] Weidner R T and Sells R L 1973 Elementary Classical Physics vol. 1 (Boston, MA: Allyn and Bacon) pp 342–3 [3] Huntley H E 1967 Dimensional Analysis (New York: Dover) pp 67–8 [4] Morse P M 1986 Vibration and Sound (New York: American Institute of Physics for the Acoustical Society of America) pp 156–63 [5] Stephens R W B and Bate A E 1966 Acoustics and Vibrational Physics (London: Edward Arnold) Figure 2. Frequency against 1/␭. pp 79–81 [6] Rossing T D, Russel D A and Brown D E 1992 Ϫ1 Acoustics of tuning forks Am. J. Phys. 60 620–6 Figure 2 is a plot of f1 in Hz against 1/␭ in m . As expected, it is a straight line graph. The slope of this [7] Morse P M and Ingard U K 1986 Acoustics line measures v. The extrapolated line passes (Princeton, NJ: Princeton University Press) pp 181–5 through the origin as expected. [8] Stumpf F B 1980 Analytical Acoustics (Ann Arbor, MI: Ann Arbor Science) pp 43–70 Conclusions [9] Fletcher N H and Rossing T D 1991 The Physics This experiment can be adoped by institutions where of Musical Instruments (New York: Springer) pp a collection of tuning forks made of the same metal 33–64 is available. It can be supplemented with a [10] Hastings R B 1965 Laboratory Physics (St discussion of the vibrations of a bar clamped at one Paul, MN: Bruce) pp 253–4 end. It gives an acceptable value of the speed of sound in aluminium. It can be used to impress on [11] Handbook of Chemistry and Physics 1992 students all the scientific calculations and (Boca Raton, FL: CRC Press) pp B–7, D–187

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