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Equation Sheet for Test #2 Equations Energy: kinetic energy KE = (1/2)mv2; potential energy PEgrav = mgh & PEspr = (1/2)Ky2 Work and Energy: Work = Force distance; W = Fd; Work done = KE Conservation of energy: KE + PE = constant = Etotal = (1/2)m(vbot)2 = mghtop Periodic motion: frequency f (Hz); period T (s); f = 1/T; amplitude A; PEmax = (1/2)KA2 Mass on spring: Fsp = - Ky, f = (1/2π)√(K/m); Pendulum: f = (1/2π)√(g/l) Relations involving v, the wave speed: - wave relation: v = f - sound wave in air: v = (331.3 + 0.6 tC) m/s -> at 20C v = 343.3 m/s - wave on a string: v = √(T/), T is string tension [N] and is mass/length [kg/m] Wave propagation: reflection (inc = refl); refraction (speed varies); diffraction (size~) Wave interference: due to path length difference, (d2 – d1) or due to frequency difference, f2 f1, ffused tone = (1/2)(f2 + f1); fbeats = f2 – f1 Harmonic Series for Standing waves: harmonic number n, fundamental is n = 1 On a string: fn = nf1, n = 2L/n, fn = v/n = (n/2L)√(T/), for n = 1,2,3… Open air-filled pipe: fn = v/n = (n/2L)v, for n = 1,2,3… Closed air-filled pipe: fn = v/n = (n/4L)v, for n = 1,3,5… Sound: Pressure p(N/m2) and Intensity I(W/m2); I = (p2)/() (p2)/(400) Intensity level LI (dB); LI = 10 log (I/I0), where I0 = 1.0 10-12 W/m2 Lp = 20 log (p/p0), where p0 = 2.0 10-5 N/m2 Relative level LI (dB); LI = LI2 - LI1 = 10 log (I2/I1); for I2/I1 = 2n, LI = n 3dB Adding Intensities LI = 10 log [(I1 + I2)/I0], Power P(W), Intensity I(W/m2), and Area(m2); Power = Intensity Area, P = I A Point Source; Intensity at distance r from a source of power P; I = P/(4r2) Loudness (phons), Pitch and Timbre; Tempered scale; f n = f0 (1.0595)n Fourier Spectrum: for complex wave of period T, f1 = 1/T, fn = nf1, and An = amplitude for triangular wave, odd harmonics only with amplitudes An = A1/n2 for square wave, odd harmonics only with amplitudes A n = A1/n Combining Tones: for f2 f1, beating phenomenon, ffused tone = (1/2)(f2 + f1); fbeats = f2 – f1 for f2 >> f1, sidebands, (f2 ± f1) Hearing: range of hearing 20 Hz to 20 kHz with maximum sensitivity for 1000-4000 Hz Speech and Singing: acoustic model of vocal tract, fn = n(v/4L) for n = 1,3,5,.. String Instruments: Standing waves of harmonic number n (the fundamental is n = 1) fn = nf1, n = 2L/n, fn = v/n = (n/2L)√T/, for n = 1,2,3… Conversion Factors and Constants Conversion Factors 1 kg = 1000 gm 1 m = 100 cm = 39.37 in 1 km = 1000 m 1 in = 2.54 cm 1 ft = 12 in = 0.305 m 1 mile = 5280 ft = 1610 m 1 s = 1000 ms 1 hour = 3600 s 1 N = 1 kgm/s2 1 J = 1 Nm = 1 kgm2/s2 Constants Acceleration due to gravity g = 9.8 m/s2 Density ( = Mass/Volume) Water = 1000 kg/m3 Nylon = 1140 kg/m3 Steel = 7700 kg/m3 Abbreviations mega M = 106 kilo k = 103 milli m = 10-3 micro = 10-6 Figures