751 7.6 Trigonometric Equations 107. Electromotive Force In an electrical circuit, suppose that the electromotive force in volts at t seconds can be modeled by the equation V = cos 2pt. Find the least value of t where 0 …t …1 2 for each value of V. (a) V = 0 (b) V = 0.5 (c) V = 0.25 108. Voltage Induced by a Coil of Wire A coil of wire rotating in a magnetic field induces a voltage modeled by the equation E = 20 sin a pt 4 - p 2b , where t is time in seconds. Find the least positive time to produce each voltage. (a) 0 (b) 1023 109. Pressure of a Plucked String If a string with a fundamental frequency of 110 Hz is plucked in the middle, it will vibrate at the odd harmonics of 110, 330, 550, . . . Hz but not at the even harmonics of 220, 440, 660, . . . Hz. The resulting pressure P caused by the string is graphed below and can be modeled by the following equation. P = 0.003 sin 220pt + 0.003 3 sin 660pt + 0.003 5 sin 1100pt + 0.003 7 sin 1540pt (Data from Benade, A., Fundamentals of Musical Acoustics, Dover Publications. Roederer, J., Introduction to the Physics and Psychophysics of Music, Second Edition, Springer-Verlag.) (a) Duplicate the graph shown here. (b) Describe the shape of the sound wave that is produced. (c) At lower frequencies, the inner ear will hear a tone only when the eardrum is moving outward. This occurs when P is negative. Determine the times over the interval 30, 0.034 when this will occur. 110. Hearing Beats in Music Musicians sometimes tune instruments by playing the same tone on two different instruments and listening for a phenomenon known as beats. Beats occur when two tones vary in frequency by only a few hertz. When the two instruments are in tune, the beats disappear. The ear hears beats because the pressure slowly rises and falls as a result of this slight variation in the frequency. (Data from Pierce, J., The Science of Musical Sound, Scientific American Books.) (a) Consider the two tones with frequencies of 220 Hz and 223 Hz and pressures P1 = 0.005 sin 440pt and P2 = 0.005 sin 446pt, respectively. A graph of the pressure P = P1 + P2 felt by an eardrum over the 1-sec interval 30.15, 1.154 is shown here. How many beats are there in 1 sec? (b) Repeat part (a) with frequencies of 220 and 216 Hz. (c) Determine a simple way to find the number of beats per second if the frequency of each tone is given. −0.005 0.005 0 0.03 For x = t, P(t) = 0.003 sin 220pt + sin 660pt + 0.003 3 sin 1100pt + 0.003 5 sin 1540pt 0.003 7 −0.01 0.01 0.15 1.15 For x = t, P(t) = 0.005 sin 440pt + 0.005 sin 446pt
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