Physics Solution Manual for 1100 and 2101

Solution substituting equation 162 into equation 173

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Unformatted text preview: θ = λ D = v 343 m/s = = 0.404 f D ( 607 Hz )( 2 × 0.700 m ) θ = sin −1 ( 0.404 ) = 23.8° 16. REASONING The diffraction angle θ depends upon the wavelength λ of the sound wave and the width D of the doorway according to sin θ = λ (Equation 17.1). We will determine D the wavelength from the speed v and frequency f of the sound wave via v = f λ (Equation 16.1). We note that, in part (a), the frequency of the sound wave is given in kilohertz (kHz), so we will use the equivalence 1 kHz = 103 Hz. SOLUTION a. Solving sin θ = λ D (Equation 17.1) for θ, we obtain 914 THE PRINCIPLE OF LINEAR SUPERPOSITION AND INTERFERENCE PHENOMENA λ D θ = sin −1 Solving v = f λ (Equation 16.1) for λ yields λ = (1), we find that (1) v . Substituting this result into Equation f λ −1 v = sin D Df θ = sin −1 (2) Therefore, when the frequency of the sound wave is 5.0 kHz = 5.0×103 Hz, the diffraction angle is 343 m/s v −1 o = sin ( = 5.1 3 Df ) ( 5.0 × 10 Hz ) 0.77 m θ = sin −1 b. When the frequency of the sound wave is 5.0×102 Hz, the diffraction angle is 343 m/s v −1 o = sin ( = 63 2 Df 0.77 m ) ( 5.0 × 10 Hz ) θ = sin −1 17. REASONING AND SOLUTION At 0 °C the speed of sound is air is given as 331 m/s in Table 16.1 in the text. This corresponds to a wavelength of λ1 = v/f = (331 m/s)/(3.00 × 103 Hz) = 0.1103 m The diffraction angle is given by Equation 17.2 as 1.22 λ –1 1.22 ( 0.1103 m ) = 50.3° = sin 0.175 m D θ1 = sin –1 For an ideal gas, the speed of sound is proportional to the square root of the Kelvin temperature, according to Equation 16.5. Therefore, the speed of sound at 29 °C is b v = 331 m / s K g 302 K = 348 m / s 273 The wavelength at this temperature is λ2 = (348 m/s)/(3.00 × 103 Hz) = 0.116 m. This gives a diffraction angle of θ2 = 54.0°. The change in the diffraction angle is thus ∆θ = 54.0° − 50.3° = 3.7° Chapter 17 Problems 18. REASONING AND SOLUTION The figure at the right shows the geometry of the situation. Diffraction horn The tone will not be heard at seats located 8.7 m at the first diffraction minimum. This occurs when sin θ = λ D = 915 v fD Stage θ θ First row of seats C x That is, the angle θ is given by v 343 m/s −1 = 27.2° = sin 4 f D (1.0 ×10 Hz)(0.075 m) θ = sin −1 From the figure at the right, we see that tan 27.2° = x 8.7 m ⇒ x = (8.7 m)(tan 27.2°) = 4.47 m 8.7 m θ x Thus, seats at which the tone cannot be heard are a distance x on either side of the center seat C. Thus, the distance between the two seats is 2x = 2(4.47 m) = 8.9 m 19. SSM REASONING The beat frequency of two sound waves is the difference between the two sound frequencies. From the graphs, we see that the period of the wave in the upper text figure is 0.020 s, so its frequency is f1 = 1/ T1 = 1/(0.020 s) = 5.0 × 101 Hz . The frequency of the wave in the lower figure is f 2 = 1/(0.024 s)=4.2 ×101 Hz . SOLUTION The beat frequency of the two sound waves is f beat = f1 − f 2 = 5.0 × 101 Hz – 4.2 × 101 Hz = 8 Hz 916 THE PRINCIPLE OF LINEAR SUPERPOSITION AND INTERFERENCE PHENOMENA 20. REASONING The time between successive beats is the period of the beat frequency. The period that corresponds to any frequency is the reciprocal of that frequency. The beat frequency itself is the magnitude of the difference between the two sound frequencies produced by the pianos. Each of the piano frequencies can be determined as the speed of sound divided by the wavelength. SOLUTION According to Equation 10.5, the period T (the time between successive beats) that corresponds to the beat frequency fbeat is T = 1/ f beat . The beat frequency is the magnitude of the difference between the two sound frequencies fA and fB, so that f beat = f A − f B . With this substitution, the expression for the period becomes 1 T= f beat = 1 fA − fB (1) The frequencies fA and fB are given by Equation 16.1 as fA = v λA and fB = v λB Substituting these expressions into Equation (1) gives T= 1 1 1 = = = 0.25 s 343 m/s 343 m/s fA − fB v v − − 0.769 m 0.776 m λ λ A B 21. REASONING When two frequencies are sounded simultaneously, the beat frequency produced is the difference between the two. Thus, knowing the beat frequency between the tuning fork and one flute tone tells us only the difference between the known frequency and the tuning-fork frequency. It does not tell us whether the tuning-fork frequency is greater or smaller than the known frequency. However, two different beat frequencies and two flute frequencies are given. Consideration of both beat frequencies will enable us to find the tuning-fork frequency. SOLUTION The fact that a 1-Hz beat frequency is heard when the tuning fork is sounded along with the 262-Hz tone implies that the tuning-fork frequency is either 263 Hz or 261 Hz. We can eliminate one of these values by considering the fact that a 3-Hz beat frequency is heard when the tuning fork is sounded along with the 266-Hz tone. This implies that the tuning-fork frequency is either 269 Hz...
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