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Unformatted text preview: 13. The wave speed v is given by v = τ μ , where τ is the tension in the rope and μ is the linear mass density of the rope. The linear mass density is the mass per unit length of rope: μ = m/L = (0.0 60 0 kg)/(2.00 m) = 0.0300 kg/m. Thus 500 N 129m s. 0.0300kg m v = = 15. (a) The wave speed is given by v = λ / T = ϖ / k , where λ is the wavelength, T is the period, ϖ is the angular frequency (2 π / T ), and k is the angular wave number (2 π / λ ). The displacement has the form y = y m sin( kx + ϖ t ), so k = 2.0 m –1 and ϖ = 30 rad/s. Thus v = (30 rad/s)/(2.0 m –1 ) = 15 m/s. (b) Since the wave speed is given by v = τ μ , where τ is the tension in the string and μ is the linear mass density of the string, the tension is ( 29 ( 29 2 2 4 1.6 10 kg m 15m s 0.036 N. v τ μ = = = 18. (a) Comparing with Eq. 162, we see that k = 20/m and ϖ = 600/s. Therefore, the speed of the wave is (see Eq. 1613) v = ϖ / k = 30 m/s. (b) From Eq. 16–26, we find 2 2 15 0.017kg m 17g m. 30 v = = = = τ μ 20. (a) The general expression for y ( x, t ) for the wave is y ( x, t ) = y m sin( kx – ϖ t ), which, at x = 10 cm, becomes y ( x = 10 cm, t ) = y m sin[ k (10 cm – ϖ t )]. Comparing this with the expression given, we find ϖ = 4.0 rad/s, or f = ϖ /2 π = 0.64 Hz. (b) Since k (10 cm) = 1.0, the wave number is k = 0.10/cm. Consequently, the wavelength is λ = 2 π / k = 63 cm. (c) The amplitude is 5.0 cm. m y = (d) In part (b), we have shown that the angular wave number is k = 0.10/cm. (e) The angular frequency is ϖ = 4.0 rad/s. (f) The sign is minus since the wave is traveling in the + x direction. Summarizing the results obtained above by substituting the values of k and ϖ into the general expression for y ( x, t ), with centimeters and seconds understood, we obtain ( , ) 5.0sin (0.10 4.0 ). y x t x t = (g) Since / / , v k = = ϖ τ μ the tension is 2 1 2 2 2 1 2 (4.0g /cm)(4.0s ) 6400g cm/s 0.064 N....
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 Spring '07
 Giammanico
 Frequency, Wavelength, Fundamental physics concepts, Standing wave

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