16 - Wave Motion

C repeat parts a and b for the function yx t sinx

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Unformatted text preview: ) for the function y(x, t ) sin(x) cos(vt ). 516 CHAPTER 16 Wave Motion ADDITIONAL PROBLEMS 49. The “wave” is a particular type of wave pulse that can sometimes be seen propagating through a large crowd gathered at a sporting arena to watch a soccer or American football match (Fig. P16.49). The particles of the medium are the spectators, with zero displacement corresponding to their being in the seated position and maximum displacement corresponding to their being in the standing position and raising their arms. When a large fraction of the spectators participate in the wave motion, a somewhat stable pulse shape can develop. The wave speed depends on people’s reaction time, which is typically on the order of 0.1 s. Estimate the order of magnitude, in minutes, of the time required for such a wave pulse to make one circuit around a large sports stadium. State the quantities you measure or estimate and their values. (a) What are the speed and direction of travel of the wave? (b) What is the vertical displacement of the string at t 0, x 0.100 m? (c) What are the wavelength and frequency of the wave? (d) What is the maximum magnitude of the transverse speed of the string? 52. Motion picture film is projected at 24.0 frames per second. Each frame is a photograph 19.0 mm in height. At what constant speed does the film pass into the projector? 53. Review Problem. A block of mass M, supported by a string, rests on an incline making an angle with the horizontal (Fig. P16.53). The string’s length is L, and its mass is m V M. Derive an expression for the time it takes a transverse wave to travel from one end of the string to the other. m, L M θ Figure P16.53 54. (a) Determine the speed of transverse waves on a string under a tension of 80.0 N if the string has a length of 2.00 m and a mass of 5.00 g. (b) Calculate the power required to generate these waves if they have a wavelength of 16.0 cm and an amplitude of 4.00 cm. Figure P16.49 WEB 50. A traveling wave propagates according to the expression y (4.0 cm) sin(2.0x 3.0t ), where x is in centimeters and t is in seconds. Determine (a) the amplitude, (b) the wavelength, (c) the frequency, (d) the period, and (e) the direction of travel of the wave. 51. The wave function for a traveling wave on a taut string is (in SI units) y(x, t ) (0.350 m) sin(10 t 3x /4) 55. Review Problem. A 2.00-kg block hangs from a rubber cord. The block is supported so that the cord is not stretched. The unstretched length of the cord is 0.500 m, and its mass is 5.00 g. The “spring constant” for the cord is 100 N/m. The block is released and stops at the lowest point. (a) Determine the tension in the cord when the block is at this lowest point. (b) What is the length of the cord in this “stretched” position? (c) Find the speed of a transverse wave in the cord if the block is held in this lowest position. 56. Review Problem. A block of mass M hangs from a rubber cord. The block is supported so that the cord is not stretched. The unstretched length of the cord is L 0 , and its mass is m, much less than M. The “spring constant” for the cord is k. The block is released and stops at the lowest point. (a) Determine the tension in the cord when the block is at this lowest point. (b) What is the length of the cord in this “stretched” position? (c) Find the speed of a transverse wave in the cord if the block is held in this lowest position. 517 Problems 57. A sinusoidal wave in a rope is described by the wave function (0.20 m) sin(0.75 x y 18 t ) where x and y are in meters and t is in seconds. The rope has a linear mass density of 0.250 kg/m. If the tension in the rope is provided by an arrangement like the one illustrated in Figure 16.12, what is the value of the suspended mass? 58. A wire of density is tapered so that its cross-sectional area varies with x, according to the equation A (1.0 10 3x 0.010) cm2 (a) If the wire is subject to a tension T, derive a relationship for the speed of a wave as a function of position. (b) If the wire is aluminum and is subject to a tension of 24.0 N, determine the speed at the origin and at x 10.0 m. 59. A rope of total mass m and length L is suspended vertically. Show that a transverse wave pulse travels the length of the rope in a time t 2√L /g. (Hint: First find an expression for the wave speed at any point a distance x from the lower end by considering the tension in the rope as resulting from the weight of the segment below that point.) 60. If mass M is suspended from the bottom of the rope in Problem 59, (a) show that the time for a transverse wave to travel the length of the rope is t √ 2 L √(M mg m) √M (b) Show that this reduces to the result of Problem 59 when M 0. (c) Show that for m V M, the expression in part (a) reduces to t √ mL Mg 61. It is stated in Problem 59 that a wave pulse travels from the bottom to the top of a rope of length L in a time t 2√L /g. Use this result to answer the following questions. (It is not nece...
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