16 - Wave Motion

00 10 6 m2 the density of the material to be 270 103

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Unformatted text preview: ssary to set up any new integrations.) (a) How long does it take for a wave pulse to travel halfway up the rope? (Give your answer as a fraction of the quantity 2√L /g.) (b) A pulse starts traveling up the rope. How far has it traveled after a time √L /g ? 62. Determine the speed and direction of propagation of each of the following sinusoidal waves, assuming that x is measured in meters and t in seconds: (a) y 0.60 cos(3.0x 15t 2) (b) y 0.40 cos(3.0x 15t 2) (c) y 1.2 sin(15t 2.0x) (d) y 0.20 sin(12t x/2 ) 63. Review Problem. An aluminum wire under zero tension at room temperature is clamped at each end. The tension in the wire is increased by reducing the temperature, which results in a decrease in the wire’s equilibrium length. What strain ( L /L) results in a transverse wave speed of 100 m/s? Take the cross-sectional area of the wire to be 5.00 10 6 m2, the density of the material to be 2.70 103 kg/m3, and Young’s modulus to be 7.00 1010 N/m2 . 64. (a) Show that the speed of longitudinal waves along a spring of force constant k is v √kL / , where L is the unstretched length of the spring and is the mass per unit length. (b) A spring with a mass of 0.400 kg has an unstretched length of 2.00 m and a force constant of 100 N/m. Using the result you obtained in (a), determine the speed of longitudinal waves along this spring. 65. A string of length L consists of two sections: The left half has mass per unit length 0/2, whereas the right half has a mass per unit length 3 3 0/2. Tension in the string is T0 . Notice from the data given that this string has the same total mass as a uniform string of length L and of mass per unit length 0 . (a) Find the speeds v and v at which transverse wave pulses travel in the two sections. Express the speeds in terms of T0 and 0 , and also as multiples of the speed v 0 (T0 / 0)1/2. (b) Find the time required for a pulse to travel from one end of the string to the other. Give your result as a multiple of t 0 L /v 0 . 66. A wave pulse traveling along a string of linear mass density is described by the relationship y [A0e bx] sin(kx t) where the factor in brackets before the sine function is said to be the amplitude. (a) What is the power (x) carried by this wave at a point x ? (b) What is the power carried by this wave at the origin? (c) Compute the ratio (x)/ (0). 67. An earthquake on the ocean floor in the Gulf of Alaska produces a tsunami (sometimes called a “tidal wave”) that reaches Hilo, Hawaii, 4 450 km away, in a time of 9 h 30 min. Tsunamis have enormous wavelengths (100 – 200 km), and the propagation speed of these waves is v √gd , where d is the average depth of the water. From the information given, find the average wave speed and the average ocean depth between Alaska and Hawaii. (This method was used in 1856 to estimate the average depth of the Pacific Ocean long before soundings were made to obtain direct measurements.) 518 CHAPTER 16 Wave Motion ANSWERS TO QUICK QUIZZES 16.1 (a) It is longitudinal because the disturbance (the shift of position) is parallel to the direction in which the wave travels. (b) It is transverse because the people stand up and sit down (vertical motion), whereas the wave moves either to the left or to the right (motion perpendicular to the disturbance). 16.2 1 cm 16.3 Only answers (f) and (h) are correct. (a) and (b) affect the transverse speed of a particle of the string, but not the wave speed along the string. (c) and (d) change the amplitude. (e) and (g) increase the time by decreasing the wave speed. 16.4 The transverse speed increases because v y, max A 2 fA. The wave speed does not change because it depends only on the tension and mass per length of the string, neither of which has been modified. The wavelength must decrease because the wave speed v f remains constant....
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This note was uploaded on 03/24/2010 for the course PHYSICS 2202 taught by Professor Mihalisin during the Spring '09 term at Temple.

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