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1. A nonuniform wire of length L and mass M has a variable lincar mass density given by u = kx, where x is the distance from one end of the wire and k is a constant. (a) Show that M = kl2 /2. (b) Show that the time t required for a pulse generated at one end of the wire to travel to the other end is given by t = /8ML/9F, where F is the tension in the wire.

1. A nonuniform wire of length L and mass M has a variable lincar mass density given by u = kx, where x is the distance from one end of the wire and k is a constant. (a) Show that M = kl2 /2. (b) Show that the time t required for a pulse generated at one end of the wire to travel to the other end is given by t = /8ML/9F, where F is the tension in the wire.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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17:34 ul 4G A cool.ntu.edu.tw 1 of 2 Unit 9 Problem Set: Wave 1. Anonuniform wire of length L and mass M has a variable linear mass density given by p = kx, where x is the distance from one end of the wire and k is a constant. (a) Show that M = kL?/2. (b) Show that the time t required for a pulse generated at one end of the wire to travel to the other end is given by t = /8ML/9F, where F is the tension in the wire. 2. A uniform circular hoop of string is rotating clockwise in the absence of gravity. The tangential speed is Find the speed of waves on the string. (Note that the answer is independent of the radius of the hoop and the linear mass density of the string!) 3. Two strings of linear mass density H1 and uz are knotted together at x = 0 and stretched to a tension F.A wave y = Asink, (x - vịt) in the string of density Hi reaches the junction between the two strings, at which it is partly transmitted into the string of density Hz and partly reflected. Call these waves Bsink2(x – vzt) and Csink, (x + vịt), respectively. (a) Assuming that kzv2 = k1V1 = w and that the displacement of the knot arising from the incident and reflected waves is the same as that arising from the transmitted wave, show that A = B + C. (b) If it is assumed that both strings near the knot have the same slope (why?) - that is, dy/dx in string 1 = dy/dx in string 2 - Show that C = A = kz+k1 A- Under what conditions is C negative? 4. The period of a pulsating variable star may be estimated by considering the star to be executing radial longitudinal pulsations in the fundamental standing wave mode; that is, the radius varies periodically with the time, with a displacement antinode at the surface. (a) Would you expect the center of the star to be a displacement node or antinode? (b) By analogy with the open organ pipe, show that the period of pulsation T is given by T = , where R is the equilibrium radius of the star and v, is the average sound speed. (c) Typical white dwarf stars are composed of material with a bulk modulus of 1.33 x 1022 Pa and a density of 1.0 x 1010 kg/m3. They have radii equal to 0.009 solar radius. What is the approximate pulsation period of a white dwarf? 5. The figure shows waves contained in a single instrument. It is used to measure the speed V of a target object transmitter and receiver of (idealized as a flat plate) that is moving directly toward the unit, by analyzing the waves reflected Target from it. (a) Apply the Doppler equations twice, first with the target as observer, and then with the target as a source, and show that the frequency fr of the reflected waves at the receiver is related to their source frequency f by fr = f ), where v is the speed of the waves. (b) In a great many practical situations, V «v. In this case, show that the equation above becomes 6. A uniform cable hangs vertically under its own weight. Show that the speed of waves on the cable is given by v = Jyg, where y is the distance from the bottom of the cable. 7. Show that the time it takes a wave to propagate up the cable in Problem 6 is t = L/g, where cable length. is the
Transcribed Image Text:17:34 ul 4G A cool.ntu.edu.tw 1 of 2 Unit 9 Problem Set: Wave 1. Anonuniform wire of length L and mass M has a variable linear mass density given by p = kx, where x is the distance from one end of the wire and k is a constant. (a) Show that M = kL?/2. (b) Show that the time t required for a pulse generated at one end of the wire to travel to the other end is given by t = /8ML/9F, where F is the tension in the wire. 2. A uniform circular hoop of string is rotating clockwise in the absence of gravity. The tangential speed is Find the speed of waves on the string. (Note that the answer is independent of the radius of the hoop and the linear mass density of the string!) 3. Two strings of linear mass density H1 and uz are knotted together at x = 0 and stretched to a tension F.A wave y = Asink, (x - vịt) in the string of density Hi reaches the junction between the two strings, at which it is partly transmitted into the string of density Hz and partly reflected. Call these waves Bsink2(x – vzt) and Csink, (x + vịt), respectively. (a) Assuming that kzv2 = k1V1 = w and that the displacement of the knot arising from the incident and reflected waves is the same as that arising from the transmitted wave, show that A = B + C. (b) If it is assumed that both strings near the knot have the same slope (why?) - that is, dy/dx in string 1 = dy/dx in string 2 - Show that C = A = kz+k1 A- Under what conditions is C negative? 4. The period of a pulsating variable star may be estimated by considering the star to be executing radial longitudinal pulsations in the fundamental standing wave mode; that is, the radius varies periodically with the time, with a displacement antinode at the surface. (a) Would you expect the center of the star to be a displacement node or antinode? (b) By analogy with the open organ pipe, show that the period of pulsation T is given by T = , where R is the equilibrium radius of the star and v, is the average sound speed. (c) Typical white dwarf stars are composed of material with a bulk modulus of 1.33 x 1022 Pa and a density of 1.0 x 1010 kg/m3. They have radii equal to 0.009 solar radius. What is the approximate pulsation period of a white dwarf? 5. The figure shows waves contained in a single instrument. It is used to measure the speed V of a target object transmitter and receiver of (idealized as a flat plate) that is moving directly toward the unit, by analyzing the waves reflected Target from it. (a) Apply the Doppler equations twice, first with the target as observer, and then with the target as a source, and show that the frequency fr of the reflected waves at the receiver is related to their source frequency f by fr = f ), where v is the speed of the waves. (b) In a great many practical situations, V «v. In this case, show that the equation above becomes 6. A uniform cable hangs vertically under its own weight. Show that the speed of waves on the cable is given by v = Jyg, where y is the distance from the bottom of the cable. 7. Show that the time it takes a wave to propagate up the cable in Problem 6 is t = L/g, where cable length. is the
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