Physics 41 Homework #2 Chapter 16. fa. Here v is the speed of the wave. 16. The speed of a wave on a massless string would be infinite!

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1 Physics 41 Hoewor # Chapter 1 Serway 7 th Conceptual: Q: 3,, 8, 11, 1, Probles P: 1, 3, 5, 9, 1, 5, 31, 35, 38, 4, 5, 57 Conceptual 3. (i) d=e, f, c, b, a (ii) Since, the saller the (the coefficient of x), the larger the wavelength. Thus, the ran according to wavelength is: d, a = b = c, e, f (iii) frequency: the larger oega (coefficient of t) the larger the frequency: f, e, d=b, a=c (iv) period is inverseley proportional to frequency so reverse (iii) (v) v Therefore the ratio of the t coefficient to the x coefficient gives the velocity: d, b = e, f, 3, 3. Higher tension aes wave speed higher. Greater linear density aes the wave ove ore slowly. va 8. Yes, aong other things it depends on. v ax A fa. Here v is the speed of the wave. 11. Each eleent of the rope ust support the weight of the rope below it. The tension increases T with height. (It increases linearly, if the rope does not stretch.) Then the wave speed v increases with height. 1. The speed of a wave on a assless string would be infinite! Probles P: 1, 3, 5, 9, 1, 5, 31, 35, 38, 4, 5, At t = 0, a transverse pulse in a wire is described by the function y x 3 where x and y are in eters. Write the function y(x, t) that describes this pulse if it is traveling in the positive x direction with a speed of 4.50 /s. P1.1 Replace x by x vt x 4.5 t (see plots) to get y x t Two points A and B on the surface of the Earth are at the sae longitude and 0.0 apart

2 in latitude. Suppose that an earthquae at point A creates a P wave that reaches point B by traveling straight through the body of the Earth at a constant speed of 7.80 /s. The earthquae also radiates a Rayleigh wave, which travels along the surface of the Earth in an analogous way to a surface wave on water, at 4.50 /s. (a) Which of these two seisic waves arrives at B first? What is the tie difference between the arrivals of the two waves at B? Tae the radius of the Earth to be 370. P1.3 (a) The longitudinal wave travels a shorter distance and is oving faster, so it will arrive at point B first. The transverse taes ore tie and carries ore energy! The wave that travels through the Earth ust travel a distance of R sin sin at a speed of Therefore, it taes /s s 817 s The wave that travels along the Earth s surface ust travel a distance of s R R rad at a speed of /s Therefore, it taes s The tie difference is 5 s 11.1 in 5. (a) Let u 10 t 3 x du dx at a point of constant phase 4 dt dt y dx 10 dt s The velocity is in the positive x -direction , sin c 4 (c) 3 : 0.7 f 10 : f 5.00 H z y vy cos 10 t 3 x t 4 (d) vy, ax s P1.9 y sin.11 x 3. t in SI units A.00 c

3 .11 rad rad s f 0.57 H z 3. v f s P1.1 (a) y () t = x () FIG. P1.1(a) 18.0 rad T s f 1.0 s f 1.0 s 75.4 rad s v f 1.0 s s (c) y A sin x t specializes to y 0.00 sin 18.0 x 75.4 ts sin at x 0, t 0 we require rad so yx, t 0.00 sin 18.0 x 75.4 t s rad 5. An astronaut on the Moon wishes to easure the local value of the free-fall acceleration by tiing pulses traveling down a wire that has an object of large ass suspended fro it. Assue a wire has a ass of 4.00 g and a length of 1.0, and that a 3.00-g object is suspended fro it. A pulse requires 3.1 s to traverse the length of the wire. Calculate gmoon fro these data. (You ay ignore the ass of the wire when calculating the tension in it.) T M g M gl L P1.5 T M g is the tension; v L t is the wave speed. M gl L Then, t 3 L g and g Mt g s 1.4 s

4 31. A steel wire and a 0.0- copper wire, both with diaeters, are connected end to end and stretched to a tension of 150 N. How long does it tae a transverse wave to travel the entire length of the two wires? P1.31 The total tie is the su of the two ties. L In each wire t L v T Let A represent the cross-sectional area of one wire. The ass of one wire can be written both as V AL and also as L. Then we have Thus, d t L 4T A d 4 1 For copper, t For steel, t s s 4150 The total tie is s 35. Sinusoidal waves 5.00 c in aplitude are to be transitted along a string that has a linear ass density of g/. If the source can deliver a axiu power of 300 W and the string is under a tension of 100 N, what is the highest frequency at which the source can operate? P1.35 A g Therefore, 1 P 300 W T 100 N P A v: T v 50.0 s 34 rad s f 55.1 H z P A v

5 38. The wave function for a wave on a taut string is y(x, t) = (0.350 )sin(10 t 3 x + /4) where x is in eters and t in seconds. (a) What is the average rate at which energy is transitted along the string if the linear ass density is 75.0 g/? What is the energy contained in each cycle of the wave? *P1.38 Coparing y 0.35sin 10t 3x 4 with y A sin x t A sin t x we 3 10 s have, 10 s, A Then v 3.33 s f f 3. (a) The rate of energy transport is P Av g 10 s s 15.1 W. The energy per cycle is E P T A g 10 s J. 4. *P1.4 The linear wave equation is Therefore, y 1 y x v t if b x vt y e y then bve b x vt y and be b xvt t x y b v e b xvt t y y v t x y and be b xvt x b, deonstrating that x vt e is a solution 5. Review proble. A bloc of ass M, supported by a string, rests on an incline aing an angle with the horizontal (Fig. P1.5). The length of the string is L and its ass is <<M. Derive an expression for the tie interval required for a transverse wave to travel fro one end of the string to the other. P1.5 Assuing the incline to be frictionless and taing the positive x-direction to be up the incline: F x T M g sin 0 or the tension in the string is T M gsin The speed of transverse waves in the string is then T M gsin v L M gl sin The tie interval for a pulse to travel the string s length is t L L L v M gl sin M gsin

Chapter 16 Solutions

Chapter 16 Solutions 16.1 Replace x by x vt = x 4.5t to get y = 6 [(x 4.5t) + 3] 16. y (c) y (c) y (c) 6 4 4 4 t = s t = 1 s t = 1.5 s 0 6 10 14 x 0 6 10 14 x 0 6 10 14 x y (c) y (c) 4 t =.5 s 4 t = 3