Answer, Key Homework 3 David McIntyre 1

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1 Answer, Key Homewor 3 Daid McIntyre 1 This print-out should hae 26 questions, chec that it is complete Multiple-choice questions may continue on the next column or page: find all choices before maing your selection The due time is Central time Chapters 16 and 17 problems 001 (part 1 of 2) 0 points A wae moing along the x axis is described by (x+ t)2 y(x, t) A e b where x is in meters and t is in seconds Gien A 4 m, 5 m/s, and b 4 m 2 Determine the speed of the wae Correct answer: 5 m/s (x+ t)2 y(x, t) A e b is of the form y(x, t) f (x + t) so it describes a wae moing in the negatie x direction at 5 m/s 002 (part 2 of 2) 0 points Determine the direction of the wae 1 y direction 2 +y direction 3 x direction correct 4 +x direction 003 (part 1 of 5) 2 points The wae function for a linearly polarized wae on a taut string is y(x, t) A sin(ω t x + φ), where A 057 m, ω 93 s 1, 49 m 1, φ 055, t is in seconds and x and y are in meters What is the speed of the wae? Correct answer: m/s Basic Concepts: λ T λ f ω Solution: For the propagation speed of a wae we hae ω (93 s1 ) (49 m 1 ) m/s 004 (part 2 of 5) 2 points What is the ertical displacement of the string at x m, t 0 1 s? Correct answer: m We can obtain the result if we simply substitute the alues for x and t into y(x, t) y(x 0, t 0 ) A sinω t x + φ] (057 m) sin (93 s 1 ) (1 s) ] (49 m 1 ) (0063 m) + (055) m 005 (part 3 of 5) 2 points What is the waelength? Correct answer: m For the wae ector we hae Thus λ λ (49 m 1 ) m

2 006 (part 4 of 5) 2 points What is the frequency of the wae? Correct answer: Hz Since ω f, we hae f ω (93 s1 ) Hz 007 (part 5 of 5) 2 points What is the maximum magnitude of the transerse speed of the string? Correct answer: 5301 m/s Taing deriatie of y oer t, we find magnitude of the transerse speed of the string: y(x, t) y t A ω cos(ω t x + φ) Thus y,max assumes its maximum alue when ω t x + φ n, n 0, 1, 2, ; ie, when cos(ω t x + φ) 1, so y,max A ω (057 m) (93 s 1 ) 5301 m/s 008 (part 1 of 3) 0 points A string of length 27 m consists of two sections The left half has mass per unit length g/m, while the right has a mass per unit length g/m Tension in the string is 855 N Notice from the data gien that this string has the same total mass as a uniform string of length 27 m and mass per unit length g/m Find the speed at which transerse wae pulses trael in the left section Correct answer: m/s The speed of wae in the left section of mass per unit length µ 1 is F 1 µ 1 Answer, Key Homewor 3 Daid McIntyre N g/m m/s 009 (part 2 of 3) 0 points Find the speed at which transerse wae pulses trael in the right section Correct answer: m/s The speed of wae in the right section of mass per unit length µ is F 2 µ N g/m m/s 010 (part 3 of 3) 0 points Find the time required for a pulse to trael from one end of the string to the other Correct answer: s The time for a pulse traeling through the string is t L + L m 2 ( m/s) + 27 m 2 ( m/s) s 011 (part 1 of 5) 2 points A sinusoidal wae traeling in the x direction (to the left) has an amplitude of 167 cm, a waelength of 158 cm and a frequency of 141 Hz The displacement of the wae at t 0 and x 0 is y 48 cm and has a positie elocity here Find the angular wae number of the wae Correct answer: rad/m

3 Answer, Key Homewor 3 Daid McIntyre 3 For a sinusoidal wae with waelength λ and frequency f, the angular wae number λ (2) (100) π 158 cm rad/m 012 (part 2 of 5) 2 points Find the period of the wae Correct answer: s The period is T 1 f Hz s 013 (part 3 of 5) 2 points Find the angular frequency of the wae Correct answer: rad/s The angular frequency is ω f (141 Hz) rad/s 014 (part 4 of 5) 2 points Find the phase elocity of the wae Correct answer: m/s The phase elocity is f λ (141 Hz) (158 cm) (001 m/cm) m/s 015 (part 5 of 5) 2 points Find the magnitude of the phase constant of the wae Correct answer: A sinusoidal wae traeling in the x direction can be written as y A sin( x + ω t φ), so at t 0 and x 0 we hae y A sin(φ) which gies the phase constant φ sin 1 y ] A ] sin 1 (48 cm) (167 cm) (part 1 of 5) 0 points A sinusoidal wae train is described by y (025 m) sin (03 m 1 ) x (40 s 1 ) t ], where x and y are in meters and t is in seconds Determine for this wae the amplitude Correct answer: 025 m For a sinusoidal wae described by y A sin( x + ω t), where its amplitude is A 025 m, angular wae number 03 m 1, angular frequency ω 40 s 1, waelength λ m and wae speed ω m/s 017 (part 2 of 5) 0 points Determine for this wae the angular frequency Correct answer: 40 s 1 See Part (part 3 of 5) 0 points Determine for this wae the wae number Correct answer: 03 m 1 See Part (part 4 of 5) 0 points Determine for this wae the waelength Correct answer: m λ (03 m 1 ) m

4 Answer, Key Homewor 3 Daid McIntyre (part 5 of 5) 0 points Determine for this wae the wae speed Correct answer: m/s ω (40 s1 ) (03 m 1 ) m/s 021 (part 1 of 1) 10 points Gien: The speed of the lightwaes in air is m/s The speed of sound waes in air is 333 m/s What is the time lapse between seeing a lightening strie and hearing the thunder if the lightening flash is 24 m away? Correct answer: s The time for the light to trael 24 m is t 1 d 1 The time for the sound to trael 24 m is Thus, the time lapse is t 2 d 2 t t 2 t ( 1 1 d 1 ) 2 1 (24 m) ( ) m/s m/s s 022 (part 1 of 3) 0 points A sinusoidal sound wae is described by the displacement s(x, t) s m cos(x ωt), where s m 459 µm, 14 rad/m, and ω 977 rad/s Find the speed of this wae Correct answer: m/s The speed of the wae is ω 977 rad/s 14 rad/m m/s 023 (part 2 of 3) 0 points Determine the instantaneous displacement of the molecules at the position x m at t 382 ms Correct answer: µm Directly from the formula for the displacement gien therefore α x ω t (14 rad/m)(00626 m) 977 rad/s s rad, s s m cos( x ω t) 459 µm cos( rad) µm 024 (part 3 of 3) 0 points Determine the maximum speed of the molecules oscillatory motion Correct answer: m/s Differentiating the equation gien with respect to time and setting the sin function equal to 1 (maximum), we obtain for the maximum speed (absolute alue of the maximum elocity) max s m ω (459 µm)(977 rad/s) µm/s m/s

5 Answer, Key Homewor 3 Daid McIntyre 5 ] (343 m/s) + ( m/s) 025 (part 1 of 2) 5 points 343 m/s In order to be able to determine her speed, 4520 Hz a sydier carries a tone generator A friend on the ground at the landing site has equipment for receiing and analyzing sound waes While the sydier is falling at terminal speed, her tone generator emits a steady tone of 1260 Hz Assume: The air is calm and that the sound speed is 343 m/s, independent of altitude If her friend on the ground (directly beneath the sydier) receies waes of frequency 2890 Hz, what is the sydier s speed of descent? Correct answer: m/s Since the dier, who is the source of the sound waes, is moing towards her friend on the ground, who is the receier of the waes, the Doppler formula taes the form So f g f e 1 dier Soling for dier, we hae dier 1 f ] e f g (343 m/s) ( dier ) f e f g m/s ] (1260 Hz) (2890 Hz) 026 (part 2 of 2) 5 points If the sydier were also carrying soundreceiing equipment sensitie enough to detect waes reflected from the ground, what frequency would she receie? Correct answer: 4520 Hz If the waes are reflected, and the sydier is moing towards them, we hae ] + dier f rec f g (2890 Hz)