s91 - Solutions to Homework #9 Chapter 28 4. Picture the...

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Unformatted text preview: Solutions to Homework #9 Chapter 28 4. Picture the Problem : The figure shows a car traveling at 17 m/s perpendicular to the line connecting two radio transmitters. The car picks up a maximum signal when it is at point A. Strategy: Set the wavelength equal to the difference in distances between the two towers. Use the Pythagorean theorem to calculate the distance between the car and each tower as a function of time. Set the difference in distance length equal to half a wavelength (for destructive interference) and solve for the time. Solution: 1. (a) Find the difference in path lengths: 450 m 150 m 300 m = =- = l 2. (b) Find the distance to the far tower as a function of time: ( 29 2 2 2 1 450 m v t = + l 3. Find the distance to the near tower as a function of time: ( 29 2 2 2 2 150 m v t = + l 4. Set the difference equal to half a wavelength: ( 29 ( 29 set 2 2 2 2 2 2 1 2 450 m 150 m 150 m 2 v t v t - = +- + = = l l 5 . Rearrange the expression, square both sides, and evaluate: ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 450 m 150 m 150 m 450 m 150 m 2 300 m 150 m 150 m v t v t v t v t v t + = + + + = + + + + 6. Rearrange the equation and square both sides again: ( 29 ( 29 ( 29 2 2 2 2 2 2 2 4 2 2 2 6 2 450 m 2 150 m 150 m 300 m 2.25 10 m 2.756 10 m v t v t- + = + = 7. Solve for t : 6 2 4 2 1 2.756 10 m 2.25 10 m 30 s 17 m/s t =- = Insight: As the car moves farther away, the path difference l decreases until it is nearly zero and the waves constructively interfere. 14. Picture the Problem : Monochromatic red light produces a two-slit interference pattern. The color of the light is then changed to blue. Strategy: Recall the principles that explain the two-slit interference pattern in order to answer the conceptual question. Solution: 1. (a) Blue light has a smaller wavelength; therefore, equations 281 and 282 show that the angles to the fringes will be smaller. We conclude that the spacing between the fringes will decrease if the color of the light is changed to blue. 2. (b) The best explanation is II . The fringe spacing decreases because blue light has a smaller wavelength than red light. Statement I erroneously assumes a greater frequency means a longer wavelength, and statement III ignores the relationship between wavelength and color. Insight: If the wavelength is increased by using infrared light instead of red light, the spacing between the fringes would increase....
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s91 - Solutions to Homework #9 Chapter 28 4. Picture the...

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