hw12_solutions - Physics 221 Homework 12 Spring 2016...

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Physics 221 Name/Section: Homework 12 Spring 2016 12.1 A wire is made out of a special steel alloy with 11 2.00 10 Pa Y = × , density ρ = 7800 kg/m 3 , radius 1.00 mm, and coefficient of linear thermal expansion α = 2.50 × 10 5 K 1 . The initial temperature of the wire is 20°C. As a constant tension of 3.53 x 10 4 N is applied to the wire, the wire is measured to be exactly 60.0 cm long. (a) What is the fundamental frequency of the wire? Assume the speed of sound is 344 m/s. (b) Where should you pluck the wire to achieve the third harmonic (second overtone) without also exciting the fundamental mode, and what is its frequency? (c) If the wire is heated to the melting point of steel (1363 o C), by how much does the fundamental frequency change? Solution: (a) The fundamental frequency for a wire fixed at both ends is given by: f 1 = v 2 L = 1 2 L T μ The mass per unit length is μ = m L = ρ V L = ρ AL L = ρπ r 2 = (7800)(3.14)(.001) 2 = 2.45 × 10 2 kg/m so the fundamental frequency is f 1 = v 2 L = 1 2 L F μ = 1 2(0.600) 3.53 × 10 4 2.45 × 10 2 = 1000 Hz. The nth harmonic is f n = n f 1 , so f 3 = (3) f 1 = 3000 Hz. The nodes for the third harmonic are 1/3 and 2/3 of the way from one end to the other, and the antinodes are at 1/6, 3/6, and 5/6 of the way from one end to the other. We must pluck an antinode to excite a given frequency. Plucking at the middle also excites the fundamental frequency, so we need to pluck at L/6 or 5L/6 (10 cm from either end) to preferentially excite just the third harmonic. (c) The initial temperature is 20°C and the final temperature is 1363 o C, so Δ L = L α Δ T = (0.600)(2.50 × 10 5 )(1343) = 0.0201 m Note that the density changes as the length of the wire changes, but the mass remains the same. The mass before heating is m = μ L = (2.45 × 10 2 )(0.600) = 1.47 × 10 2 kg
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Physics 221 Name/Section: Homework 12 Spring 2016 Then the new fundamental frequency is f = v 2 L = 1 2 L T μ = 1 2 L TL m = 1 2 T mL = 1 2 3.53 × 10 4 (.0147)(0.6201) = 984 Hz 12.2 (a) In which of these scenarios do you hear a higher pitched sound? Explain. (i) A train with its horn sounding is stationary and you move directly towards it with a speed v 0 .
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