18 - roofner(bar784 Homework 18 Weathers(17104 This...

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roofner (bar784) – Homework 18 – Weathers – (17104) 1 This print-out should have 12 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points A particle executes simple harmonic motion with an amplitude of 3 . 73 cm. At what positive displacement from the midpoint of its motion does its speed equal one half of its maximum speed? Correct answer: 3 . 23027 cm. Explanation: The potential energy of a simple harmonic oscillator at displacement x from the equilib- rium point is U osc = 1 2 k x 2 = 1 2 m ω 2 x 2 , since k = m ω 2 . When the particle is at maximum displacement A , the energy is all potential: U = 1 2 m ω 2 A 2 . At other points x , the energy might be both kinetic (speed v ) and potential K + U = 1 2 m v 2 + 1 2 m ω 2 x 2 . Conservation of energy gives 1 2 m v 2 + 1 2 m ω 2 x 2 = 1 2 m ω 2 A 2 , or v 2 + ω 2 x 2 = ω 2 A 2 . The speed v is v = A ω sin( ω t ) , and the sine is never more than 1, meaning v max = ω A . We are asked for the displacement at half this speed, v = ωA 2 , so conservation of energy is now parenleftbigg ω A 2 parenrightbigg 2 + ω 2 x 2 = ω 2 A 2 , or 1 4 A 2 + x 2 = A 2 , from which we see x = ± 3 2 A = ± 3 2 (3 . 73 cm) = ± 3 . 23027 cm . 002 (part 1 of 3) 10.0 points A block of unknown mass is attached to a spring of spring constant 6 . 7 N / m and under- goes simple harmonic motion with an ampli- tude of 12 . 9 cm. When the mass is halfway between its equilibrium position and the end- point, its speed is measured to be 40 cm / s. Calculate the mass of the block. Correct answer: 0 . 522631 kg. Explanation: Let : k = 6 . 7 N / m , A = 12 . 9 cm , and v = 40 cm / s . If the maximum displacement (amplitude) is A , the halfway displacement is A 2 . By energy conservation, K i + U i = F f + U f 0 + 1 2 k A 2 = 1 2 m v 2 + 1 2 k parenleftbigg A 2 parenrightbigg 2 k A 2 = m v 2 + 1 4 k A 2 m = 3 k A 2 4 v 2 = 3 (6 . 7 N / m) (0 . 129 m) 2 4 (0 . 4 m / s) 2 = 0 . 522631 kg . 003 (part 2 of 3) 10.0 points
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roofner (bar784) – Homework 18 – Weathers – (17104) 2 Find the period of the motion. Correct answer: 1 . 75485 s. Explanation: ω = radicalbigg k m = radicalBigg 6 . 7 N / m 0 . 522631 kg = 3 . 58047 rad / s , so the period is T = 2 π ω = 2 π 3 . 58047 rad / s = 1 . 75485 s . 004 (part 3 of 3) 10.0 points Calculate the maximum acceleration of the block. Correct answer: 1 . 65375 m / s 2 . Explanation: Simple harmonic motion is described by x = A cos ωt , so the acceleration is a = ω 2 A cos ωt . The maximum of the cosine function is 1, so the maximum acceleration is a max = ω 2 A = (3 . 58047 rad / s) 2 (0 . 129 m) = 1 . 65375 m / s 2 . This happens when the block is at its turning point (maximum displacement). 005 10.0 points A 0 . 353 kg mass is attached to a spring and undergoes simple harmonic motion with a pe- riod of 0 . 12 s. The total energy of the system is 4 . 3 J.
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