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Unformatted text preview: Overton, Mays Homework 17 Due: Mar 4 2005, 4:00 am Inst: Turner 1 This print-out should have 13 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points A particle of mass 5 . 63 kg is attached to two identical springs on a horizontal frictionless tabletop as shown. The springs have spring constant 59 . 1 N / m and equilibrium length L = 1 . 85 m. L L x x k k m Top View If the mass is pulled 0 . 496 m to the right and then released, what is its speed when it reaches the equilibrium point x = 0? Correct answer: 0 . 299369 m / s. Explanation: First calculate the potential energy of the springs. When the mass moves a distance x , the length of each spring changes from L to p L 2 + x 2 , so each exerts a force F = k p L 2 + x 2- L toward its fixed end. The y-components can- cel out and the x-components add to F x = 2 F x L 2 + x 2 =- 2 k x + 2 k L x L 2 + x 2 . We choose U = 0 at x = 0. Then at any point U =- Z x F x dx = k x 2 + 2 k L L- p L 2 + x 2 = (59 . 1 N / m)(0 . 496 m) 2 + 2(59 . 1 N / m)(1 . 85 m) 1 . 85 m- q (1 . 85 m) 2 + (0 . 496 m) 2 = 0 . 252285 J . From conservation of energy ( K + U ) i = ( K + U ) f , we obtain 0 + U = m v 2 f 2 + 0 . Therefore v f = r 2 U m = s 2(0 . 252285 J) 5 . 63 kg = . 299369 m / s . 002 (part 1 of 1) 10 points Note: The pendulum bob is released at a height below the height of the peg. A pendulum made of a string of length 11 . 7 m and a spherical bob of mass 4 kg and negligible raius swings in a vertical plane. The pendulum is released from an angular position 43 from vertical as shown in the figure below. The string hits a peg located a distance 7 m below the point of suspension and swings about the peg up to an angle on the other side of the peg. Then, the bob proceeds to oscillate back and forth between these two angular positions. The acceleration of gravity is 9 . 8 m / s 2 . 9 . 8 m / s 2 4 kg 1 1 . 7 m 43 7m What is the maximum angle which the pendulum will swing to the right after hitting the peg as shown above? Correct answer: 70 . 6558 . Explanation: Overton, Mays Homework 17 Due: Mar 4 2005, 4:00 am Inst: Turner 2 Let : m = 4 kg , = 11 . 7 m , d = 7 m , = 43 , and g = 9 . 8 m / s 2 . The radius of the sphere about the peg r = - d = 4 . 7 m . (1) Relative to the the point of suspension, the initial potential energy is U i =- m g cos . (2) The final potential energy at the maximum angle which the pendulum will swing to the right after hitting the peg is U f =- m g d- m g r cos . (3) Setting Eqs. 2 and 3 equal, we have m g cos = m g d + m g r cos cos - d = r cos , so (4) = arccos cos...
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