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Unformatted text preview: WebAssign Problems 1. (a) A hanging spring stretches by 35.0 cm when an object of mass 450 g is hung on it at rest. In this situation, we define its position as x = 0. The object is pulled down an additional 18.0 cm and released from rest to oscillate without friction. What is its position x at a time 84.4 s later? (b) What If? A hanging spring stretches by 35.5 cm when an object of mass 440 g is hung on it at rest. We define this new position as x = 0. This object is also pulled down an additional 18.0 cm and released from rest to oscillate without friction. Find its position 84.4 s later. Does this circumstance reveal a fundamental difficulty in calculating the future? (c) Find the distance traveled by the vibrating object in part (a). (d) Find the distance traveled by the object in part (b). (e) Why are the answers to (a) and (b) different by such a large percentage when the two systems and initial conditions are so similar? Solution: (a) The spring constant of this spring is 2 0.45 kg 9.8 m s 12.6 N m 0.35 m F k x = = = we take the x-axis pointing downward, so = 2 12.6 kg cos 18.0 cm cos 84.4 s 18.0 cm cos446.6 rad 15.8 cm 0.45 kg s x A t = = = = (b) By the same steps, 2 0.44 kg 9.8 m s 12.1 N m 0.355 m k = = 12.1 cos 18.0 cm cos 84.4 18.0 cm cos443.5 rad 15.9 cm 0.44 k x A t m = = = = - (c) Now 446.6 rad 71 2 0.497 rad = + . In each cycle the object moves ( ) 4 18 72 cm = , so it has moved ( ) ( ) 71 72 cm 18 15.8 cm 51.1 m +- = . (d) ( ) 443.5 rad 70 2 3.62 rad = + Distance moved ( ) 70 72 cm 18 15.9 cm 50.7 m = + + = (e) The answers to (c) and (d) are not very different given the difference in the data about the two vibrating systems. But when we ask about details of the future, the imprecision in our knowledge about the present makes it impossible to make precise predictions. The two oscillations start out in phase but get completely out of phase. 2. A piston in a gasoline engine is in simple harmonic motion. If the extremes of its position relative to its center point are 5.00 cm, find the maximum velocity and acceleration of the piston when the engine is running at the rate of 3 600 rev/min. Compare the maximum acceleration of the piston to the acceleration of free fall....
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