MP56- Simple Harmonic Motion Energy

MP56- Simple Harmonic Motion Energy - MasteringPhysics:...

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Unformatted text preview: MasteringPhysics: Assignment Print View 4/28/08 3:04 PM [ Print View ] PHYS 2211 ABCDE Spring 08 MP56: Simple Harmonic Motion Energy Due at 11:59pm on Tuesday, April 22, 2008 View Grading Details Energy of Harmonic Oscillators Learning Goal: To learn to apply the law of conservation of energy to the analysis of harmonic oscillators. Systems in simple harmonic motion, or harmonic oscillators, obey the law of conservation of energy just like all other systems do. Using energy considerations, one can analyze many aspects of motion of the oscillator. Such an analysis can be simplified if one assumes that mechanical energy is not dissipated. In other words, , where is the total mechanical energy of the system, is the kinetic energy, and is the potential energy. As you know, a common example of a harmonic oscillator is a mass attached to a spring. In this problem, we will consider a horizontally moving block attached to a spring. Note that, since the gravitational potential energy is not changing in this case, it can be excluded from the calculations. For such a system, the potential energy is stored in the spring and is given by , where is the force constant of the spring and is the distance from the equilibrium position. The kinetic energy of the system is, as always, , where is the mass of the block and is the speed of the block. . We will also assume that there are no resistive forces; that is, Consider a harmonic oscillator at four different moments, labeled A, B, C, and D, as shown in the figure . Assume that the force constant , the mass of the block, , and the amplitude of vibrations, , are given. Answer the following questions. Part A Which moment corresponds to the maximum potential energy of the system? Hint A.1 Consider the position of the block Hint not displayed ANSWER: A B C D Part B Which moment corresponds to the minimum kinetic energy of the system? Hint B.1 How does the velocity change? Hint not displayed http://session.masteringphysics.com/myct/assignmentPrint?assignmentID=1114178 Page 1 of 4 MasteringPhysics: Assignment Print View 4/28/08 3:04 PM ANSWER: A B C D from equilibrium, the spring is stretched (or compressed) the most, and the . At that moment, of course, When the block is displaced a distance block is momentarily at rest. Therefore, the maximum potential energy is . Recall that . Therefore, . In general, the mechanical energy of a harmonic oscillator equals its potential energy at the maximum or minimum displacement. Part C Consider the block in the process of oscillating. ANSWER: at the equilibrium position. at the amplitude displacement. moving to the right. moving to the left. moving away from equilibrium. moving toward equilibrium. If the kinetic energy of the block is increasing, the block must be Part D Which moment corresponds to the maximum kinetic energy of the system? Hint D.1 Consider the velocity of the block Hint not displayed ANSWER: A B C D Part E Which moment corresponds to the minimum potential energy of the system? Hint E.1 Consider the distance from equilibrium Hint not displayed ANSWER: A B C D When the block is at the equilibrium position, the spring is not stretched (or compressed) at all. At that moment, of course, . Meanwhile, the block is at its maximum speed ( ). The maximum kinetic energy can then be written as . Recall that and that at the equilibrium position. Therefore, . Recalling what we found out before, , we can now conclude that , or . http://session.masteringphysics.com/myct/assignmentPrint?assignmentID=1114178 Page 2 of 4 MasteringPhysics: Assignment Print View 4/28/08 3:04 PM Part F At which moment is Hint F.1 ? Consider the potential energy Hint not displayed ANSWER: A B C D Part G Find the kinetic energy Hint G.1 of the block at the moment labeled B. How to approach the problem Hint not displayed Part G.2 Find the potential energy Part not displayed Express your answer in terms of ANSWER: = and . Energy of a Spring An object of mass attached to a spring of force constant oscillates with simple harmonic motion. The maximum . displacement from equilibrium is Part A What is the system's potential energy when its kinetic energy is equal to Hint A.1 How to approach the problem Hint not displayed Part A.2 Find the fraction of total energy that is potential energy Part not displayed Part A.3 Find the total energy of the system Part not displayed ANSWER: ? and the total mechanical energy of the system is Part B What is the object's velocity when its potential energy is Hint B.1 How to approach the problem Hint not displayed Part B.2 Find the kinetic energy Part not displayed Hint B.3 Formula for the velocity in terms of position Hint not displayed Part B.4 Find the object's position Part not displayed ANSWER: ? http://session.masteringphysics.com/myct/assignmentPrint?assignmentID=1114178 Page 3 of 4 MasteringPhysics: Assignment Print View ANSWER: 4/28/08 3:04 PM Summary 2 of 2 items complete (105.67% avg. score) 2.11 of 2 points http://session.masteringphysics.com/myct/assignmentPrint?assignmentID=1114178 Page 4 of 4 ...
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