Chapter 15 - 15 Oscillatory Motion CHAPTER OUTLINE 15.1 15.2 15.3 15.4 Motion of an Object Attached to a Spring Mathematical Representation of

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15 CHAPTER OUTLINE 15.1 Motion of an Object Attached to a Spring 15.2 Mathematical Representation of Simple Harmonic Motion 15.3 Energy of the Simple Harmonic Oscillator 15.4 Comparing Simple Harmonic Motion with Uniform Circular Motion 15.5 The Pendulum 15.6 Damped Oscillations 15.7 Forced Oscillations Oscillatory Motion ANSWERS TO QUESTIONS Q15.1 Neither are examples of simple harmonic motion, although they are both periodic motion. In neither case is the acceleration proportional to the position. Neither motion is so smooth as SHM. The ball’s acceleration is very large when it is in contact with the floor, and the student’s when the dismissal bell rings. Q15.2 You can take φπ = , or equally well, =− . At t = 0, the particle is at its turning point on the negative side of equilibrium, at xA . Q15.3 The two will be equal if and only if the position of the particle at time zero is its equilibrium position, which we choose as the origin of coordinates. Q15.4 (a) In simple harmonic motion, one-half of the time, the velocity is in the same direction as the displacement away from equilibrium. (b) Velocity and acceleration are in the same direction half the time. (c) Acceleration is always opposite to the position vector, and never in the same direction. Q15.5 No. It is necessary to know both the position and velocity at time zero. Q15.6 The motion will still be simple harmonic motion, but the period of oscillation will be a bit larger. The effective mass of the system in ω = F H G I K J k m eff 12 will need to include a certain fraction of the mass of the spring. 439
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440 Oscillatory Motion Q15.7 We assume that the coils of the spring do not hit one another. The frequency will be higher than f by the factor 2 . When the spring with two blocks is set into oscillation in space, the coil in the center of the spring does not move. We can imagine clamping the center coil in place without affecting the motion. We can effectively duplicate the motion of each individual block in space by hanging a single block on a half-spring here on Earth. The half-spring with its center coil clamped—or its other half cut off—has twice the spring constant as the original uncut spring, because an applied force of the same size would produce only one-half the extension distance. Thus the oscillation frequency in space is 1 2 2 2 12 π F H G I K J F H G I K J = k m f . The absence of a force required to support the vibrating system in orbital free fall has no effect on the frequency of its vibration. Q15.8 No; Kinetic, Yes; Potential, No. For constant amplitude, the total energy 1 2 2 kA stays constant. The kinetic energy 1 2 2 mv would increase for larger mass if the speed were constant, but here the greater mass causes a decrease in frequency and in the average and maximum speed, so that the kinetic and potential energies at every point are unchanged.
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This homework help was uploaded on 04/13/2008 for the course PHYS 211 taught by Professor Shannon during the Spring '08 term at MSU Bozeman.

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Chapter 15 - 15 Oscillatory Motion CHAPTER OUTLINE 15.1 15.2 15.3 15.4 Motion of an Object Attached to a Spring Mathematical Representation of

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