SM_PDF_chapter12 - Oscillatory Motion CHAPTER OUTLINE 12.1...

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331 Oscillatory Motion CHAPTER OUTLINE 12.1 Motion of a Particle Attached to a Spring 12.2 Mathematical Representation of Simple Harmonic Motion 12.3 Energy Considerations in Simple Harmonic Motion 12.4 The Simple Pendulum 12.5 The Physical Pendulum 12.6 Damped Oscillations 12.7 Forced Oscillations 12.8 Context Connection— Resonance in Structures ANSWERS TO QUESTIONS Q12.1 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. Q12.2 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. Q12.3 You can take φ π = , or equally well, φ π = − . At t = 0 , the particle is at its turning point on the negative side of equilibrium, at x A = − . Q12.4 No. It is necessary to know both the position and velocity at time zero. Q12.5 (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. Q12.6 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 1 2 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.
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332 Oscillatory Motion Q12.7 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. Q12.8 Since the acceleration is not constant in simple harmonic motion, none of the equations in Table 2.2 are valid. Equation Information given by equation x t A t a f b g = + cos ω φ position as a function of time v t A t a f b g = − + ω ω φ sin velocity as a function of time v x A x a f e j = ± ω 2 2 1 2 velocity as a function of position a t A t a f b g = − + ω ω φ 2 cos acceleration as a function of time a x x t a f
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