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Unformatted text preview: 15 Oscillatory Motion CHAPTER OUTLINE 15.1 Motion of an Object Attached to a Spring 15.2 The Particle in 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 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 balls acceleration is very large when it is in contact with the fl oor, and the students when the dismissal bell rings. *Q15.2 (i) Answer (c). At 120 cm we have the midpoint between the turning points, so it is the equilibrium position and the point of maximum speed. (ii) Answer (a). In simple harmonic motion the acceleration is a maximum when the excursion from equilibrium is a maximum. (iii) Answer (a), by the same logic as in part (ii). (iv) Answer (c), by the same logic as in part (i). (v) Answer (c), by the same logic as in part (i). (vi) Answer (e). The total energy is a constant. 395 Q15.3 You can take = , or equally well, = . At t = 0, the particle is at its turning point on the negative side of equilibrium, at x A = . *Q15.4 The amplitude does not affect the period in simple harmonic motion; neither do constant forces that offset the equilibrium position. Thus a, b, e, and f all have equal periods. The period is pro- portional to the square root of mass divided by spring constant. So c, with larger mass, has larger period than a. And d with greater stiffness has smaller period. In situation g the motion is not quite simple harmonic, but has slightly smaller angular frequency and so slightly longer period. Thus the ranking is c &gt; g &gt; a = b = e = f &gt; d. *Q15.5 (a) Yes. In simple harmonic motion, one-half of the time, the velocity is in the same direction as the displacement away from equilibrium. (b) Yes. Velocity and acceleration are in the same direction half the time. (c) No. Acceleration is always opposite to the position vector, and never in the same direction. 13794_15_ch15_p395-426.indd 395 13794_15_ch15_p395-426.indd 395 12/11/06 2:36:42 PM 12/11/06 2:36:42 PM 396 Chapter 15 *Q15.6 Answer (e). 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 clampedor its other half cut offhas 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...
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