Solucionario

olucionario15
Oscillatory Motion
CHAPTER OUTLINE
15.1 15.2 15.3 15.4 Motion of an Object Attached to a Spring The Particle in Simple Harmonic Motion Energy of the Simple Harmonic Oscillator Comparing Simple Harmonic Motion with Uniform Circular Motion The Pendulum Damped Oscillations Forced Oscillations

ANSWERS TO QUESTIONS
Q15.1 Neither are examples of simple harmonic motion, althoughthey 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. 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. Answer(a). In simple harmonic motion the acceleration is a maximum when the excursion from equilibrium is a maximum. Answer (a), by the same logic as in part (ii). Answer (c), by the same logic as in part (i). Answer (c), by the same logic as in part (i). Answer (e). The total energy is a constant.

15.5 15.6 15.7

*Q15.2 (i)

(ii)

(iii) (iv) (v) (vi) Q15.3

You can take φ = π , or equallywell, φ = −π . 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 proportional 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 > g > a = b = e = f > 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.Yes. Velocity and acceleration are in the same direction half the time. No. Acceleration is always opposite to the position vector, and never in the same direction.

(b) (c)

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*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 clamped—or its other half cutoff—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 12 ⎛ 1 ⎞ ⎛ 2k ⎞ = 2 f . The absence of a force required to the oscillation frequency in space is ⎝ 2π ⎠ ⎝ m ⎠ support the vibrating system in orbital free fall has no effect on the frequency of its vibration. *Q15.7 Answer (c). The equilibriumposition is 15 cm below the starting point. The motion is symmetric about the equilibrium position, so the two turning points are 30 cm apart. Q15.8 Since the acceleration is not constant in simple harmonic motion, none of the equations in Table 2.2 are valid. Equation x ( t ) = A cos (ω t + φ ) v ( t ) = −ω A sin (ω t + φ ) v ( x ) = ±ω ( A − x
2 2 2 2 12

Information given by equationposition as a function of time velocity as a function of time velocity as a function of position acceleration as a function of time acceleration as a function of position

a ( t ) = −ω A cos (ω t + φ ) a ( t ) = −ω x ( t )

)

The angular frequency ω appears in every equation. It is a good idea to figure out the value of angular frequency early in the solution to a problem about vibration, and...