Movimiento Oscilatorio

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

15.5 15.6 15.7

Q15.2

Q15.3

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. Velocity and acceleration are in the same direction halfthe time. Acceleration is always opposite to the position vector, and never in the same direction.

(b) (c) Q15.5 Q15.6

No. It is necessary to know both the position and velocity at time zero. The motion will still be simple harmonic motion, but the period of oscillation will be a bit larger. The

F kI effective mass of the system in ω = G H m JK
eff

12

will need to include a certainfraction of the mass of the

spring.

439

440 Q15.7

Oscillatory Motion

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 isFG 1 IJ FG 2 k IJ H 2π K H m K

12

= 2 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 kinetic energy 1 2 kA stays constant. The 2

1 mv 2 would increase for larger mass if the speed were constant, but here the greater2 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. Q15.9 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 ωt + φ
2

af b g af b g va x f = ±ω e A − x j aat f = −ω A cosbωt + φ g aat f = −ω xat f
v t = −ωA sin ωt +φ
2 2 2 12

Information given by equation position 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

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 to storeit in calculator memory. Q15.10 Lf Li 2 Li and T f = = = 2Ti . The period gets larger by g g g mass has no effect on the period of a simple pendulum. We have Ti = (a) Period decreases. (b) Period increases. (c) 2 times. Changing the

Q15.11 Q15.12

No change.

No, the equilibrium position of the pendulum will be shifted (angularly) towards the back of the car. The period of oscillation will...