Answered: Draw a phase portrait for an…

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter12: Oscillatory Motion

Section: Chapter Questions

Problem 12OQ

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Similar questions

Plot a velocity resonance curve for a driven, damped oscillator with Q = 6, and show that the full width of the curve between the points corresponding to is approximately equal to ω0/6.
A small object is attached to the end of a string to form a simple pendulum. The period of its harmonic motion is measured for small angular displacements and three lengths. For lengths of 1.000 m, 0.750 m, and 0.500 m, total time intervals for 50 oscillations of 99.8 s, 86.6 s, and 71.1s are measured with a stopwatch. (a) Determine the period of motion for each length. (b) Determine the mean value of g obtained from these three independent measurements and compare it with the accepted value. (c) Plot T2 versus L and obtain a value for g from the slope of your best-fit straight-line graph. (d) Compare the value found in part (c) with that obtained in part (b).
For a damped, driven oscillator, show that the average kinetic energy is the same at a frequency of a given number of octaves* above the kinetic energy resonance as at a frequency of the same number of octaves below resonance.
(a) If frequency is not constant for some oscillation, can the oscillation be simple harmonic motion? (b) Can you mink of any examples of harmonic motion where the frequency may depend on the amplitude?
A suspension bridge oscillates with an effective force constant of 1.00108 N/m . (a) How much energy is needed to make it oscillate with an amplitude of 0.100 m? (b) If soldiers march across the bridge with a cadence equal to the bridge’s natural frequency and impart 1.00104 J of energy each second, how long does it take for the bridge’s oscillations to go from 0.100 m to 0.500 m amplitude.
A simple harmonic oscillator has amplitude A and period T. Find the minimum time required for its position to change from x = A to x = A/2 in terms of the period T.
Obtain an expression for the fraction of a complete period that a simple harmonic oscillator spends within a small interval Δx at a position x. Sketch curves of this function versus x for several different amplitudes. Discuss the physical significance of the results. Comment on the areas under the various curves.
Allow the motion in the preceding problem to take place in a resisting medium. After oscillating for 10 s, the maximum amplitude decreases to half the initial value. Calculate (a) the damping parameter β, (b) the frequency υ1 (compare with the undamped frequency υ0), and (c) the decrement of the motion.
Consider the simplified single-piston engine in Figure CQ12.13. Assuming the wheel rotates with constant angular speed, explain why the piston rod oscillates in simple harmonic motion. Figure CQ12.13
Figure 12.4 shows two curves representing particles undergoing simple harmonic motion. The correct description of these two motions is that the simple harmonic motion of particle B is (a) of larger angular frequency and larger amplitude than that of particle A, (b) of larger angular frequency and smaller amplitude than that of particle A, (c) of smaller angular frequency and larger amplitude than that of particle A, or (d) of smaller angular frequency and smaller amplitude than that of particle A. Figure 12.4 (Quick Quiz 12.3) Two xt graphs for particles undergoing simple harmonic motion. The amplitudes and frequencies are different for the two particles.
In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the expression x=5.00cos(2t+6) where x is in centimeters and t is in seconds. At t = 0, find (a) the position of the piston, (b) its velocity, and (c) its acceleration. Find (d) the period and (e) the amplitude of the motion.
The mechanical energy of an undamped blockspring system is constant as kinetic energy transforms to elastic potential energy and vice versa. For comparison, explain what happens to the energy of a damped oscillator in terms of the mechanical, potential, and kinetic energies.

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Draw a phase portrait for an oscillating, damped spring.