Classical Dynamics of Particles and Systems
5th Edition
ISBN: 9780534408961
Author: Stephen T. Thornton, Jerry B. Marion
Publisher: Cengage Learning
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Textbook Question
Chapter 3, Problem 3.22P
Let the initial position and speed of an overdamped, nondriven oscillator be x0 and v0, respectively.
- (a) Show that the values of the amplitudes A1 and A2 in Equation 3.44 have the values
- (b) Show that when A1 = 0, the phase paths of Figure 3-11 must be along the dashed curve given by
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the general solution to a harmonic oscillator are
related.
There are two common forms for the general
solution for the position of a harmonic oscillator as
a function of time t:
1. x(t) = A cos (wt + p) and
2. x(t) = C cos (wt) + S sin (wt).
Either of these equations is a general solution of a
second-order differential equation (F= mā);
hence both must have at least two--arbitrary
constants--parameters that can be adjusted to fit
the solution to the particular motion at hand. (Some
texts refer to these arbitrary constants as boundary
values.)
Part D
Find analytic expressions for the arbitrary constants A and in Equation 1 (found in Part A) in terms of the constants C and Sin
Equation 2 (found in Part B), which are now considered as given parameters.
Express the amplitude A and phase (separated by a comma) in terms of C and S.
► View Available Hint(s)
Α, φ =
V
ΑΣΦ
?
Chapter 3 Solutions
Classical Dynamics of Particles and Systems
Ch. 3 - Prob. 3.1PCh. 3 - Allow the motion in the preceding problem to take...Ch. 3 - Prob. 3.3PCh. 3 - Prob. 3.4PCh. 3 - Obtain an expression for the fraction of a...Ch. 3 - Two masses m1 = 100 g and m2 = 200 g slide freely...Ch. 3 - Prob. 3.7PCh. 3 - Prob. 3.8PCh. 3 - A particle of mass m is at rest at the end of a...Ch. 3 - If the amplitude of a damped oscillator decreases...
Ch. 3 - Prob. 3.11PCh. 3 - Prob. 3.12PCh. 3 - Prob. 3.13PCh. 3 - Prob. 3.14PCh. 3 - Reproduce Figures 3-10b and c for the same values...Ch. 3 - Prob. 3.16PCh. 3 - For a damped, driven oscillator, show that the...Ch. 3 - Show that, if a driven oscillator is only lightly...Ch. 3 - Prob. 3.19PCh. 3 - Plot a velocity resonance curve for a driven,...Ch. 3 - Let the initial position and speed of an...Ch. 3 - Prob. 3.26PCh. 3 - Prob. 3.27PCh. 3 - Prob. 3.28PCh. 3 - Prob. 3.29PCh. 3 - Prob. 3.30PCh. 3 - Prob. 3.31PCh. 3 - Obtain the response of a linear oscillator to a...Ch. 3 - Calculate the maximum values of the amplitudes of...Ch. 3 - Consider an undamped linear oscillator with a...Ch. 3 - Prob. 3.35PCh. 3 - Prob. 3.36PCh. 3 - Prob. 3.37PCh. 3 - Prob. 3.38PCh. 3 - Prob. 3.39PCh. 3 - An automobile with a mass of 1000 kg, including...Ch. 3 - Prob. 3.41PCh. 3 - An undamped driven harmonic oscillator satisfies...Ch. 3 - Consider a damped harmonic oscillator. After four...Ch. 3 - A grandfather clock has a pendulum length of 0.7 m...
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