Classical Dynamics of Particles and Systems
5th Edition
ISBN: 9780534408961
Author: Stephen T. Thornton, Jerry B. Marion
Publisher: Cengage Learning
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Question
Chapter 3, Problem 3.38P
To determine
The response of a damped oscillator to a forcing function of the form
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A spring/mass/dashpot system has mass 5 kg, damping constant 70 kg/sec and spring constant 845
kg/sec/sec. Express the ODE for the system in the form
a"+ 2px' + wr = 0
Identify the natural (undamped) frequency of the spring:
wo 3=
(square Hz)
Identify the parameter p:
(Hz)
Now assume that the system has the oscillating forcing function cos(wod) with the same frequendy as the
spring's natural frequency.
+ 14a'+ 169a =
cos(wat)
Find the general solution.
A spring/mass/dashpot system has mass 9 kg, damping constant 288 kg/sec and spring constant 3249
kg/sec/sec. Express the ODE for the system in the form
x"+2px'+wx = 0
Identify the natural (undamped) frequency of the spring:
(square Hz)
wo =
Identify the parameter p:
p=
(Hz)
Now assume that the system has the oscillating forcing function cos(wot) with the same frequency as the
spring's natural frequency. Complexify the ODE and use the real part as a particular solution:
x"+32x+361x = cos(wot)
Xp=
(meters)
A particle of mass m is in a one-dimensional potential field, where its potential energy depends
on the coordinate x as U(x) = U,(1 – cos(ax)), U, and a are some constants.
Find the period of small oscillations of the particle near the equilibrium position.
d U(x).
Note: F(x) =
dx
oscillations are small→ small deviation from the equilibrium position!
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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