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 3.27P
To determine
Show that Equation 3.89 can be expressed as
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f(r) = f(r) f(r)
where
|f(r) = − ▼ |
Απ
and
V fir)
d³r
|r-r'|
f₁(r) = Vx [ — ƒ d³r' -
Απ
V' X f(r')
]
r-r
Consider f(k), the Fourier transform of f(r):
1
+00
f(r)
d³k eik r f(k)
(2x)-00
Using the results above, find expressions for the Fourier transforms of the curlfree (longitudinal) and divergenceless (transverse) parts of f(r). Your results should tell you why these
parts have the names that they do!
It may help to know that the Fourier transform of 1/|r-r'| is (47/k²)e-ik·r', and that the Fourier transform of the Dirac delta function ô(r-r') is e-ik·r'
The Fourier transform of a function f (x) is defined as:
f (w) = L dx
f (ω) -.
f (x)e-iwx
Similarly, the inverse Fourier transform of a function
f (w) whose Fourier transform is known is found as:
1
f(x) = L f (w) e-iwx dw
So, find the Fourier transform f (w) for a> 0 of the
function given below, and using this result, calculate the
inverse Fourier transform to verify the form f (x) given to
you:
for x > 0
for x <0
e-ax,
f (x) = }{
Ex. 946. Given x(t)=220.0 on the interval t=[-2.80,0] milliseconds and x(t)=0
elsewhere.
Determine the Fourier transform of x(t), which is X(f) where f
Determine X (f)_real, X(f)_imaginary,
is frequency in Hertz.
X(f)_amplitude,
and X(f) radians, all evaluated at f=80.0 Hertz. ans:4
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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