CLASSICA
DYNAMIC
OF PARTICLES
AND SYSTEMS
FIFTH EDITION
Stephen T. Thornton
Professor ofPhysics, University of Virginia
Je...
To
Dr. Kathryn C. Thornton
Astronaut and Wife
As she soars and walks through space,
May her life be safe andfulfilling,
An...
Preface
Of the five editions of this text, this is the third edition that I have prepared. In
doing so, I have attempted t...
PREFACE
Special Feature
vii
The author has kept one popular feature ofJerry Marion's original book: the ad-
dition of hist...
PREFACE
Steve Mellema, Gustavus Adolphus
College
Adrian Melott, University ofKansas
William A. Mendoza, Jacksonville
Unive...
Contents
1 Matrices, Vectors, and Vector Calculus 1
1.1 Introduction 1
1.2 Concept of a Scalar 2
1.3 Coordinate Transforma...
CONTENTS xiii
6.4 The "Second Form" of the Euler Equation 216
6.5 Functions with Several Dependent Variables 218
6.6 Euler...
CONTENTS
12.9 The Loaded String 498
Problems 507
xv
13 Continuous Systems; Waves 512
13.1 Introduction 512
13.2 Continuous...
THOMSON
---+---BROOKS/COLE
Acquisitions Editor: Chris Hall
Assistant Editor: Alyssa White
Editorial Assistant: Seth Dobrin...
vi PREFACE
who have used this text, it would not be possible to produce a textbook of signif-
icant value to the physics c...
viii
William L. Alford, Auburn University
Philip Baldwin, University ofAkron
Robert P. Bauman, University of
Alabama, Birm...
xii
2.5
2.6
2.7
Conservation Theorems 76
Energy 82
Limitations of Newtonian Mechanics 88
Problems 90
CONTENTS
3 Oscillatio...
xiv
9.4
9.5
9.6
9.7
9.8
9.9
9.10
9.11
Angular Momentum of the System 336
Energy of the System 339
Elastic Collisions of Tw...
XVI CONTENTS
C Ordinary Differential Equations of Second Order 599
C.1 Linear Homogeneous Equations 599
C.2 Linear Inhomog...
CHAPTER
Matrices, Vectors,
and Vector Calculus
1.1 Introduction
Physical phenomena can be discussed concisely and elegantl...
2 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
We do not attempt a complete exposition of vector methods; instead, we
consid...
1.3 COORDINATE TRANSFORMATIONS 3
1.3 Coordinate Transformations
Consider a point Pwith coordinates (Xl> X2, xs) with respe...
4 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
Therefore, for Figure 1-2, we have
All = COS(X~, Xl) = cos (}
A" = cos(x], x,...
1.3 COORDINATE TRANSFORMATIONS
-
5
A point Pis represented in the (Xl> X2' XS) system by P(2, 1,3). In another coor-
dinat...
6 1 I MATRICES, VECTORS, AND VECTOR CALCULUS
Notice that the rotation operator preserves the length of the position vector...
1.4 PROPERTIES OF ROTATION MATRICES 7
The coordinate rotation may be specified by giving the cosines of all the angles
bet...
8 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
and Equation 1.13 is the orthogonality condition. The transformation matrix A...
1.5 MATRIX OPERATIONS 9
(1.16)
Therefore, the coordinates of the point P with respect to the rotated axes may
be found (se...
10 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
Equations l.I8a-d completely specify the operation of matrix multiplication
...
1.5 MATRIX OPERATIONS
-Find the product AB of the two matrices listed below:
A = (-:
1
-:)2
-1 -3
B=
C~ -n
11
Solution. We...
12
but
thus
1 / MATRICES, VECTORS, AND VECTOR CALCULUS
AB =1= BA
1.6 Further DeImitions
A transposed matrix is a matrix de...
1.6 FURTHER DEFINITIONS
Then
13
(1.26)
Using the orthogonality relation (Equation 1.13), we find
Ail + Ai2 = A~I + A~2 = 1...
14 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
1.7 Geometrical Significance
of Transformation Matrices
Consider coordinate ...
1.7 GEOMETRICAL SIGNIFICANCE OF TRANSFORMATION MATRICES 15
90° rotation
about X<j-axis
I
X3
""------I xl
FIGURE 1-6 Coordi...
16 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
Therefore, the two rotations already described may be represented by a singl...
1.7 GEOMETRICAL SIGNIFICANCE OF TRANSFORMATION MATRICES 17
Next, consider the coordinate rotation pictured in Figure 1-10 ...
18 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
Xl
(Inversion)
•
FIGURE 1-11 An object undergoes an inversion, which is a re...
1.7 GEOMETRICAL SIGNIFICANCE OF TRANSFORMATION MATRICES 19
But, because A and IL are orthogonal, At = A-I and ILt = 1L-1• ...
20
-
1 I MATRICES, VECTORS, AND VECTOR CALCULUS
Show that I A2 1 = 1 and 1 A6 1 = - 1.
Solution.
1 0
IA2 1 - 0 0
o -1
o 0
...
1.10 SCALAR PRODUCT OF TWO VECTORS
cP + 1/1 = 1/1 + c/J Commutative law
cP + (1/1 + ~) = (c/J + 1/1) + ~ Associative law
M...
22 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
, ,,,,, ,,,
',A
I
I
I
I
I
Xl
FIGURE 1-12 A vector A is shown in coordinate s...
FIGURE 1-13 The vector Ais the position vector of point (Xl> X2' X3), and vector B is
the position vector of point (Xl> X2...
24 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
There are several variants of the symbols for unit vectors; examples of the ...
1.12 VECTOR PRODUCT OF TWO VECTORS
1.12 Vector Product of Two Vectors
25
We next consider another method of combining two ...
26 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
c
B
A
FIGURE 1-14 The magnitude of the vector C determined by C = A X B has ...
1.12 VECTOR PRODUCT OF TWO VECTORS 27
1.72 can also be written as A . (B X D) = (A X B) . D, indicating that the scalar
an...
carried out. We can now use an important property of the eijk (see Problem 1-22):
(1.78)
We therefore have
(A X B) • (C x ...
1.13 DIFFERENTIATION OF A VECTOR WITH RESPECT TO A SCAlAR
1.13 Differentiation of a Vector
with Respect to a Scalar
29
(1....
30 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
(1.86b)
The derivatives of vector sums and products obey the rules of ordina...
1.14 EXAMPLES OF DERIVATIVES-VELOCITYAND ACCELERATION 31
not necessarily constant in time, and the components of the time ...
32 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
Equations 1.92 and 1.93 are perhaps easier to see by referring to Figure 1-1...
1.14 EXAMPLES OF DERIVATIVE5-VELOCIlYAND ACCELERATION 33
(The expressions for plane polar coordinates result from Equation...
34 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
(1.102)
The time derivatives of the unit vectors are found by taking the der...
1.15 ANGULAR VELOCITY
o
co
35
FIGURE 1-18 A particle moving ccw about an axis according to the right-hand rule
has an angu...
36 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
09
FIGURE 1-19 The position vector r changes to r + 5r by an infinitesimal
r...
1.16 GRADIENT OPERATOR 37
It is the fact that 80 is a vector that allows angular velocity to be represented
by a vector, b...
38 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
Substituting Equation 1.113 into Equation 1.110, we obtain
ac/JI = ~A ,.ac/J...
1.16 GRADIENT OPERATOR 39
y
o
(a)
y
O.2~O.20.1 O~~
oL.....-----'------'------'-------1.- X
x
(b)
FIGURE 1-20 (a) A three-d...
40 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
The maximum value of dc/J results when Vc/J and ds are in the same direc-
ti...
1.17 INTEGRATION OF VECTORS 41
FIGURE 1-21 The differential da is an element of area of the surface. Its direction is
norm...
42 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
c
ds
B
FIGURE 1-22 The element ds is an element oflength along the given pat...
PROBLEMS
Surface S
43
Volume V
FIGURE 1-23 The differential da is an element of area on a surface S that surrounds
a close...
44 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
1-6. Show that Equation 1.15 can be obtained by using the requirement that t...
PROBLEMS
1-14. Consider the following matrices:
45
2
3
o
-1)1 ,
1
1
-1
1
c = (~
Find the following
(a) IABI (b) AC (c) ABC...
46 1 / MATRICES, VECTORS, AND VECTOR CALCULUS
1-23. Use the Eijk notation and derive the identity
(A x B) X (C x D) = (ABD...
PROBLEMS
1-34. Evaluate the integral
47
1-35. Show that the volume common to the intersecting cylinders defined by x2 + y2...
CHAPTER.........
Newtonian Mechanics-
Single Particle
2.1 Introduction
The science of mechanics seeks to provide a precise...
-
2.2 NEWTON'S LAWS 49
of the laws in various situations.* But the logical structure of the science of me-
chanics is not ...
50 2 / NEWTONIAN MECHANICS-SINGLE PARTICLE
except insofar as it doesn't." This is hardly fair to Newton, who meant somethi...
2.2 NEWTON'S LAWS 51
(2.4a)
(2.3)
III'. Iftwo bodies constitute an ideal, isolated system, then the accelerations ofthese ...
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Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)
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Classical dynamics of particles and systems 5 ed thornton & marion (thomson learning 2004 672s)

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