CHAPTER
0
Contents
Preface
v
Problems Solved in Student Solutions Manual
vii
1
Matrices, Vectors, and Vector Calculus
1
2
Newtonian Mechanics—Single Particle
29
3
Oscillations
4
Nonlinear Oscillations and Chaos
5
Gravitation
6
Some Methods in The Calculus of Variations
7
Hamilton’s Principle—Lagrangian and Hamiltonian Dynamics
8
Central-Force Motion
9
Dynamics of a System of Particles
10
Motion in a Noninertial Reference Frame
11
Dynamics of Rigid Bodies
12
Coupled Oscillations
13
Continuous Systems; Waves
435
14
Special Theory of Relativity
461
79
127
149
165
181
233
277
333
353
397
iii
iv
CONTENTS
CHAPTER
0
Preface
This Instructor’s Manual contains the solutions to all the end-of-chapter problems (but not the
appendices) from Classical Dynamics of Particles and Systems, Fifth Edition, by Stephen T.
Thornton and Jerry B. Marion. It is intended for use only by instructors using Classical Dynamics
as a textbook, and it is not available to students in any form. A Student Solutions Manual
containing solutions to about 25% of the end-of-chapter problems is available for sale to
students. The problem numbers of those solutions in the Student Solutions Manual are listed on
the next page.
As a result of surveys received from users, I continue to add more worked out examples in
the text and add additional problems. There are now 509 problems, a significant number over
the 4th edition.
The instructor will find a large array of problems ranging in difficulty from the simple
“plug and chug” to the type worthy of the Ph.D. qualifying examinations in classical mechanics.
A few of the problems are quite challenging. Many of them require numerical methods. Having
this solutions manual should provide a greater appreciation of what the authors intended to
accomplish by the statement of the problem in those cases where the problem statement is not
completely clear. Please inform me when either the problem statement or solutions can be
improved. Specific help is encouraged. The instructor will also be able to pick and choose
different levels of difficulty when assigning homework problems. And since students may
occasionally need hints to work some problems, this manual will allow the instructor to take a
quick peek to see how the students can be helped.
It is absolutely forbidden for the students to have access to this manual. Please do not
give students solutions from this manual. Posting these solutions on the Internet will result in
widespread distribution of the solutions and will ultimately result in the decrease of the
usefulness of the text.
The author would like to acknowledge the assistance of Tran ngoc Khanh (5th edition),
Warren Griffith (4th edition), and Brian Giambattista (3rd edition), who checked the solutions of
previous versions, went over user comments, and worked out solutions for new problems.
Without their help, this manual would not be possible. The author would appreciate receiving
reports of suggested improvements and suspected errors. Comments can be sent by email to
stt@virginia.edu, the more detailed the better.
Stephen T. Thornton
Charlottesville, Virginia
v
vi
PREFACE
CHAPTER
1
Matrices, Vectors,
and Vector Calculus
1-1.
x2 = x2′
x1′
45˚
x1
45˚
x3
x3′
Axes x′1 and x′3 lie in the x1 x3 plane.
The transformation equations are:
x1′ = x1 cos 45° − x3 cos 45°
x2′ = x2
x3′ = x3 cos 45° + x1 cos 45°
x1′ =
1
1
x1 −
x3
2
2
x2′ = x2
x3′ =
1
1
x1 −
x3
2
2
So the transformation matrix is:







1
2
0
1
2
0 −
1
0
1 
2

0 
1 

2 
1
2
CHAPTER 1
1-2.
a)
x3
D
E
γ
O
β
x2
B
α
A
C
x1
From this diagram, we have
OE cos α = OA
OE cos β = OB
(1)
OE cos γ = OD
Taking the square of each equation in (1) and adding, we find
2
2
2
OE cos 2 α + cos 2 β + cos 2 γ  = OA + OB + OD
2
(2)
But
2
2
2
2
2
2
OA + OB = OC
(3)
and
OC + OD = OE
(4)
Therefore,
2
2
2
OA + OB + OD = OE
2
(5)
Thus,
cos 2 α + cos 2 β + cos 2 γ = 1
(6)
b)
x3
D
E
D′
O
A A′
x1
E′
θ
B′
B
C
x2
C′
First, we have the following trigonometric relation:
OE + OE′ − 2OE OE′ cos θ = EE′
2
2
2
(7)
3
MATRICES, VECTORS, AND VECTOR CALCULUS
But,
2
2
2

 
 

EE′ = OB′ − OB  + OA′ − OA  + OD′ − OD 

 
 

2
2

 

= OE′ cos β ′ − OE cos β  + OE′ cos α ′ − OE cos α 

 



+ OE′ cos γ ′ − OE cos γ 


2
2
(8)
or,
2
2
2
EE′ = OE′ cos 2 α ′ + cos 2 β ′ + cos 2 γ ′  + OE  cos 2 α + cos 2 β + cos 2 γ 
− 2OE′ OE cos α cos α ′ + cos β cos β ′ + cos γ cos γ ′ 
= OE′ 2 + OE2 − 2OE OE′ cos α cos α ′ + cos β cos β ′ + cos γ cos γ ′ 
(9)
Comparing (9) with (7), we find
cos θ = cos α cos α ′ + cos β cos β ′ + cos γ cos γ ′
(10)
1-3.
x3
e3
O
e1
e2
e2′
e3
A
x2
e2
e1′
e3′
e1
x1
Denote the original axes by x1 , x2 , x3 , and the corresponding unit vectors by e1 , e2 , e3 . Denote
the new axes by x′1 , x′2 , x′3 and the corresponding unit vectors by e1′ , e2′ , e3′ . The effect of the
rotation is e1 → e3′ , e2 → e1′ , e3 → e2′ . Therefore, the transformation matrix is written as:
 cos ( e1′ , e1 ) cos ( e1′ , e2 ) cos ( e1′ , e3 )  0 1 0 


λ = cos ( e′2 , e1 ) cos ( e′2 , e2 ) cos ( e′2 , e3 )  = 0 0 1 

 1 0 0 

 cos ( e′3 , e1 ) cos ( e′3 , e2 ) cos ( e′3 , e3 )  
1-4.
a) Let C = AB where A, B, and C are matrices. Then,
Cij = ∑ Aik Bkj
(1)
k
(C )
t
ij
= C ji = ∑ Ajk Bki = ∑ Bki Ajk
k
k
4
CHAPTER 1
( )
Identifying Bki = Bt
ik
( )
and Ajk = At
,
kj
(C ) = ∑ (B ) ( A )
t
t
ij
t
ik
k
(2)
kj
or,
C t = ( AB) = Bt At
(3)
( AB) B−1 A−1 = I = ( B−1 A−1 ) AB
(4)
( AB) B−1 A−1 = AIA−1 = AA−1 = I
(5)
(B
(6)
t
b) To show that ( AB) = B−1 A−1 ,
−1
That is,
1-5.
−1
)
A−1 ( AB) = B−1 IB = B−1B = I
Take λ to be a two-dimensional matrix:
λ =
λ11 λ12
= λ11λ 22 − λ12 λ 21
λ 21 λ 22
(1)
Then,
2
2
2
2 2
2
λ = λ112 λ 22
− 2λ11λ 22 λ12 λ 21 + λ12
λ 21
+ ( λ11
λ 21 + λ122 λ 22
) − ( λ112 λ 212 + λ122 λ 222 )
2
(
)
(
) (
2
2
2
2
2
= λ 22
λ112 + λ122 + λ 21
λ112 + λ122 − λ112 λ 21
+ 2λ11λ 22λ12 λ 21 + λ12
λ 22
(
)(
)
2
2
2
2
= λ11
+ λ12
λ 22
+ λ 21
− ( λ11λ 21 + λ12 λ 22 )
But since λ is an orthogonal transformation matrix,
2
∑λ λ
ij
)
(2)
kj
= δ ik .
j
Thus,
2
2
λ112 + λ122 = λ 21
+ λ 22
=1
(3)
λ11λ 21 + λ12 λ 22 = 0
Therefore, (2) becomes
λ =1
2
1-6.
(4)
The lengths of line segments in the x j and x ′j systems are
L=
∑x
j
2
j
; L′ =
∑ x′
i
i
2
(1)
5
MATRICES, VECTORS, AND VECTOR CALCULUS
If L = L ′ , then
∑ x = ∑ x′
2
j
2
i
(2)
xi′ = ∑ λ ij x j
(3)
j
i
The transformation is
j
Then,

∑ x = ∑  ∑ λ
2
j
j
i
ik
k


xk   ∑ λ iA xA 

 A
(4)


= ∑ xk xA  ∑ λik λ iA 
 i

k ,A
But this can be true only if
∑λ
ik
λ iA = δ k A
(5)
i
which is the desired result.
1-7.
x3
(0,0,1)
(1,0,1)
(0,0,0)
x1
(0,1,1)
(1,1,1)
(1,0,0)
(0,1,0)
x2
(1,1,0)
There are 4 diagonals:
D1 , from (0,0,0) to (1,1,1), so (1,1,1) – (0,0,0) = (1,1,1) = D1 ;
D 2 , from (1,0,0) to (0,1,1), so (0,1,1) – (1,0,0) = (–1,1,1) = D 2 ;
D 3 , from (0,0,1) to (1,1,0), so (1,1,0) – (0,0,1) = (1,1,–1) = D 3 ; and
D 4 , from (0,1,0) to (1,0,1), so (1,0,1) – (0,1,0) = (1,–1,1) = D 4 .
The magnitudes of the diagonal vectors are
D1 = D 2 = D 3 = D 4 = 3
The angle between any two of these diagonal vectors is, for example,
D1 ⋅ D 2
(1,1,1) ⋅ ( −1,1,1) 1
= cos θ =
=
D1 D 2
3
3
6
CHAPTER 1
so that
1
θ = cos−1   = 70.5°
 3
Similarly,
D1 ⋅ D 3
D ⋅D
D ⋅D
D ⋅D
D ⋅D
1
= 1 4 = 2 3 = 2 4 = 3 4 =±
D1 D 3
D1 D 4
D2 D3
D2 D 4
D3 D 4
3
1-8.
Let θ be the angle between A and r. Then, A ⋅ r = A2 can be written as
Ar cos θ = A2
or,
r cos θ = A
(1)
This implies
QPO =
π
2
Therefore, the end point of r must be on a plane perpendicular to A and passing through P.
1-9.
a)
A = i + 2j − k
B = −2i + 3j + k
A − B = 3i − j − 2k
2
2
A − B = ( 3) + ( −1) + (−2)2 
12
A − B = 14
b)
B
θ
A
component of B along A
The length of the component of B along A is B cos θ.
A ⋅ B = AB cos θ
B cos θ =
A ⋅ B −2 + 6 − 1 3
6
=
=
or
A
2
6
6
The direction is, of course, along A. A unit vector in the A direction is
1
( i + 2j − k )
6
(2)
7
MATRICES, VECTORS, AND VECTOR CALCULUS
So the component of B along A is
1
( i + 2j − k )
2
cos θ =
c)
A⋅B
3
3
3
=
=
; θ = cos −1
AB
6 14 2 7
2 7
θ 71°
i
j
k
2 −1
1 −1
1 2
−j
+k
A × B = 1 2 −1 = i
3 1
−2 1
−2 3
−2 3 1
d)
A × B = 5i + j + 7 k
A − B = 3i − j − 2k
e)
A + B = −i + 5j
i
j
k
( A − B ) × ( A + B ) = 3 −1 − 2
−1
5
0
( A − B) × ( A + B) = 10i + 2j + 14k
1-10.
r = 2b sin ω t i + b cos ω t j
v = r = 2bω cos ω t i − bω sin ω t j
a)
a = v = −2bω 2 sin ω t i − bω 2 cos ω t j = −ω 2 r
speed = v =  4b 2ω 2 cos 2 ω t + b 2ω 2 sin 2 ω t 
= bω  4 cos 2 ω t + sin 2 ω t 
12
speed = bω  3 cos 2 ω t + 1
b)
At t = π 2ω , sin ω t = 1 , cos ω t = 0
So, at this time, v = − bω j , a = −2bω 2 i
So, θ 90°
12
12
8
CHAPTER 1
1-11.
a) Since ( A × B) i = ∑ ε ijk Aj Bk , we have
jk
(A × B) ⋅ C = ∑∑ ε ijk Aj Bk Ci
j,k
i
= C1 ( A2 B3 − A3 B2 ) − C2 ( A1B3 − A3 B1 ) + C3 ( A1B2 − A2 B1 )
C1
= A1
B1
C2
A2
B2
C3
A1
A3 = − C1
B3
B1
A2
C2
B2
A3 A1
C3 = B1
B3
C1
B3
A2
B2
C2
(1)
A3
B3 = A ⋅ ( B × C )
C3
We can also write
C1
C2
C3
B1
B2
(A × B) ⋅ C = − B1
A1
B2
A2
B3 = C1
A3 A1
C2
A2
C3 = B ⋅ ( C × A )
A3
(2)
We notice from this result that an even number of permutations leaves the determinant
unchanged.
b) Consider vectors A and B in the plane defined by e1 , e2 . Since the figure defined by A, B,
C is a parallelepiped, A × B = e3 × area of the base, but e3 ⋅ C = altitude of the parallelepiped.
Then,
C ⋅ ( A × B) = ( C ⋅ e3 ) × area of the base
= altitude × area of the base
= volume of the parallelepiped
1-12.
O
c
C
a
h
a–c
b
A
b–a
c–b
B
The distance h from the origin O to the plane defined by A, B, C is
9
MATRICES, VECTORS, AND VECTOR CALCULUS
h=
=
=
a ⋅ ( b − a ) × ( c − b)
( b − a ) × ( c − b)
a ⋅ ( b × c − a × c + a × b)
b×c−a×c+a×b
a⋅b× c
a×b+b×c+c×a
(1)
The area of the triangle ABC is:
A=
1-13.
1
1
1
( b − a ) × ( c − b) = ( a − c ) × ( b − a ) = ( c − b ) × ( a − c )
2
2
2
Using the Eq. (1.82) in the text, we have
A × B = A × ( A × X ) = ( X ⋅ A ) A − ( A ⋅ A ) X = φ A − A2 X
from which
X=
( B × A ) + φA
A2
1-14.
a)
 1 2 −1  2 1 0   1 −2 1 


 

AB =  0 3 1  0 −1 2  =  1 −2 9 
 2 0 1   1 1 3   5 3 3 
Expand by the first row.
AB = 1
−2 9
1 9
1 −2
+2
+1
3 3
5 3
5 3
AB = −104
b)
 1 2 −1  2 1   9 7 


 

AC = 0 3 1   4 3  = 13 9 
 2 0 1   1 0   5 2 
 9 7


AC = 13 9 
 5 2 
(2)
10
CHAPTER 1
5
 1 2 −1  8



ABC = A ( BC ) = 0 3 1   −2 −3 
 2 0 1   9 4 
c)
 −5 −5 


ABC =  3 −5 
 25 14 
AB − Bt At = ?
d)
 1 −2 1 


AB =  1 −2 9 
 5 3 3 
(from part a )
 2 0 1  1 0 2  1 1 5


 

B A =  1 −1 1   2 3 0  =  −2 −2 3 
 0 2 3   −1 1 1   1 9 3 
t
t
 0 −3 −4 


AB − B A =  3 0
6
 4 −6 0 
t
1-15.
t
If A is an orthogonal matrix, then
At A = 1
1 0 0 1 0 0  1 0 0


 

0 a a  0 a − a  = 0 1 0 
 0 − a a  0 a a  0 0 1 
1 0

2
0 2a
0 0
a=
1
2
0  1 0 0
 

0  = 0 1 0 
2a 2  0 0 1 
11
MATRICES, VECTORS, AND VECTOR CALCULUS
1-16.
x3
P
r
θ
r
a
θ a
x2
x1
r cos θ
r ⋅ a = constant
ra cos θ = constant
It is given that a is constant, so we know that
r cos θ = constant
But r cos θ is the magnitude of the component of r along a.
The set of vectors that satisfy r ⋅ a = constant all have the same component along a; however, the
component perpendicular to a is arbitrary.
Thus the surface represented by r ⋅ a = constant
is a plane perpendicular to a.
1-17.
a
B
C
θ
c
b
A
Consider the triangle a, b, c which is formed by the vectors A, B, C. Since
C= A−B
C = ( A − B) ⋅ ( A − B)
2
(1)
= A 2 − 2A ⋅ B + B 2
or,
C = A2 + B2 − 2 AB cos θ
2
which is the cosine law of plane trigonometry.
1-18.
Consider the triangle a, b, c which is formed by the vectors A, B, C.
a
B
c
α
C
γ
β
A
b
(2)
12
CHAPTER 1
C=A−B
(1)
C × B = ( A − B) × B
(2)
so that
but the left-hand side and the right-hand side of (2) are written as:
C × B = BC sin α e3
(3)
( A − B) × B = A × B − B × B = A × B = AB sin γ e3
(4)
and
where e3 is the unit vector perpendicular to the triangle abc. Therefore,
BC sin α = AB sin γ
(5)
or,
C
A
=
sin γ sin α
Similarly,
C
A
B
=
=
sin γ sin α sin β
(6)
which is the sine law of plane trigonometry.
1-19.
x2
a
a2
b
b2
α
β
a1
b1
x1
a) We begin by noting that
a − b = a 2 + b 2 − 2ab cos (α − β )
2
(1)
We can also write that
a − b = ( a1 − b1 ) + ( a2 − b2 )
2
2
2
= ( a cos α − b cos β ) + ( a sin α − b sin β )
2
(
)
(
2
)
= a 2 sin 2 α + cos 2 α + b 2 sin 2 β + cos 2 β − 2 ab ( cos α cos β + sin α sin β )
= a 2 + b 2 − 2ab ( cos α cos β + sin α sin β )
(2)
13
MATRICES, VECTORS, AND VECTOR CALCULUS
Thus, comparing (1) and (2), we conclude that
cos (α − β ) = cos α cos β + sin α sin β
b)
(3)
Using (3), we can find sin (α − β ) :
sin (α − β ) = 1 − cos 2 (α − β )
= 1 − cos 2 α cos 2 β − sin 2 α sin 2 β − 2cos α sin α cos β sin β
(
)
(
)
= 1 − cos 2 α 1 − sin 2 β − sin 2 α 1 − cos 2 β − 2cos α sin α cos β sin β
= sin 2 α cos 2 β − 2sin α sin β cos α cos β + cos 2 α sin 2 β
=
( sin α cos β − cos α sin β )2
(4)
so that
sin (α − β ) = sin α cos β − cos α sin β
(5)
1-20.
a) Consider the following two cases:
When i ≠ j
δ ij = 0
but ε ijk ≠ 0 .
When i = j
δ ij ≠ 0
but ε ijk = 0 .
Therefore,
∑ε
ijk
δ ij = 0
ij
b) We proceed in the following way:
When j = k, ε ijk = ε ijj = 0 .
Terms such as ε j11 ε A11 = 0 . Then,
∑ε
ijk
ε Ajk = ε i12 ε A12 + ε i13 ε A13 + ε i 21 ε A 21 + ε i 31 ε A 31 + ε i 32 ε A 32 + ε i 23 ε A 23
jk
Now, suppose i = A = 1 , then,
∑= ε
jk
123
ε 123 + ε 132 ε 132 = 1 + 1 = 2
(1)
14
CHAPTER 1
∑= ε
for i = A = 2 ,
213
ε 213 + ε 231 ε 231 = 1 + 1 = 2 . For i = A = 3 ,
jk
A = 2 gives
∑= ε
312
ε 312 + ε 321 ε 321 = 2 . But i = 1,
jk
∑ = 0 . Likewise for i = 2, A = 1 ; i = 1, A = 3 ; i = 3, A = 1 ; i = 2, A = 3 ; i = 3, A = 2 .
jk
Therefore,
∑ε
ijk
ε Ajk = 2δ iA
(2)
j,k
c)
∑ε
ijk
ε ijk = ε 123 ε 123 + ε 312 ε 312 + ε 321 ε 321 + ε 132 ε 132 + ε 213 ε 213 + ε 231 ε 231
ijk
= 1 ⋅ 1 + 1 ⋅ 1 + ( −1) ⋅ ( −1) + ( −1) ⋅ ( −1) + ( −1) ⋅ ( −1) + (1) ⋅ (1)
or,
∑ε
ijk
ε ijk = 6
(3)
ijk
( A × B)i = ∑ ε ijk Aj Bk
1-21.
(1)
jk
( A × B) ⋅ C = ∑
i
∑ε
ijk
Aj Bk Ci
(2)
jk
By an even permutation, we find
ABC = ∑ ε ijk Ai Bj Ck
ijk
1-22.
To evaluate
∑ε
ijk
ε Amk we consider the following cases:
k
a)
i= j:
∑ε
ijk
k
b)
i=A:
∑ε
ε Amk = ∑ ε iik ε Amk = 0 for all i , A , m
k
ijk
k
ε Amk = ∑ ε ijk ε imk = 1 for j = m and k ≠ i , j
k
= 0 for j ≠ m
c)
i = m:
∑ε
ijk
k
ε Amk = ∑ ε ijk ε Aik = 0 for j ≠ A
k
= −1 for j = A and k ≠ i , j
d)
j=A:
∑ε
k
ijk
ε Amk = ∑ ε ijk ε jmk = 0 for m ≠ i
k
= −1 for m = i and k ≠ i , j
(3)
MATRICES, VECTORS, AND VECTOR CALCULUS
e)
∑ε
j = m:
15
ε Amk = ∑ ε ijk ε Ajk = 0 for i ≠ A
ijk
k
k
= 1 for i = A and k ≠ i , j
f)
A = m:
∑ε
k
g)
ijk
ε Amk = ∑ ε ijk ε AAk = 0 for all i , j , k
k
i ≠ A or m : This implies that i = k or i = j or m = k.
Then,
∑ε
ijk
ε Amk = 0 for all i , j , A , m
k
h)
j ≠ A or m :
∑ε
ijk
ε Amk = 0 for all i , j , A , m
k
Now, consider δ iA δ jm − δ im δ jA and examine it under the same conditions. If this quantity
behaves in the same way as the sum above, we have verified the equation
∑ε
ijk
ε Amk = δ iA δ jm − δ im δ jA
k
a)
i = j : δ iA δ im − δ im δ iA = 0 for all i , A , m
b)
i = A : δ ii δ jm − δ im δ ji = 1 if j = m, i ≠ j , m
= 0 if j ≠ m
c)
i = m : δ iA δ ji − δ ii δ jA = −1 if j = A , i ≠ j , A
= 0 if j ≠ A
d)
j = A : δ iA δ Am − δ im δ AA = −1 if i = m, i ≠ A
= 0 if i ≠ m
e)
j = m : δ iA δ mm − δ im δ mA = 1 if i = A , m ≠ A
= 0 if i ≠ A
f)
A = m : δ iA δ jA − δ il δ jA = 0 for all i , j , A
g)
i ≠ A , m : δ iA δ jm − δ im δ jA = 0 for all i , j , A , m
h)
j ≠ A , m : δ iA δ jm − δ im δ iA = 0 for all i , j , A , m
Therefore,
∑ε
ijk
ε Amk = δ iA δ jm − δ im δ jA
k
Using this result we can prove that
A × ( B × C ) = ( A ⋅ C ) B − ( A ⋅ B) C
(1)
16
CHAPTER 1
First ( B × C ) i = ∑ ε ijk Bj Ck . Then,
jk
[ A × (B × C) ]A = ∑ ε Amn Am ( B × C ) n = ∑ ε Amn Am ∑ ε njk BjCk
mn
=
mn
∑ε
Amn
ε njk Am Bj Ck =
jkmn
∑ε
jk
Amn
ε jkn Am Bj Ck
jkmn


= ∑  ∑ ε lmn ε jkn  Am Bj Ck

jkm  n
(
)
= ∑ δ jlδ km − δ kAδ jm Am Bj Ck
jkm




= ∑ Am BA Cm − ∑ Am BmCA = BA  ∑ AmCm  − CA  ∑ Am Bm 
 m

 m

m
m
= ( A ⋅ C ) BA − ( A ⋅ B) CA
Therefore,
A × ( B × C ) = ( A ⋅ C ) B − ( A ⋅ B) C
1-23.
Write
( A × B) j = ∑ ε jAm AA Bm
Am
( C × D) k = ∑ ε krs Cr Ds
rs
Then,
(2)
17
MATRICES, VECTORS, AND VECTOR CALCULUS



[ ( A × B) × (C × D)]i = ∑ ε ijk  ∑ ε jAm AA Bm   ∑ ε krs Cr Ds 
Am
jk
=
∑ε
ijk
rs
ε jAm ε krs AA BmCr Ds
jk Amrs
=
∑ε
jAm
jAmrs
=


 ∑ ε ijk ε rsk  AA BmCr Ds
k
∑ ε (δ
jAm
ir
)
δ js − δ is δ jr AA BmCr Ds
jAmrs
(
= ∑ ε jAm AA BmCi Dj − AA Bm Di C j
jAm
)




=  ∑ ε jAm Dj AA Bm  Ci −  ∑ ε jAm C j AA Bm  Di
 jAm

 jAm

= (ABD)Ci − (ABC)Di
Therefore,
[( A × B) × (C × D) ] = (ABD)C − (ABC)D
1-24.
Expanding the triple vector product, we have
e × ( A × e) = A ( e ⋅ e) − e ( A ⋅ e)
(1)
A ( e ⋅ e) = A
(2)
A = e ( A ⋅ e) + e × ( A × e)
(3)
But,
Thus,
e(A · e) is the component of A in the e direction, while e × (A × e) is the component of A
perpendicular to e.
18
CHAPTER 1
1-25.
er
eφ
θ
φ
eθ
The unit vectors in spherical coordinates are expressed in terms of rectangular coordinates by
eθ = ( cos θ cos φ , cos θ sin φ , − sin θ ) 


eφ = ( − sin φ , cos φ , 0 )


er = ( sin θ cos φ , sin θ sin φ , cos θ ) 
(1)
Thus,
(
e θ = −φ cos θ sin φ − θ sin θ cos φ , φ cos θ cos φ − θ sin θ sin φ , − θ cos θ
)
= −θer + φ cos θ eφ
(2)
Similarly,
(
e φ = −φ cos φ , − φ sin φ , 0
)
= −φ cos θ eθ − φ sin θ er
(3)
e r = φ sin θ eφ + θeθ
(4)
Now, let any position vector be x. Then,
x = rer
(
(5)
)
x = re r + re r = r φ sin θ eφ + θeθ + re r
= rφ sin θ eφ + rθeθ + rer
(
(6)
(
)
)
cos θ + rφ sin θ e + rφ sin θ e + rθ + rθ e + rθe + x = rφ sin θ + rθφ
rer + re r
φ
φ
θ
θ
(
(
)
)
cos θ + rφ sin θ e + = 2rφ sin θ + 2rθφ
r − rφ 2 sin 2 θ − rθ 2 er
φ
(
)
+ 2rθ + rθ − rφ 2 sin θ cos θ eθ
or,
(7)
19
MATRICES, VECTORS, AND VECTOR CALCULUS
( )
1 d 2 2

r − rθ 2 − rφ 2 sin 2 θ  er + 
r θ − rφ sin θ cos θ  eθ
x = a = 
 r dt

 1

d 2
+
r φ sin 2 θ  eφ
 r sin θ dt

(
1-26.
)
(8)
When a particle moves along the curve
r = k (1 + cos θ )
(1)
we have
r = − kθ sin θ



2
r = − k θ cos θ + θ sin θ  
Now, the velocity vector in polar coordinates is [see Eq. (1.97)]
(2)
v = rer + rθ eθ
(3)
so that
v 2 = v = r 2 + r 2θ 2
2
(
)
= k 2θ 2 sin 2 θ + k 2 1 + 2 cos θ + cos 2 θ θ 2
= k 2θ 2 2 + 2 cos θ 
(4)
and v 2 is, by hypothesis, constant. Therefore,
θ =
v2
2k (1 + cos θ )
(5)
v
2kr
(6)
2
Using (1), we find
θ =
Differentiating (5) and using the expression for r , we obtain
θ =
v 2 sin θ
v 2 sin θ
=
2
4r 2
4 k 2 (1 + cos θ )
(7)
The acceleration vector is [see Eq. (1.98)]
(
)
(
)
a = r − rθ 2 er + rθ + 2rθ eθ
so that
(8)
20
CHAPTER 1
a ⋅ er = r − rθ 2
(
)
= − k θ 2 cos θ + θ sin θ − k (1 + cos θ ) θ 2


θ 2 sin 2 θ
= − k θ 2 cos θ +
+ (1 + cos θ ) θ 2 
2 (1 + cos θ )




1 − cos 2 θ
= − kθ 2  2 cos θ +
+ 1
2 (1 + cos θ ) 

3
= − kθ 2 (1 + cos θ )
2
(9)
or,
3 v2
4 k
(10)
3 v 2 sin θ
4 k 1 + cos θ
(11)
( a ⋅ er ) 2 + ( a ⋅ eθ ) 2
(12)
3 v2
4 k
(13)
a ⋅ er = −
In a similar way, we find
a ⋅ eθ = −
From (10) and (11), we have
a =
or,
a =
1-27.
2
1 + cos θ
Since
r × (v × r) = (r ⋅ r) v − (r ⋅ v) r
we have
d
d
r × ( v × r )] = [(r ⋅ r ) v − ( r ⋅ v) r ]
[
dt
dt
= (r ⋅ r) a + 2 (r ⋅ v) v − (r ⋅ v) v − ( v ⋅ v) r − (r ⋅ a) r
(
= r 2a + ( r ⋅ v ) v − r v2 + r ⋅ a
)
(1)
Thus,
(
d
[ r × ( v × r )] = r 2a + ( r ⋅ v ) v − r r ⋅ a + v 2
dt
)
(2)
21
MATRICES, VECTORS, AND VECTOR CALCULUS
1-28.
grad ( ln r ) = ∑
i
∂
( ln r ) ei
∂x i
(1)
where
∑x
r =
2
i
(2)
i
Therefore,
∂
1
ln r ) =
(
r
∂x i
xi
∑x
2
i
i
=
xi
r
(3)
2
so that
grad ( ln r ) =

1 
xe
2 ∑ i i

r  i
(4)
r
r2
(5)
or,
grad ( ln r ) =
1-29. Let r 2 = 9 describe the surface S1 and x + y + z 2 = 1 describe the surface S2 . The angle θ
between S1 and S2 at the point (2,–2,1) is the angle between the normals to these surfaces at the
point. The normal to S1 is
(
)
(
grad ( S1 ) = grad r 2 − 9 = grad x 2 + y 2 + z 2 − 9
= ( 2xe1 + 2 ye2 + 2ze3 )
x = 2, y = 2, z = 1
)
(1)
= 4e1 − 4e2 + 2e3
In S2 , the normal is:
(
)
grad ( S2 ) = grad x + y + z 2 − 1
= ( e1 + e2 + 2ze3 )
= e1 + e2 + 2e3
Therefore,
x = 2, y =−2, z = 1
(2)
22
CHAPTER 1
cos θ =
=
grad ( S1 ) ⋅ grad ( S2 )
grad ( S1 ) grad ( S2 )
( 4e1 − 4e2 + 2e3 ) ⋅ (e1 + e2 + 2e3 )
(3)
6 6
or,
cos θ =
4
(4)
6 6
from which
θ = cos −1
3
1-30.
grad ( φψ ) = ∑ ei
i =1
(5)
 ∂ψ ∂ φ 
∂ ( φψ )
ψ
= ∑ ei φ
+
∂x i
i
 ∂x i ∂x i 
= ∑ ei φ
i
6
= 74.2°
9
∂ψ
∂φ
ψ
+ ∑ ei
∂x i
∂x i
i
Thus,
grad ( φψ ) = φ grad ψ + ψ grad φ
1-31.
a)
12
 
∂rn
∂ 
n
2
 ∑ xj  
= ∑ ei
grad r = ∑ ei
∂x i
∂xi  j
 
i =1

n
3
n
2
n
= ∑ ei 2xi  ∑ x 2j 
2 j

i
n

2
= ∑ ei xi n  ∑ x 2j 
 j

i
−1
−1
= ∑ ei x i n r ( n − 2)
(1)
i
Therefore,
grad r n = nr ( n − 2) r
(2)
23
MATRICES, VECTORS, AND VECTOR CALCULUS
b)
3
grad f ( r ) = ∑ ei
i =1
∂f ( r ) 3 ∂ f ( r ) ∂ r
= ∑ ei
∂x i
∂ r ∂x i
i =1

∂ 
= ∑ ei
x 2j 
∑

∂x i  j

i


= ∑ ei xi  ∑ x 2j 
 j

i
= ∑ ei
i
12
−1 2
∂f ( r )
∂r
∂f ( r )
∂r
x i ∂f
r dr
(3)
Therefore,
grad f ( r ) =
r ∂f ( r )
r ∂r
(4)
c)
12
 
∂ 2 ln r
∂2  
2
∇ ( ln r ) = ∑
= ∑ 2  ln  ∑ x j  
∂xi2
∂x i   j
 
i

2
−1 2
1



 ⋅ 2xi  ∑ x 2j 

2




∂
j
=∑


12


i ∂x i 

2


 ∑ xj 
 j



∂
=∑
i ∂x i
−1
 
 
 xi  ∑ x 2j  
  j
 



= ∑ ( − xi )( 2xi )  ∑ x 2j 
 j

i
(
)( )
= ∑ −2x 2j r 2
i
=−
−2
−2

∂x 
+ ∑ i  ∑ x 2j 

i ∂x i  j
−1
1
+ 3 2 
r 
2r 2 3
1
+ 2 = 2
4
r
r
r
(5)
or,
∇ 2 ( ln r ) =
1
r2
(6)
24
1-32.
CHAPTER 1
Note that the integrand is a perfect differential:
2ar ⋅ r + 2br ⋅ r = a
d
d
( r ⋅ r ) + b ( r ⋅ r )
dt
dt
(1)
Clearly,
∫ ( 2ar ⋅ r + 2br ⋅ r ) dt = ar
1-33.
2
+ br 2 + const.
(2)
Since
d  r  rr − rr r rr
=
= − 2
dt  r 
r2
r r
(1)
we have
∫
d r 
 r rr 
 r − r 2  dt = ∫ dt  r  dt
(2)
r
 r rr 
 r − r 2  dt = r + C
(3)
from which
∫
where C is the integration constant (a vector).
1-34.
First, we note that
(
)
d
=A
×A
+A×A
A×A
dt
(1)
But the first term on the right-hand side vanishes. Thus,
∫ ( A × A ) dt = ∫ dt ( A × A ) dt
(2)
∫ ( A × A ) dt = A × A + C
(3)
d
so that
where C is a constant vector.
25
MATRICES, VECTORS, AND VECTOR CALCULUS
1-35.
y
x
z
We compute the volume of the intersection of the two cylinders by dividing the intersection
volume into two parts. Part of the common volume is that of one of the cylinders, for example,
the one along the y axis, between y = –a and y = a:
( )
V1 = 2 π a 2 a = 2π a 3
(1)
The rest of the common volume is formed by 8 equal parts from the other cylinder (the one
along the x-axis). One of these parts extends from x = 0 to x = a, y = 0 to y = a 2 − x 2 , z = a to
z = a 2 − x 2 . The complementary volume is then
a2 − x 2
a
V2 = 8 ∫ dx ∫
0
0
dy ∫
a2 − x 2
a
dz
= 8 ∫ dx a 2 − x 2  a 2 − x 2 − a 
0


a
a

x 3 a3
x
= 8  a2 x −
− sin −1 
3
2
a 0

=
16 3
a − 2π a 3
3
(2)
Then, from (1) and (2):
V = V1 + V2 =
16 a 3
3
(3)
26
CHAPTER 1
1-36.
z
d
y
c2 = x2 + y2
x
The form of the integral suggests the use of the divergence theorem.
∫
S
A ⋅ da = ∫ ∇ ⋅ A dv
V
(1)
Since ∇ ⋅ A = 1 , we only need to evaluate the total volume. Our cylinder has radius c and height
d, and so the answer is
∫
V
dv = π c 2 d
(2)
1-37.
z
R
y
x
To do the integral directly, note that A = R3er , on the surface, and that da = daer .
∫
S
A ⋅ da = R 3
∫
S
da = R3 × 4π R 2 = 4π R 5
(1)
To use the divergence theorem, we need to calculate ∇ ⋅ A . This is best done in spherical
coordinates, where A = r 3er . Using Appendix F, we see that
∇⋅A =
1 ∂ 2
r A r = 5r 2
r 2 ∂r
(
)
(2)
Therefore,
∫
V
π
2π
0
0
∇ ⋅ A dv = ∫ sin θ dθ ∫
R
( )
dφ ∫ r 2 5r 2 dr = 4π R5
0
Alternatively, one may simply set dv = 4π r 2 dr in this case.
(3)
27
MATRICES, VECTORS, AND VECTOR CALCULUS
1-38.
z
z = 1 – x2 – y2
y
C
x2 + y2 = 1
x
By Stoke’s theorem, we have
∫ (∇ × A) ⋅ da = ∫
S
C
A ⋅ ds
(1)
The curve C that encloses our surface S is the unit circle that lies in the xy plane. Since the
element of area on the surface da is chosen to be outward from the origin, the curve is directed
counterclockwise, as required by the right-hand rule. Now change to polar coordinates, so that
we have ds = dθ eθ and A = sin θ i + cos θ k on the curve. Since eθ ⋅ i = − sin θ and eθ ⋅ k = 0 , we
have
∫
C
A ⋅ ds = ∫
2π
0
( − sin θ ) dθ = −π
2
1-39.
a) Let’s denote A = (1,0,0); B = (0,2,0); C = (0,0,3). Then AB = (−1, 2, 0) ; AC = (−1, 0, 3) ; and
AB × AC = (6, 3, 2) . Any vector perpendicular to plane (ABC) must be parallel to AB × AC , so
the unit vector perpendicular to plane (ABC) is n = (6 7 , 3 7 , 2 7 )
b) Let’s denote D = (1,1,1) and H = (x,y,z) be the point on plane (ABC) closest to H. Then
DH = ( x − 1, y − 1, z − 1) is parallel to n given in a); this means
x−1 6
= =2
y−1 3
and
x −1 6
= =3
z−1 2
Further, AH = ( x − 1, y , z) is perpendicular to n so one has 6( x − 1) + 3 y + 2 z = 0 .
Solving these 3 equations one finds
H = ( x , y , z) = (19 49 , 34 49 , 39 49) and DH =
5
7
1-40.
a)
At the top of the hill, z is maximum;
0=
∂z
= 2 y − 6 x − 18
∂x
and
0=
∂z
= 2 x − 8 y + 28
∂y
(2)
28
CHAPTER 1
so x = –2 ; y = 3, and the hill’s height is max[z]= 72 m. Actually, this is the max value of z,
because the given equation of z implies that, for each given value of x (or y), z describes an
upside down parabola in term of y ( or x) variable.
b) At point A: x = y = 1, z = 13. At this point, two of the tangent vectors to the surface of the
hill are
t1 = (1, 0,
∂z
) = (1, 0, −8)
∂x (1,1)
and
t2 = (0,1,
∂z
) = (0,1, 22)
∂y (1,1)
Evidently t1 × t2 = (8, −22,1) is perpendicular to the hill surface, and the angle θ between this
and Oz axis is
cos θ =
(0, 0,1) ⋅ (8, −22,1)
8 + 22 + 1
2
2
2
=
1
23.43
so θ = 87.55 degrees.
c) Suppose that in the α direction ( with respect to W-E axis), at point A = (1,1,13) the hill is
steepest. Evidently, dy = (tan α )dx and
dz = 2xdy + 2 ydx − 6 xdx − 8 ydy − 18 dx + 28 dy = 22(tan α − 1)dx
then
tan β =
dx 2 + dy 2
dz
=
dx cos α
−1
=
22(tan α − 1)dx 22 2 cos (α + 45)
The hill is steepest when tan β is minimum, and this happens when α = –45 degrees with
respect to W-E axis. (note that α = 135 does not give a physical answer).
1-41.
A ⋅ B = 2a( a − 1)
then A ⋅ B = 0 if only a = 1 or a = 0.
CHAPTER
2
Newtonian Mechanics—
Single Particle
2-1.
The basic equation is
F = mi xi
a)
F ( xi , t ) = f ( xi ) g ( t ) = mi xi : Not integrable
b)
F ( xi , t ) = f ( xi ) g ( t ) = mi xi
mi
(2)
dxi
= f ( xi ) g ( t )
dt
g (t)
dxi
=
dt : Integrable
f ( xi )
mi
c)
2-2.
(1)
F ( xi , xi ) = f ( xi ) g ( xi ) = mi xi : Not integrable
(3)
(4)
Using spherical coordinates, we can write the force applied to the particle as
F = Fr er + Fθ eθ + Fφ eφ
(1)
But since the particle is constrained to move on the surface of a sphere, there must exist a
reaction force − Fr er that acts on the particle. Therefore, the total force acting on the particle is
Ftotal = Fθ eθ + Fφ eφ = mr
(2)
r = Re r
(3)
The position vector of the particle is
where R is the radius of the sphere and is constant. The acceleration of the particle is
a = r = Re r
(4)
29
30
CHAPTER 2
We must now express er in terms of er , eθ , and eφ . Because the unit vectors in rectangular
coordinates, e1 , e2 , e3 , do not change with time, it is convenient to make the calculation in
terms of these quantities. Using Fig. F-3, Appendix F, we see that
er = e1 sin θ cos φ + e2 sin θ sin φ + e3 cos θ 

eθ = e1 cos θ cos φ + e2 cos θ sin φ − e3 sin θ 


eφ = −e1 sin φ + e2 cos φ

(5)
Then
(
(
)
)
er = e1 −φ sin θ sin φ + θ cos θ cos φ + e2 θ cos θ sin φ + φ sin θ cos φ − e3 θ sin θ
(6)
= eφ φ sin θ + eθ θ
Similarly,
eθ = −er θ + eφ φ cos θ
(7)
eφ = −er φ sin θ − eθ φ cos θ
(8)
And, further,
(
)
(
(
)
er = −er φ 2 sin 2 θ + θ 2 + eθ θ − φ 2 sin θ cos θ + eφ 2θφ cos θ + φ sin θ
)
(9)
which is the only second time derivative needed.
The total force acting on the particle is
Ftotal = mr = mRer
(10)
and the components are
(
Fθ = mR θ − φ 2 sin θ cos θ
(
)
Fφ = mR 2θφ cos θ + φ sin θ
)
(11)
31
NEWTONIAN MECHANICS—SINGLE PARTICLE
2-3.
y
P
v0
α
β
x
The equation of motion is
F=ma
(1)
The gravitational force is the only applied force; therefore,
Fx = mx = 0


Fy = my = − mg 
(2)
Integrating these equations and using the initial conditions,
x ( t = 0 ) = v0 cos α 

y ( t = 0 ) = v0 sin α 
(3)
We find
x ( t ) = v0 cos α


y ( t ) = v0 sin α − gt 
(4)
So the equations for x and y are
x ( t ) = v0 t cos α


1 2 
y ( t ) = v0 t sin α − gt 
2

(5)
Suppose it takes a time t0 to reach the point P. Then,
cos β = v0 t0 cos α



1
sin β = v0 t0 sin α − gt02 
2

(6)
Eliminating between these equations,


2v sin α 2v0
1
+
cos α tan β  = 0
gt0  t0 − 0
2
g
g


from which
(7)
32
CHAPTER 2
t0 =
2v0
( sin α − cos α tan β )
g
(8)
2-4. One of the balls’ height can be described by y = y0 + v0 t − gt 2 2 . The amount of time it
takes to rise and fall to its initial height is therefore given by 2v0 g . If the time it takes to cycle
the ball through the juggler’s hands is τ = 0.9 s , then there must be 3 balls in the air during that
time τ. A single ball must stay in the air for at least 3τ, so the condition is 2v0 g ≥ 3τ , or
v0 ≥ 13.2 m ⋅ s −1 .
2-5.
flightpath
N
er
mg
plane
point of maximum
acceleration
(
)
a) From the force diagram we have N − mg = mv 2 R er . The acceleration that the pilot feels is
(
)
N m = g + mv R er , which has a maximum magnitude at the bottom of the maneuver.
2
b) If the acceleration felt by the pilot must be less than 9g, then we have
(
3 ⋅ 330 m ⋅ s −1
v2
R≥
=
8g
8 ⋅ 9.8 m ⋅ s −2
)
12.5 km
(1)
A circle smaller than this will result in pilot blackout.
2-6.
Let the origin of our coordinate system be at the tail end of the cattle (or the closest cow/bull).
a) The bales are moving initially at the speed of the plane when dropped. Describe one of
these bales by the parametric equations
x = x 0 + v0 t
(1)
33
NEWTONIAN MECHANICS—SINGLE PARTICLE
y = y0 −
1 2
gt
2
(2)
where y0 = 80 m , and we need to solve for x0 . From (2), the time the bale hits the ground is
τ = 2 y0 g . If we want the bale to land at x (τ ) = −30 m , then x0 = x (τ ) − v0τ . Substituting
v0 = 44.4 m ⋅ s -1 and the other values, this gives x0
210 m behind the cattle.
−210 m . The rancher should drop the bales
b) She could drop the bale earlier by any amount of time and not hit the cattle. If she were late
by the amount of time it takes the bale (or the plane) to travel by 30 m in the x-direction, then
she will strike cattle. This time is given by ( 30 m ) v0 0.68 s .
2-7.
Air resistance is always anti-parallel to the velocity. The vector expression is:
W=
1
1
 v
cw ρ Av 2  −  = − cw ρ Avv
2
2
 v
(1)
Including gravity and setting Fnet = ma , we obtain the parametric equations
x = − bx x 2 + y 2
(2)
y = −by x 2 + y 2 − g
(3)
where b = cw ρ A 2m . Solving with a computer using the given values and ρ = 1.3 kg ⋅ m -3 , we
find that if the rancher drops the bale 210 m behind the cattle (the answer from the previous
problem), then it takes 4.44 s to land 62.5 m behind the cattle. This means that the bale
should be dropped at 178 m behind the cattle to land 30 m behind. This solution is what is
plotted in the figure. The time error she is allowed to make is the same as in the previous
problem since it only depends on how fast the plane is moving.
80
y (m)
60
40
20
0
–200
–180
–160
–140
–120
x (m)
With air resistance
No air resistance
–100
–80
–60
–40
34
CHAPTER 2
2-8.
y
v0
P
Q
h
α
x
From problem 2-3 the equations for the coordinates are
x = v0 t cos α
y = v0 t sin α −
1 2
gt
2
(1)
(2)
In order to calculate the time when a projective reaches the ground, we let y = 0 in (2):
v0 t sin α −
t=
1 2
gt = 0
2
2v0
sin α
g
(3)
(4)
Substituting (4) into (1) we find the relation between the range and the angle as
x=
The range is maximum when 2α =
π
2
v02
sin 2α
g
, i.e., α =
π
4
. For this value of α the coordinates become




v
1
x = 0 t − gt 2 

2
2
x=
(5)
v0
t
2
(6)
Eliminating t between these equations yields
x2 −
v02
v2
x+ 0 y=0
g
g
(7)
We can find the x-coordinate of the projectile when it is at the height h by putting y = h in (7):
x2 −
v02
v2 h
x+ 0 =0
g
g
(8)
This equation has two solutions:

v02 v02
−
v02 − 4 gh 
2g 2g


v02 v02

2
x2 =
v0 − 4 gh 
+
2g 2g

x1 =
(9)
35
NEWTONIAN MECHANICS—SINGLE PARTICLE
where x1 corresponds to the point P and x2 to Q in the diagram. Therefore,
d = x2 − x1 =
v0
g
v02 − 4 gh
(10)
2-9.
a) Zero resisting force ( Fr = 0 ):
The equation of motion for the vertical motion is:
F = ma = m
dv
= − mg
dt
(1)
Integration of (1) yields
v = − gt + v0
(2)
where v0 is the initial velocity of the projectile and t = 0 is the initial time.
The time tm required for the projectile to reach its maximum height is obtained from (2). Since
tm corresponds to the point of zero velocity,
v ( tm ) = 0 = v0 − gtm ,
(3)
we obtain
tm =
v0
g
(4)
b) Resisting force proportional to the velocity ( Fr = − kmv ) :
The equation of motion for this case is:
F=m
dv
= − mg − kmv
dt
(5)
where –kmv is a downward force for t < tm′ and is an upward force for t > tm′ . Integrating, we
obtain
v (t) = −
g kv0 + g − kt
e
+
k
k
(6)
For t = tm , v(t) = 0, then from (6),
v0 =
(
)
g ktm
e −1
k
(7)
which can be rewritten as
 kv 
ktm = ln 1 + 0 
g 

Since, for small z (z
1) the expansion
(8)
36
CHAPTER 2
1
1
ln (1 + z ) = z − z 2 + z 3
2
3
(9)
is valid, (8) can be expressed approximately as
v
tm = 0
g
2
 kv

1  kv 
1 − 0 +  0  − …
2g 3  2g 


(10)
which gives the correct result, as in (4) for the limit k → 0.
2-10. The differential equation we are asked to solve is Equation (2.22), which is x = − kx .
Using the given values, the plots are shown in the figure. Of course, the reader will not be able
to distinguish between the results shown here and the analytical results. The reader will have to
take the word of the author that the graphs were obtained using numerical methods on a
computer. The results obtained were at most within 10 −8 of the analytical solution.
v vs t
v (m/s)
10
5
0
5
10
15
20
25
30
20
25
30
t (s)
x vs t
x (m)
100
50
0
0
5
10
15
t (s)
v vs x
v (m/s)
10
5
0
0
20
40
60
80
100
x (m)
2-11.
The equation of motion is
m
d2 x
= − kmv 2 + mg
dt 2
This equation can be solved exactly in the same way as in problem 2-12 and we find
(1)
37
NEWTONIAN MECHANICS—SINGLE PARTICLE
 g − kv02 
1
log 
2
2k
 g − kv 
x=
(2)
where the origin is taken to be the point at which v = v0 so that the initial condition is
x ( v = v0 ) = 0 . Thus, the distance from the point v = v0 to the point v = v1 is
 g − kv02 
1
s ( v0 → v1 ) =
log 
2 
2k
 g − kv1 
2-12.
(3)
The equation of motion for the upward motion is
m
d2 x
= − mkv 2 − mg
2
dt
(1)
Using the relation
d 2 x dv dv dx
dv
=
=
=v
2
dt
dt dx dt
dx
(2)
v dv
= − dx
kv 2 + g
(3)
we can rewrite (1) as
Integrating (3), we find
(
)
1
log kv 2 + g = − x + C
2k
(4)
where the constant C can be computed by using the initial condition that v = v0 when x = 0:
(
)
C=
1
log kv02 + g
2k
x=
kv 2 + g
1
log 02
kv + g
2k
(6)
d2 x
= − mkv 2 + mg
dt 2
(7)
(5)
Therefore,
Now, the equation of downward motion is
m
This can be rewritten as
v dv
= dx
− kv 2 + g
(8)
Integrating (8) and using the initial condition that x = 0 at v = 0 (w take the highest point as the
origin for the downward motion), we find
38
CHAPTER 2
g
1
log
g − kv 2
2k
x=
(9)
At the highest point the velocity of the particle must be zero. So we find the highest point by
substituting v = 0 in (6):
xh =
kv 2 + g
1
log 0
g
2k
(10)
Then, substituting (10) into (9),
kv 2 + g 1
g
1
=
log 0
log
2k
2k
g
g − kv 2
(11)
g 2
v0
k
v=
g
v02 +
k
(12)
Solving for v,
We can find the terminal velocity by putting x → ∞ in(9). This gives
g
k
vt =
(13)
Therefore,
v=
2-13.
v0 vt
(14)
v02 + vt2
The equation of motion of the particle is
m
(
)
(1)
dv
= − k ∫ dt
+ a2
(2)
dv
= − mk v 3 + a 2 v
dt
Integrating,
∫ v(v
2
)
and using Eq. (E.3), Appendix E, we find
 v2 
1
ln  2
= − kt + C
2
2
2a
a + v 
(3)
v2
= C ′ e − At
a2 + v 2
(4)
Therefore, we have
39
NEWTONIAN MECHANICS—SINGLE PARTICLE
where A ≡ 2a 2 k and where C′ is a new constant. We can evaluate C′ by using the initial
condition, v = v0 at t = 0:
C′ =
v02
a 2 + v02
(5)
Substituting (5) into (4) and rearranging, we have
 a 2C ′e − At 
v=
− At 
 1 − C ′e 
12
=
dx
dt
(6)
Now, in order to integrate (6), we introduce u ≡ e − At so that du = –Au dt. Then,
x=
∫
=−
 a 2C ′e − At 
 1 − C e − At 
′


a C′
A
∫
12
a
dt =
A
∫
 C ′u 
 1 − C ′u 
12
du
u
du
(7)
−C ′u2 + u
Using Eq. (E.8c), Appendix E, we find
x=
a
sin −1 (1 − 2C ′u) + C ′′
A
(8)
Again, the constant C″ can be evaluated by setting x = 0 at t = 0; i.e., x = 0 at u = 1:
C ′′ = −
a
sin −1 (1 − 2C ′ )
A
(9)
Therefore, we have
(
)
a
sin −1 −2C ′ e − At + 1 − sin −1 ( −2C ′ + 1) 

A
x=
Using (4) and (5), we can write
x=
2
2

1  −1  −v 2 + a 2 
−1  − v0 + a 
−
sin
sin

 2

2
2 
2 
2ak 
 v +a 
 v0 + a  
(10)
From (6) we see that v → 0 as t → ∞. Therefore,
 −v 2 + a2 
π
lim sin −1  2
= sin −1 (1) =
2 
t →∞
2
 v +a 
(11)
Also, for very large initial velocities,
 −v 2 + a2 
π
lim sin −1  20 2  = sin −1 ( −1) = −
v0 →∞
2
 v0 + a 
Therefore, using (11) and (12) in (10), we have
(12)
40
CHAPTER 2
x (t → ∞) =
π
(13)
2ka
and the particle can never move a distance greater than π 2ka for any initial velocity.
2-14.
α
β
d
y
x
a) The equations for the projectile are
x = v0 cos α t
y = v0 sin α t −
1 2
gt
2
Solving the first for t and substituting into the second gives
y = x tan α −
gx 2
1
2 v02 cos 2 α
Using x = d cos β and y = d sin β gives
d sin β = d cos β tan α −
gd 2 cos 2 β
2v02 cos 2 α
 gd cos 2 β

− cos β tan α + sin β 
0 = d 2
2
 2v0 cos α

Since the root d = 0 is not of interest, we have
d=
=
2 ( cos β tan α − sin β ) v02 cos 2 α
g cos 2 β
2v02 cos α ( sin α cos β − cos α sin β )
g cos 2 β
d=
2v02 cos α sin (α − β )
g cos 2 β
(1)
41
NEWTONIAN MECHANICS—SINGLE PARTICLE
b) Maximize d with respect to α
2v02
d
 − sin α sin (α − β ) + cos α cos (α − β )  cos ( 2α − β )
( d) = 0 =
dα
g cos 2 β 
cos ( 2α − β ) = 0
2α − β =
α=
π
4
+
π
2
β
2
c) Substitute (2) into (1)
dmax =
2v02 
π β
π β
cos  +  sin  −  
2



 4 2
g cos β 
4 2
Using the identity
sin A − sin B = 2 cos
1
1
( A + B) sin ( A − B)
2
2
we have
π
dmax
sin − sin β v 2  1 − sin β 
2v02
2
=
⋅
= 0

2
g cos β
2
g  1 − sin 2 β 
dmax =
v02
g (1 + sin β )
2-15.
mg sin θ
mg
θ
The equation of motion along the plane is
m
dv
= mg sin θ − kmv 2
dt
(1)
Rewriting this equation in the form
1
dv
= dt
k g sin θ − v 2
k
(2)
42
CHAPTER 2
We know that the velocity of the particle continues to increase with time (i.e., dv dt > 0 ), so that
( g k ) sin θ > v2 . Therefore, we must use Eq. (E.5a), Appendix E, to perform the integration. We
find
v


tanh −1 
=t+C
g
g

sin θ
sin θ 
k
 k

1
k
1
(3)
The initial condition v(t = 0) = 0 implies C = 0. Therefore,
v=
g
sin θ tanh
k
(
)
gk sin θ t =
dx
dt
(4)
We can integrate this equation to obtain the displacement x as a function of time:
x=
g
sin θ ∫ tanh
k
(
)
gk sin θ t dt
Using Eq. (E.17a), Appendix E, we obtain
x=
(
)
ln cosh gk sin θ t
g
sin θ
+ C′
k
gk sin θ
(5)
The initial condition x(t = 0) = 0 implies C′ = 0. Therefore, the relation between d and t is
d=
1
ln cosh
k
(
gk sin θ t
)
(6)
From this equation, we can easily find
t=
( )
cosh −1 e dk
gk sin θ
(7)
2-16. The only force which is applied to the article is the component of the gravitational force
along the slope: mg sin α. So the acceleration is g sin α. Therefore the velocity and displacement
along the slope for upward motion are described by:
v = v0 − ( g sin α ) t
x = v0 t −
1
( g sin α ) t 2
2
(1)
(2)
where the initial conditions v ( t = 0 ) = v0 and x ( t = 0 ) = 0 have been used.
At the highest position the velocity becomes zero, so the time required to reach the highest
position is, from (1),
t0 =
At that time, the displacement is
v0
g sin α
(3)
43
NEWTONIAN MECHANICS—SINGLE PARTICLE
x0 =
1 v02
2 g sin α
(4)
For downward motion, the velocity and the displacement are described by
v = ( g sin α ) t
x=
(5)
1
( g sin α ) t 2
2
(6)
where we take a new origin for x and t at the highest position so that the initial conditions are
v(t = 0) = 0 and x(t = 0) = 0.
We find the time required to move from the highest position to the starting position by
substituting (4) into (6):
t′ =
v0
g sin α
(7)
t=
2v0
g sin α
(8)
Adding (3) and (7), we find
for the total time required to return to the initial position.
2-17.
v0
35˚
60 m
0.7 m
Fence
The setup for this problem is as follows:
x = v0 t cos θ
y = y0 + v0 t sin θ −
(1)
1 2
gt
2
(2)
where θ = 35 and y0 = 0.7 m . The ball crosses the fence at a time τ = R ( v0 cos θ ) , where
R = 60 m. It must be at least h = 2 m high, so we also need h − y0 = v0τ sin θ − gτ 2 2 . Solving for
v0 , we obtain
v02 =
which gives v0
25.4 m ⋅ s −1 .
gR 2
2 cos θ  R sin θ − ( h − y0 ) cos θ 
(3)
44
CHAPTER 2
2-18.
a) The differential equation here is the same as that used in Problem 2-7. It must be solved for
many different values of v0 in order to find the minimum required to have the ball go over the
fence. This can be a computer-intensive and time-consuming task, although if done correctly is
easily tractable by a personal computer. This minimum v0 is 35.2 m ⋅ s −1 , and the trajectory is
shown in Figure (a). (We take the density of air as ρ = 1.3 kg ⋅ m −3 .)
15
y (m)
10
5
0
0
10
20
30
40
50
60
x (m)
With air resistance
No air resistance
fence height
fence range
b) The process here is the same as for part (a), but now we have v0 fixed at the result just
obtained, and the elevation angle θ must be varied to give the ball a maximum height at the
fence. The angle that does this is 0.71 rad = 40.7°, and the ball now clears the fence by 1.1 m.
This trajectory is shown in Figure (b).
20
y (m)
15
10
5
0
0
10
20
30
x (m)
Flight Path
fence height
fence range
40
50
60
45
NEWTONIAN MECHANICS—SINGLE PARTICLE
2-19.
The projectile’s motion is described by
x = ( v0 cos α ) t



1
y = ( v0 sin α ) t − gt 2 
2

(1)
where v0 is the initial velocity. The distance from the point of projection is
r = x2 + y2
(2)
Since r must always increase with time, we must have r > 0 :
r=
xx + yy
>0
r
(3)
Using (1), we have
xx + yy =
1 2 3 3
g t − g ( v0 sin α ) t 2 + v02t
2
2
(4)
Let us now find the value of t which yields xx + yy = 0 (i.e., r = 0 ):
t=
3 v0 sin α v0
±
9 sin 2 α − 8
2
g
2g
(5)
For small values of α, the second term in (5) is imaginary. That is, r = 0 is never attained and the
value of t resulting from the condition r = 0 is unphysical.
Only for values of α greater than the value for which the radicand is zero does t become a
physical time at which r does in fact vanish. Therefore, the maximum value of α that insures
r > 0 for all values of t is obtained from
9 sin 2 α max − 8 = 0
(6)
or,
sin α max =
2 2
3
(7)
so that
α max ≅ 70.5°
2-20.
(8)
If there were no retardation, the range of the projectile would be given by Eq. (2.54):
v02
sin 2θ 0
g
The angle of elevation is therefore obtained from
R=
(1)
46
CHAPTER 2
sin 2θ 0 =
Rg
v02
(1000 m ) × ( 9.8 m/sec 2 )
=
(140 m/sec ) 2
= 0.50
(2)
so that
θ 0 = 15°
(3)
Now, the real range R′, in the linear approximation, is given by Eq. (2.55):
 4 kV 
R′ = R 1 −

3g 

=
v02 sin 2θ  4 kv0 sin θ 
1 −

3g
g


(4)
Since we expect the real angle θ to be not too different from the angle θ 0 calculated above, we
can solve (4) for θ by substituting θ 0 for θ in the correction term in the parentheses. Thus,
sin 2θ =
g R′
 4kv0 sin θ 0 
v 1 −

3g


(5)
2
0
Next, we need the value of k. From Fig. 2-3(c) we find the value of km by measuring the slope of
the curve in the vicinity of v = 140 m/sec. We find km ≅ (110 N ) ( 500 m/s ) ≅ 0.22 kg/s . The
curve is that appropriate for a projectile of mass 1 kg, so the value of k is
k
0.022 sec −1
(6)
Substituting the values of the various quantities into (5) we find θ = 17.1° . Since this angle is
somewhat greater than θ 0 , we should iterate our solution by using this new value for θ 0 in (5).
We then find θ = 17.4° . Further iteration does not substantially change the value, and so we
conclude that
θ = 17.4°
If there were no retardation, a projectile fired at an angle of 17.4° with an initial velocity of
140 m/sec would have a range of
R=
(140 m/sec ) 2 sin 34.8°
9.8 m/sec 2
1140 m
47
NEWTONIAN MECHANICS—SINGLE PARTICLE
2-21.
x3
v0
α
x2
x1
Assume a coordinate system in which the projectile moves in the x2 − x3 plane. Then,
x2 = v0 t cos α



1
x3 = v0 t sin α − gt 2 
2

(1)
or,
r = x 2 e2 + x 3 e 3
1


= ( v0 t cos α ) e2 +  v0 t sin α − gt 2  e3


2
(2)
The linear momentum of the projectile is
p = mr = m ( v0 cos α ) e2 + ( v0 sin α − gt ) e3 
(3)
and the angular momentum is
(
)
L = r × p = ( v0 t cos α ) e2 + v0 t sin α − gt 2 e3  × m ( v0 cos α ) e2 + ( v0 sin α − gt ) e3 
(4)
Using the property of the unit vectors that ei × e j = e3 ε ijk , we find
(
)
1
mg v0 t 2 cos α e1
2
(5)
L = − ( mg v0 t cos α ) e1
(6)
L=
This gives
Now, the force acting on the projectile is
F = − mg e3
so that the torque is
1


 
N = r × F = ( v0 t cos α ) e2 +  v0 t sin α − gt 2  e3  ( − mg ) e3

 
2

= − ( mg v0 t cos α ) e1
which is the same result as in (6).
(7)
48
CHAPTER 2
2-22.
z
B
E
y
e
x
Our force equation is
F = q ( E + v × B)
(1)
a) Note that when E = 0, the force is always perpendicular to the velocity. This is a centripetal
acceleration and may be analyzed by elementary means. In this case we have also v ⊥ B so that
v × B = vB .
mv 2
= qvB
r
macentripetal =
(2)
Solving this for r
r=
mv v
=
qB ω c
(3)
with ω c ≡ qB/m .
b) Here we don’t make any assumptions about the relative orientations of v and B, i.e. the
velocity may have a component in the z direction upon entering the field region. Let
r = xi + yj + zk , with v = r and a = r . Let us calculate first the v × B term.
i
j
k
v × B = x y z = B ( yi − xj)
0 0 B
(4)
(
(5)
The Lorentz equation (1) becomes
)
F = mr = qByi + q Ey − Bx j + qEz k
Rewriting this as component equations:
x=
qB
y = ωc y
m
y=−
z=
qEy
Ey 

qB
x+
= −ω c  x − 
m
m
B

qEz
m
(6)
(7)
(8)
49
NEWTONIAN MECHANICS—SINGLE PARTICLE
The z-component equation of motion (8) is easily integrable, with the constants of integration
given by the initial conditions in the problem statement.
z ( t ) = z0 + z0 t +
qEz 2
t
2m
(9)
c) We are asked to find expressions for x and y , which we will call vx and vy , respectively.
Differentiate (6) once with respect to time, and substitute (7) for v y
Ey 

vx = ω c vy = −ω c2  vx − 
B

(10)
or
vx + ω c2 vx = ω c2
Ey
(11)
B
This is an inhomogeneous differential equation that has both a homogeneous solution (the
solution for the above equation with the right side set to zero) and a particular solution. The
most general solution is the sum of both, which in this case is
vx = C1 cos (ω c t ) + C2 sin (ω c t ) +
Ey
B
(12)
where C1 and C2 are constants of integration. This result may be substituted into (7) to get v y
v y = −C1ω c cos (ω c t ) − C2ω c sin (ω c t )
(13)
vy = −C1 sin (ω c t ) + C2 cos (ω c t ) + K
(14)
where K is yet another constant of integration. It is found upon substitution into (6), however,
that we must have K = 0. To compute the time averages, note that both sine and cosine have an
average of zero over one of their periods T ≡ 2π ω c .
x =
Ey
B
,
y =0
(15)
d) We get the parametric equations by simply integrating the velocity equations.
x=
y=
C1
ωc
C1
ωc
sin (ω c t ) −
cos (ω c t ) +
C2
ωc
C2
ωc
cos (ω c t ) +
Ey
B
t + Dx
sin (ω c t ) + Dy
(16)
(17)
where, indeed, Dx and Dy are constants of integration. We may now evaluate all the C’s and
D’s using our initial conditions x ( 0 ) = − A ω c , x ( 0 ) = Ey B , y ( 0 ) = 0 , y ( 0 ) = A . This gives us
C1 = Dx = Dy = 0 , C2 = A and gives the correct answer
x (t) =
−A
ωc
cos (ω c t ) +
Ey
B
t
(18)
50
CHAPTER 2
y(t) =
A
ωc
sin (ω c t )
(19)
These cases are shown in the figure as (i) A > Ey B , (ii) A < Ey B , and (iii) A = Ey B .
(i)
(ii)
(iii)
F ( t ) = ma ( t ) = kte − at
2-23.
(1)
with the initial conditions x(t) = v(t) = 0. We integrate to get the velocity. Showing this explicitly,
k
∫ v(0)a (t ) dt = m ∫ 0te
v( t )
t
−α t
dt
(2)
Integrating this by parts and using our initial conditions, we obtain
v(t) =
k  1 1 
1  −α t 
−
+
t


 e 
m  α 2 α  α 

(3)
By similarly integrating v(t), and using the integral (2) we can obtain x(t).
x (t) =
k  2
1
1 
2

− 3 + 2 + 2  t +  e −α t 



α α
α
m α

(4)
To make our graphs, substitute the given values of m = 1 kg, k = 1 N ⋅ s −1 , and α = 0.5 s −1 .
x ( t ) = te − t 2
(5)
v ( t ) = 4 − 2 ( t + 2) e − t 2
(6)
α ( t ) = −16 + 4t + 4 ( t + 4 ) e − t 2
(7)
51
NEWTONIAN MECHANICS—SINGLE PARTICLE
x(t)
100
50
0
0
5
10
t
15
20
0
5
10
t
15
20
0
5
10
t
15
20
v(t)
4
2
0
a(t)
1
0.5
0
2-24.
Ff
N
sθ
y
mg
co
N′
mg
mg
sin
y
x
θ
d = length of incline
s = distance skier travels
along level ground
θ
x
Ff
B
mg
While on the plane:
∑ Fy = N − mg cosθ = my = 0
∑ Fx = mg sin θ − Ff ;
so N = mg cos θ
Ff = µ N = µ mg cos θ
mg sin θ − µ mg cos θ = mx
So the acceleration down the plane is:
a1 = g ( sin θ − µ cos θ ) = constant
52
CHAPTER 2
While on level ground: N ′ = mg ; Ff = − µ mg
So
∑ Fx = mx becomes − µmg = mx
The acceleration while on level ground is
a2 = − µ g = constant
For motion with constant acceleration, we can get the velocity and position by simple
integration:
x=a
v = x = at + v0
x − x 0 = v0 t +
(1)
1 2
at
2
(2)
Solving (1) for t and substituting into (2) gives:
v − v0
=t
a
x − x0 =
v0 ( v − v0 )
a
1 ( v − v0 )
+ ⋅
2
a
2
or
2a ( x − x0 ) = v 2 − v02
Using this equation with the initial and final points being the top and bottom of the incline
respectively, we get:
2a1d = VB2
VB = speed at bottom of incline
Using the same equation for motion along the ground:
2a2 s = −VB2
(3)
Thus
a1d = − a2 s
a1 = g ( sin θ − µ cos θ )
So
gd ( sin θ − µ cos θ ) = µ gs
Solving for µ gives
µ=
d sin θ
d cos θ + s
Substituting θ = 17°, d = 100 m, s = 70 m gives
µ = 0.18
Substituting this value into (3):
a2 = − µ g
53
NEWTONIAN MECHANICS—SINGLE PARTICLE
−2µ gs = −VB2
VB = 2µ gs
VB = 15.6 m/sec
2-25.
a)
At A, the forces on the ball are:
N
mg
The track counters the gravitational force and provides centripetal acceleration
N − mg = mv 2 R
Get v by conservation of energy:
Etop = Ttop + Utop = 0 + mgh
EA = TA + U A =
1
mv 2 + 0
2
Etop = EA → v = 2 gh
So
N = mg + m2 gh R
2h 

N = mg  1 + 

R
b)
At B the forces are:
N
45˚
mg
N = mv 2 R + mg cos 45°
= mv 2 R + mg
Get v by conservation of energy. From a), Etotal = mgh .
At B, E =
1
mv 2 + mgh ′
2
2
(1)
54
CHAPTER 2
R
2
R
R cos 45˚ = R
R 45˚
2
h′
R=
R
+ h′
2
or
1 

h′ = R  1 −


2
So Etotal = TB + UB becomes:
1  1

2
mgh = mgR  1 −
 + mv

2 2
Solving for v 2

1 

2
2  gh − gR  1 −
  = v

2


Substituting into (1):
 2h  3

N = mg  + 
− 2 

R  2
c) From b) vB2 = 2 g  h − R + R
2 
(
v =  2 g h − R + R
)
2 
12
d) This is a projectile motion problem
45˚
45˚
B
A
Put the origin at A.
The equations:
x = x0 + vx 0 t
y = y0 + v y 0 t −
1 2
gt
2
become
x=
R
v
+ B t
2
2
y = h′ +
Solve (3) for t when y = 0 (ball lands).
vB
1
t − gt 2
2
2
(2)
(3)
55
NEWTONIAN MECHANICS—SINGLE PARTICLE
gt 2 − 2 vBt − 2h ′ = 0
2 vB ± 2vB2 + 8 gh′
t=
2g
We discard the negative root since it gives a negative time. Substituting into (2):
x=
2
R
v  2 vB ± 2vB + 8 gh ′ 

+ B 
2g
2
2


Using the previous expressions for vB and h′ yields
x=
e)
(
)
3


2 − 1 R + h +  h 2 − R2 + 2 R2 
2


12
U ( x) = mgy( x ) , with y(0) = h , so U ( x) has the shape of the track.
2-26. All of the kinetic energy of the block goes into compressing the spring, so that
mv 2 2 = kx 2 2 , or x = v m k 2.3 m , where x is the maximum compression and the given
values have been substituted. When there is a rough floor, it exerts a force µk mg in a direction
that opposes the block’s velocity. It therefore does an amount of work µk mgd in slowing the
block down after traveling across the floor a distance d. After 2 m of floor, the block has energy
mv 2 2 − µk mgd , which now goes into compressing the spring and still overcoming the friction
on the floor, which is kx 2 2 + µk mgx . Use of the quadratic formula gives
x=−
µ mg
k
mv 2 2µ mgd
 µ mg 
+ 
+
−
 k 
k
k
2
Upon substitution of the given values, the result is
(1)
1.12 m.
2-27.
0.6 m
To lift a small mass dm of rope onto the table, an amount of work dW = ( dm) g ( z0 − z ) must be
done on it, where z0 = 0.6 m is the height of the table. The total amount of work that needs to be
done is the integration over all the small segments of rope, giving
z0
W = ∫ (µ dz) g( z0 − z) =
0
µ gz02
2
When we substitute µ = m L = ( 0.4 kg ) ( 4 m ) , we obtain W
0.18 J .
(1)
56
CHAPTER 2
2-28.
v4
m
v
v3
v
M
before
collision
after
collision
The problem, as stated, is completely one-dimensional. We may therefore use the elementary
result obtained from the use of our conservation theorems: energy (since the collision is elastic)
and momentum. We can factor the momentum conservation equation
m1v1 + m2 v2 = m1v3 + m2 v4
(1)
out of the energy conservation equation
1
1
1
1
m1v12 + m2 v22 = m1v32 + m2 v42
2
2
2
2
(2)
v1 + v3 = v2 + v4
(3)
and get
This is the “conservation” of relative velocities that motivates the definition of the coefficient of
restitution. In this problem, we initially have the superball of mass M coming up from the
ground with velocity v = 2gh , while the marble of mass m is falling at the same velocity.
Conservation of momentum gives
Mv + m ( −v ) = Mv3 + mv4
(4)
and our result for elastic collisions in one dimension gives
v + v3 = ( − v ) + v 4
solving for v3 and v4 and setting them equal to
(5)
2 ghitem , we obtain
3 −α 
h
=
 1 + α 
2
hmarble
 1 − 3α 
hsuperball = 
h
 1 + α 
(6)
2
(7)
where α ≡ m M . Note that if α < 1 3 , the superball will bounce on the floor a second time after
the collision.
57
NEWTONIAN MECHANICS—SINGLE PARTICLE
2-29.
N
Ff
mg cos θ
y
mg
mg sin θ
θ
x
θ = tan −1 0.08 = 4.6°
∑F
y
= N − mg cos θ
= my = 0
N = mg cos θ
∑F
x
= mg sin θ − Ff
= mx
Ff = µ N = µ mg cos θ
so
mx = mg sin θ − µ mg cos θ
x = g ( sin θ − µ cos θ )
Integrate with respect to time
x = gt ( sin θ − µ cos θ ) + x0
(1)
Integrate again:
x = x0 + x0 t +
1 2
gt ( sin θ − µ cos θ )
2
Now we calculate the time required for the driver to stop for a given x0 (initial speed) by
solving Eq. (1) for t with x = 0 .
t′ = −
x0
−1
sin θ − µ cos θ )
(
g
Substituting this time into Eq. (2) gives us the distance traveled before coming to a stop.
1
( x ′ − x0 ) = x0t ′ + 2 gt ′ 2 ( sin θ − µ cos θ )
∆x = −
∆x =
x02
1 x2
−1
−1
sin θ − µ cos θ ) + g 02 ( sin θ − µ cos θ )
(
2 g
g
x02
−1
µ cos θ − sin θ )
(
2g
(2)
58
CHAPTER 2
We have θ = 4.6° , µ = 0.45 , g = 9.8 m/sec 2 .
For x0 = 25 mph = 11.2 m/sec , ∆x = 17.4 meters .
If the driver had been going at 25 mph, he could only have skidded 17.4 meters.
Therefore, he was speeding
How fast was he going?
∆x ≥ 30 meters gives x0 ≥ 32.9 mph .
T = t1 + t2
2-30.
where
T = total time = 4.021 sec.
t1 = the time required for the balloon to reach the ground.
t2 = the additional time required for the sound of the splash to reach the first
student.
We can get t1 from the equation
y = y0 + y 0 t −
1 2
gt ;
2
y0 = y 0 = 0
When t = t1 , y = –h; so (h = height of building)
−h = −
t=
1 2
gt1
2
or
t1 =
2h
g
distance sound travels h
=
speed of sound
v
Substituting into (1):
T=
2h h
+
g v
or
h
2h
+
−T =0
v
g
This is a quadratic equation in the variable
−
h=
Substituting
2
±
g
2 4T
+
g v
2
v
=
h . Using the quadratic formula, we get:
v
2g

2 gT 
 −1 ± 1 +

V 

V = 331 m/sec
g = 9.8 m/sec 2
T = 4.021 sec
and taking the positive root because it is the physically acceptable one, we get:
(1)
59
NEWTONIAN MECHANICS—SINGLE PARTICLE
h = 8.426 m1 2
h = 71 meters
2-31. For x0 ≠ 0 , example 2.10 proceeds as is until the equations following Eq. (2.78).
Proceeding from there we have
α B = x0 ≠ 0
α A = z0
so
z
x
( x − x0 ) = α0 cos α t + α0 sin α t
( y − y0 ) = y 0 t
x
z
( z − z0 ) = − α0 cos α t + α0 sin α t
Note that
( x − x 0 ) + ( z − z0 ) =
2
2
z02
α2
+
x02
α2
Thus the projection of the motion onto the x–z plane is a circle of radius
(x
α
1
2
0
+ z02
)
12
.
So the motion is unchanged except for a change in the
12
m 2
radius of the helix. The new radius is
x0 + z02 .
qB0
(
)
2-32.
The forces on the hanging mass are
T
mg
The equation of motion is (calling downward positive)
mg − T = ma
The forces on the other mass are
or
T = m ( g − a)
(1)
60
CHAPTER 2
T
N
y
x
2mg cos θ
Ff
2mg
2mg sin θ
θ
The y equation of motion gives
N − 2mg cos θ = my = 0
or
N = 2mg cos θ
(
The x equation of motion gives Ff = µ k N = 2µk mg cos θ
)
T − 2mg sin θ − 2µ k mg cos θ = ma
(2)
Substituting from (1) into (2)
mg − 2mg sin θ − 2µ k mg cos θ = 2ma
When θ = θ 0 , a = 0. So
g − 2 g sin θ 0 − 2µk g cos θ 0 = 0
1
= sin θ 0 + µ k cos θ 0
2
(
= sin θ 0 + µ k 1 − sin 2 θ 0
)
12
Isolating the square root, squaring both sides and rearranging gives
(1 + µ ) sin
2
k
2
θ 0 − sin θ 0  − µ k2  = 0
1
4

Using the quadratic formula gives
sin θ 0 =
2-33.
1 ± µk 3 + 4 µ k2
(
2 1 + µ k2
)
The differential equation to solve is
 v  2 
cW ρ Av 2
− g = g    − 1
y=
2m
  vt 

where vt = 2mg cw ρ A is the terminal velocity. The initial conditions are y0 = 100 m , and
v0 = 0 . The computer integrations for parts (a), (b), and (c) are shown in the figure.
(1)
61
y (m)
NEWTONIAN MECHANICS—SINGLE PARTICLE
100
100
100
50
50
50
0
0
2
4
6
0
0
5
v (m/s)
t (s)
10
15
0
0
5
t (s)
10
0
5
t (s)
10
0
5
t (s)
10
t (s)
0
0
–20
–5
0
–5
–10
0
2
4
6
0
5
a (m/s2)
t (s)
10
15
t (s)
0
0
0
–5
–5
–5
0
2
4
t (s)
6
0
5
10
15
t (s)
d) Taking ρ = 1.3 kg ⋅ m −3 as the density of air, the terminal velocities are 32.2, 8.0, and 11.0 (all
m ⋅ s -1 ) for the baseball, ping-pong ball, and raindrop, respectively. Both the ping-pong ball and
the raindrop essentially reach their terminal velocities by the time they hit the ground. If we
rewrite the mass as average density times volume, then we find that vt ∝ ρmaterial R . The
differences in terminal velocities of the three objects can be explained in terms of their densities
and sizes.
e) Our differential equation shows that the effect of air resistance is an acceleration that is
inversely proportional to the square of the terminal velocity. Since the baseball has a higher
terminal velocity than the ping-pong ball, the magnitude of its deceleration is smaller for a
given speed. If a person throws the two objects with the same initial velocity, the baseball goes
farther because it has less drag.
f) We have shown in part (d) that the terminal velocity of a raindrop of radius 0.004 m will be
larger than for one with radius 0.002 m ( 9.0 m ⋅ s -1 ) by a factor of 2 .
2-34.
FR
y
mg
Take the y-axis to be positive downwards. The initial conditions are y = y = 0 at t = 0.
62
CHAPTER 2
FR = α v
a)
The equation of motion is
my = m
dv
= mg − α v
dt
m dv
= dt
mg − α v
Integrating gives: −
m
α
ln ( mg − α v ) = t + C
Evaluate C using the condition v = 0 at t = 0:
−
−
So
or −
m
α
ln ( mg − α v ) +
m
α
m
ln ( mg ) = C
α
ln ( mg ) = t
 mg − α v 
 αv 
= ln 
 = ln 1 −

m
 mg 
 mg 
αt
Take the exponential of both sides and solve for v:
e −α r m = 1 −
αv
αv
mg
= 1 − e −α t m
mg
v=
mg
dy =
mg
α
α
(1 − e
(1 − e
−α t m
−α t m
)
(1)
) dt
Integrate again:
y+C =
mg  m −α t m 
t + e

α  α
y = 0 at t = 0, so:
C=
y=
mg  m 
= m2 g α 2
α  α 
mg  m
m

− + t + e −α t m 

α  α
α

(2)
Solve (3) for t and substitute into (4):
1−
αv
mg
= e −α t m
(3)
63
NEWTONIAN MECHANICS—SINGLE PARTICLE
t=−
y=
 αv 
ln 1 −

α  mg 
m
mg  m m  α v  m  α v   mg  v m  α v  
 − − ln 1 −
 − − ln 1 −
 + 1 −
 =

α  α α  mg  α  mg   α  g α  mg  
y=−
b)
(4)
mg  α v  
m
ln 1 −
v +

α 
α
 mg  
FR = β v 2
The equation of motion becomes:
m
dv
= mg − β v 2
dt
m dv
= dt
mg − β v 2
Integrate and apply the initial condition v = 0 at t = 0:
dv
∫ mg
β
From integral tables
∫a
2
−v
=
2
β
m∫
dt
dx
1
x
= tanh −1 ; so
2
−x
a
a
1
v β
tanh −1 = t + C where a ≡
a
a m
1
α
mg
β
tanh −1 0 = 0 = 0 + C
so:
1
v β
tanh −1 = t
a
a m
Solving for v:
v = a tanh
aβ t
m
dy
aβt
= a tanh
dt
m
From integral tables ∫ tanh u du = ln cosh u
So y + C =
m
β
ln cosh
aβ t
m
(5)
64
CHAPTER 2
Apply the conditions at y = 0 and t = 0
C=
m
ln ( cosh 0 ) =
β
m
β
ln 1 = 0
So
m
y=
β
ln cosh
aβ t
m
(6)
Solving (5) for t:
t=
m
αβ
tanh −1
v
a
Substituting into (6):
y=
Use the identity: tanh −1 u = cosh −1

ln cosh  tanh −1
β

m
1
1 − u2
(In our case u < 1 as it should be because
v
a 
, where u < 1 .
v
βv2
=
; and the condition that u < 1 just says that
a
mg
gravity is stronger than the retarding force, which it must be.) So
y=

ln cosh  cosh −1

β
m
y=−
 m
2
 = ln 1 − β v mg
2

β
1 − β v mg 
(
1
(
m
ln 1 − β v 2 mg
2β
)
)
−1 2
65
NEWTONIAN MECHANICS—SINGLE PARTICLE
2-35.
14
12
y (km)
10
8
6
4
2
0
0
5
10
15
20
25
30
35
x (km)
Range (km)
30
20
10
0
0
0.02
0.04
0.06
0.08
k (1/s)
We are asked to solve Equations (2.41) and (2.42), for the values k = 0, 0.005, 0.01, 0.02, 0.04, and
0.08 (all in s−1 ), with initial speed v0 = 600 m ⋅ s −1 and angle of elevation θ = 60° . The first figure
is produced by numerical solution of the differential equations, and agrees closely with Figure
2-8. Figure 2-9 can be most closely reproduced by finding the range for our values of k, and
plotting them vs. k. A smooth curve could be drawn, or more ranges could be calculated with
more values of k to fill in the plot, but we chose here to just connect the points with straight
lines.
66
CHAPTER 2
2-36.
y
θ
h
x
R
Put the origin at the initial point. The equations for the x and y motion are then
x = v0 ( cos θ ) t
y = v0 ( sin θ ) t −
1 2
gt
2
Call τ the time when the projectile lands on the valley floor. The y equation then gives
− h = v0 ( sin θ ) τ −
1 2
gτ
2
Using the quadratic formula, we may find τ
v02 sin 2 θ + 2 gh
v0 sin θ
+
τ=
g
g
(We take the positive since τ > 0 .) Substituting τ into the x equation gives the range R as a
function of θ.
R=
v02


cos θ  sin θ + sin 2 θ + x 2 
g
(1)
where we have defined x 2 ≡ 2 gh v02 . To maximize R for a given h and v0 , we set dR dθ = 0 . The
equation we obtain is
cos 2 θ − sin 2 θ − sin θ sin 2 θ + x 2 +
sin θ cos 2 θ
sin 2 θ + x 2
=0
(2)
Although it can give x = x (θ ) , the above equation cannot be solved to give θ = θ ( x ) in terms of
the elementary functions. The optimum θ for a given x is plotted in the figure, along with its
respective range in units of v02 g . Note that x = 0, which among other things corresponds to
h = 0, gives the familiar result θ = 45° and R = v02 g .
67
NEWTONIAN MECHANICS—SINGLE PARTICLE
50
45˚
40
θ
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
x
R/(v02/g)
10
5
1
0
0
1
2
3
4
5
6
7
8
x
v=α x
2-37.
Since
dv dx = −α x 2
dv dv dx dv
=
=
v then
dt dx dt dx
F=m
dv
dv
α   α 
= mv
= m   − 2 
dt
dx
 x  x 
F ( x ) = − mα 2 x 3
2-38.
v ( x ) = ax − n
a)
F=m
(
)(
dv
dv dx
dv
=m
= mv
= m ax − n − nax − n −1
dt
dx dt
dx
)
F ( x ) = − mna 2 x − ( 2 n + 1)
b)
v ( x) =
dx
= ax − n
dt
x n dx = adt
Integrate:
x n+1
= at + C
n+1
C = 0 using given initial conditions
9
10
68
CHAPTER 2
x n + 1 = ( n + 1) at
x = [ ( n + 1) at ]
1 ( n + 1)
c) Substitute x(t) into F(x):
F ( t ) = − mna 2 {( n + 1) at}

1 ( n + 1)
F ( t ) = − mna 2 [ ( n + 1) at ]


− ( 2 n + 1)
− ( 2 n + 1) ( n + 1)
2-39.
a)
F = −α e β v
dv
α
= − e βv
dt
m
∫e
−
− βv
1
dv = −
e − βv = −
β
α
m∫
α
m
dt
t+C
v = v0 at t = 0, so
−
−
(e
β
1
1
e − β v0 = C
β
− βv
)
− e − β v0 = −
α
m
t
Solving for v gives
v (t) = −
 αβ t − β v0 
ln 
+e

β  m
1
b) Solve for t when v = 0
αβ t
m
t=
+ e − β v0 = 1
m
1 − e − β v0 
αβ 
c) From a) we have
dx = −
 αβ t − β v0 
ln 
+e
 dt
β  m
1
69
NEWTONIAN MECHANICS—SINGLE PARTICLE
Using
∫ ln ( ax + b) dx =
ax + b
ln ( ax + b ) − x we obtain
a
  αβ t
  αβ t − β v0  
+ e − β v0  ln 
+e


 
1  m
m


− t
x+C =− 
β
αβ m



Evaluating C using x = 0 at t = 0 gives
C=
v0 m
αβ
e − β v0
So
x=−
mv0
αβ
e − β v0 +
t
β
−
m  αβ t − β v0   αβ t − β v0 
+e
 ln  m + e

αβ 2  m
Substituting the time required to stop from b) gives the distance required to stop
x=
m 1
− β v0
 −e
αβ  β

1 
 v0 + β  


2-40.
y
(x(t),y(t))
an
at
x
Write the velocity as v(t) = v(t)T(t). It follows that
a (t) =
dv dv
dT
T+v
=
= at T + an N
dt dt
dt
(1)
where N is the unit vector in the direction of dT dt . That N is normal to T follows from
0 = d dt ( T ⋅ T ) . Note also an is positive definite.
a) We have v = x 2 + y = Aα 5 − 4 cos α t . Computing from the above equation,
2
at =
dv 2 Aα 2 sin α t
=
dt
5 − 4 cos α t
(2)
We can get an from knowing a in addition to at . Using a = x 2 + y = Aα 2 , we get
2
an = a 2 − at2 = Aα 2
2 cos α t − 1
5 − 4 cos α t
b) Graphing an versus t shows that it has maxima at α t = nπ , where an = Aα 2 .
(3)
70
CHAPTER 2
2-41.
a)
As measured on the train:
Ti = 0 ; Tf =
1
mv 2
2
∆T =
b)
1
mv 2
2
As measured on the ground:
Ti =
1
1
2
mu 2 ; T f = m ( v + u )
2
2
∆T =
1
mv 2 + mvu
2
c) The woman does an amount of work equal to the kinetic energy gain of the ball as
measured in her frame.
W=
1
mv 2
2
d) The train does work in order to keep moving at a constant speed u. (If the train did no
work, its speed after the woman threw the ball would be slightly less than u, and the speed of
the ball relative to the ground would not be u + v.) The term mvu is the work that must be
supplied by the train.
W = mvu
2-42.
θ
Rθ
b
R
θ
From the figure, we have h(θ ) = (R + b 2) cos θ + Rθ sin θ , and the potential is U (θ ) = mgh(θ ) .
Now compute:
dU
 b

= mg  − sin θ + Rθ cos θ 
dθ
 2

(1)
d 2U
b


= mg  R −  cos θ − Rθ sin θ 
2
2
dθ


(2)
71
NEWTONIAN MECHANICS—SINGLE PARTICLE
The equilibrium point (where dU dθ = 0 ) that we wish to look at is clearly θ = 0. At that point,
we have d 2U dθ 2 = mg ( R − b 2) , which is stable for R > b 2 and unstable for R < b/ 2 . We can
use the results of Problem 2-46 to obtain stability for the case R = b 2 , where we will find that
the first non-trivial result is in fourth order and is negative. We therefore have an equilibrium at
θ = 0 which is stable for R > b 2 and unstable for R ≤ b 2 .
2-43.
F = − kx + kx 3 α 2
U ( x ) = − ∫ F dx =
1 2 1 x4
kx − k 2
2
4 α
To sketch U(x), we note that for small x, U(x) behaves like the parabola
behavior is determined by −
1 2
kx . For large x, the
2
1 x4
k
4 α2
U(x)
E0
E1
E2
x4 x5
x1 x2 x3
x
E3 = 0
E4
E=
1
mv 2 + U ( x )
2
For E = E0 , the motion is unbounded; the particle may be anywhere.
For E = E1 (at the maxima in U(x)) the particle is at a point of unstable equilibrium. It may
remain at rest where it is, but if perturbed slightly, it will move away from the equilibrium.
What is the value of E1 ? We find the x values by setting
dU
=0.
dx
0 = kx − kx 3 α 2
x = 0, ± α are the equilibrium points
U ( ±α ) = E1 =
1 2 1 2 1 2
kα − kα = kα
2
4
4
For E = E2 , the particle is either bounded and oscillates between − x2 and x2 ; or the particle
comes in from ±∞ to ± x3 and returns to ±∞.
72
CHAPTER 2
For E3 = 0 , the particle is either at the stable equilibrium point x = 0, or beyond x = ± x4 .
For E4 , the particle comes in from ±∞ to ± x5 and returns.
2-44.
θ
T
T
T
m1
m2
m1 g
m2 g
From the figure, the forces acting on the masses give the equations of motion
m1 x 1 = m1 g − T
(1)
m2 x 2 = m2 g − 2T cos θ
(2)
where x2 is related to x1 by the relation
( b − x1 )2 − d2
x2 =
(3)
4
and cos θ = d ( b − x1 ) 2  . At equilibrium, x 1 = x 2 = 0 and T = m1 g . This gives as the equilibrium
values for the coordinates
x10 = b −
4 m1 d
(4)
4 m12 − m22
m2 d
x20 =
(5)
4 m12 − m22
We recognize that our expression x10 is identical to Equation (2.105), and has the same
requirement that m2 m1 < 2 for the equilibrium to exist. When the system is in motion, the
descriptive equations are obtained from the force laws:
m1 ( x 1 − g ) =
m2 ( b − x1 )
4 x2
(x 2 − g)
(6)
To examine stability, let us expand the coordinates about their equilibrium values and look at
their behavior for small displacements. Let ξ1 ≡ x1 − x10 and ξ2 ≡ x2 − x20 . In the calculations,
take terms in ξ1 and ξ2 , and their time derivatives, only up to first order. Equation (3) then
becomes ξ2 −(m1 m2 )ξ1 . When written in terms of these new coordinates, the equation of
motion becomes
ξ1 = −
(
g 4 m12 − m22
)
32
4 m1m2 ( m1 + m2 ) d
ξ1
(7)
73
NEWTONIAN MECHANICS—SINGLE PARTICLE
which is the equation for simple harmonic motion. The equilibrium is therefore stable, when it
exists.
2-45. and 2-46.
Expand the potential about the equilibrium point
1  di u  i
∑  i x
i = n + 1 i !  dx  0
(1)
dU
1  d( n + 1)U 
= −  ( n + 1)  x n
dx
n!  dx
0
(2)
U ( x) =
∞
The leading term in the force is then
F( x ) = −
The force is restoring for a stable point, so we need F ( x > 0 ) < 0 and F ( x < 0 ) > 0 . This is never
true when n is even (e.g., U = kx 3 ), and is only true for n odd when  d( n + 1)U dx( n + 1)  0 < 0 .
2-47.
We are given U ( x) = U 0 ( a x + x a ) for x > 0 . Equilibrium points are defined by
dU dx = 0 , with stability determined by d 2U dx 2 at those points. Here we have
dU
 a 1
= U0  − 2 + 
dx
a
 x
(1)
 d 2U   2U0 
 dx 2  =  a 3  > 0

a
(2)
which vanishes at x = a. Now evaluate
indicating that the equilibrium point is stable.
25
U(x)/U0
20
15
10
5
0
0
0.5
1
x/a
1.5
2
74
CHAPTER 2
2-48. In the equilibrium, the gravitational force and the eccentric force acting on each star
must be equal
mG
πd
⇒τ =
=
2d
v
Gm2 mv 2
=
⇒v=
d2
d /2
2-49.
2π d 3 2
mG
The distances from stars to the center of mass of the system are respectively
r1 =
dm2
m1 + m2
and
r2 =
dm1
m1 + m2
At equilibrium, like in previous problem, we have
Gm1m2 m1v12
=
⇒ v1 =
d2
r1
Gm2 2
2π r1
2π d 3 2
⇒τ =
=
d(m1 + m2 )
v1
G(m1 + m2 )
The result will be the same if we consider the equilibrium of forces acting on 2nd star.
2-50.
a)
d  m0 v
dt 
v2
 1− 2

c
t

 m0 v
= F⇒∫d


v2
0

 1− 2


c
m0 v

=
= Ft ⇒ v(t) =

v2

1− 2

c
t
⇒ x(t) = ∫ v(t)dt =
0
Ft
m02 +

c2 
F 2t 2
2
m
+
− m0 

0
2

F
c
b)
v
t
c) From a) we find
t=
vm0
F 1−
Now if
F
= 10 , then
m0
F 2t 2
c2
v2
c2
75
NEWTONIAN MECHANICS—SINGLE PARTICLE
when v = c 2 , we have t =
c
= 0.55 year
10 3
when v = 99% c, we have t =
99c
= 6.67 years
10 199
2-51.
a)
m
mv0
dv
dv
b
= − bv 2 ⇒ ∫ 2 = − ∫ dt ⇒ v(t) =
dt
v
m
btv0 + m
Now let v(t) = v0/1000 , one finds t =
999m
= 138.7 hours .
v0 b
v
t
t
b)
x(t ) = ∫ vdt =
0
m  btv0 + m 
ln 

b 
m 
We use the value of t found in question a) to find the corresponding distance
x(t ) =
m
ln(1000) = 6.9 km
b
2-52.
a)
F( x ) = −
4U x 
dU
x2 
= − 20  1 − 2 
dx
a 
a 
b)
U
x
When F = 0, there is equilibrium; further when U has a local minimum (i.e. dF dx < 0 ) it is
stable, and when U has a local maximum (i.e. dF dx > 0 ) it is unstable.
76
CHAPTER 2
So one can see that in this problem x = a and x = –a are unstable equilibrium positions, and x = 0
is a stable equilibrium position.
c)
Around the origin, F ≈ −
4U 0 x
≡ − kx ⇒ ω =
a2
k
=
m
4U 0
ma 2
d) To escape to infinity from x = 0, the particle needs to get at least to the peak of the potential,
2
2U 0
mvmin
= U max = U 0 ⇒ vmin =
m
2
e) From energy conservation, we have
2
mv 2 U 0 x 2 mvmin
dx
+ 2 =
⇒
=v=
a
dt
2
2
2U 0 
x2 
 1 − 2 
m
a
We note that, in the ideal case, because the initial velocity is the escape velocity found in d),
ideally x is always smaller or equal to a, then from the above expression,


a  exp  t


m
dx
ma
a+x
ln
(
)
=
⇒
=
x
t
2U 0 ∫0 
8U 0 a − x

x2 

 1 − 2 
exp  t

a


x
t=
2
8U 0  
 −1
ma 2  
8U 0  
 +1
ma 2  
x
t
2-53.
F is a conservative force when there exists a non-singular potential function U(x) satisfying
F(x) = –grad(U(x)). So if F is conservative, its components satisfy the following relations
∂Fx ∂Fy
=
∂y
∂x
and so on.
a) In this case all relations above are satisfied, so F is indeed a conservative force.
Fx = −
bx 2
∂U
= ayz + bx + c ⇒ U = − ayzx −
− cx + f1 ( y , z)
2
∂x
(1)
where f1 ( y , z) is a function of only y and z
Fy = −
∂U
= axz + bz ⇒ U = − ayzx − byz + f 2 ( x , z)
∂y
(2)
77
NEWTONIAN MECHANICS—SINGLE PARTICLE
where f 2 ( x , z) is a function of only x and z
Fz = −
∂U
= axy + by + c ⇒ U = − ayzx − byz + f 3 ( y , z)
∂z
(3)
then from (1), (2), (3) we find that
U = − axyz − byz − cx 2 −
bx 2
+C
2
where C is a arbitrary constant.
b)
is
Using the same method we find that F in this case is a conservative force, and its potential
U = − z exp( − x) − y ln z + C
c) Using the same method we find that F in this case is a conservative force, and its potential
is ( using the result of problem 1-31b):
U = − a ln r
2-54.
a) Terminal velocity means final steady velocity (here we assume that the potato reaches this
velocity before the impact with the Earth) when the total force acting on the potato is zero.
mg = kmv
and consequently
v = g k = 1000 m/s .
b)
x
0
dv F
dx
dv
vdv
= = −( g + kv) ⇒
= dt = −
⇒ ∫ dx = − ∫
⇒
dt m
v
g + kv
g + kv
0
v0
xmax =
2-55.
g
v0 g
+ 2 ln
= 679.7 m
k k
g + kv0
where v0 is the initial velocity of the potato.
Let’s denote vx 0 and vy 0 the initial horizontal and vertical velocity of the pumpkin.
Evidently, vx 0 = vy 0 in this problem.
m
vx 0 − vxf
dvx
dv
dx
= Fx = − mkvx ⇒ −
= − dt = x ⇒ x f =
dt
vx
kvx
k
(1)
where the suffix f always denote the final value. From the second equality of (1), we have
− dt =
dvx
− kt
⇒ vxf = vx 0 e f
kvx
(2)
vx 0 (
− kt
1− e f )
k
(3)
Combining (1) and (2) we have
xf =
78
CHAPTER 2
Do the same thing with the y-component, and we have
m
dvy
dt
= Fy = − mg − mkvy ⇒ −
dvy
g + kvyf vy 0 − vyf
dy
g
= − dt =
⇒ 0 = y f = 2 ln
+
vy
g + kvy
k
g + kvy 0
k
and
− dt =
dvy
g + kvy
⇒ g + kvyf = ( g + kv0 f ) e
− kt f
(4)
(5)
From (4) and (5) with a little manipulation, we obtain
1− e
− kt f
=
gkt f
g + kvy 0
(3) and (6) are 2 equations with 2 unknowns, t f and k. We can eliminate t f , and obtain an
equation of single variable k.
xf =
Putting x f = 142 m and vx 0 = vy 0 =
vx 0 (
− kt g + kv
( gv )
1 − e f ( y0 ) x0 )
k
v0
= 38.2 m/s we can numerically solve for k and obtain
2
K= 0.00246 s −1 .
(6)
CHAPTER
3
Oscillations
3-1.
a)
1
ν0 =
2π
k
1
=
m 2π
10 4 dyne/cm 10
=
10 2 gram
2π
gram ⋅ cm
sec 2 ⋅ cm = 10 sec −1
gram
2π
or,
ν 0 ≅ 1.6 Hz
τ0 =
1
ν0
=
(1)
2π
sec
10
or,
τ 0 ≅ 0.63 sec
b)
E=
(2)
1 2 1
kA = × 10 4 × 32 dyne-cm
2
2
so that
E = 4.5 × 10 4 erg
(3)
c) The maximum velocity is attained when the total energy of the oscillator is equal to the
kinetic energy. Therefore,
1
2
mvmax
= 4.5 × 10 4 erg
2
v max =
2 × 4.5 × 10 4
100
79
80
CHAPTER 3
or,
vmax = 30 cm/sec
(4)
3-2.
a) The statement that at a certain time t = t1 the maximum amplitude has decreased to onehalf the initial value means that
xen = A0 e − βt1 =
1
A0
2
(1)
or,
e − βt1 =
1
2
(2)
so that
β=
ln 2 0.69
=
t1
t1
(3)
Since t1 = 10 sec ,
β = 6.9 × 10 −2 sec −1
b)
(4)
According to Eq. (3.38), the angular frequency is
ω 1 = ω 02 − β 2
(5)
where, from Problem 3-1, ω 0 = 10 sec −1 . Therefore,
ω 1 = (10 ) − ( 6.9 × 10 −2 )
2
2
2
 1

≅ 10 1 − ( 6.9) × 10 −6  sec −1
 2

(6)
so that
ν1 =
10
(1 − 2.40 × 10 −5 ) sec −1
2π
(7)
which can be written as
ν1 = ν 0 (1 − δ )
(8)
δ = 2.40 × 10 −5
(9)
where
That is, ν1 is only slightly different from ν 0 .
81
OSCILLATIONS
c) The decrement of the motion is defined to be e βτ 1 where τ 1 = 1 ν 1 . Then,
e βτ 1
3-3.
1.0445
The initial kinetic energy (equal to the total energy) of the oscillator is
m = 100 g and v0 = 1 cm/sec .
1
mv02 , where
2
a) Maximum displacement is achieved when the total energy is equal to the potential energy.
Therefore,
1
1
mv02 = kx02
2
2
x0 =
m
10 2
1
v0 =
×1=
cm
4
k
10
10
or,
x0 =
1
cm
10
(1)
b) The maximum potential energy is
U max =
1 2 1
kx0 = × 10 4 × 10 −2
2
2
or,
U max = 50 ergs
(2)
3-4.
a) Time average:
The position and velocity for a simple harmonic oscillator are given by
x = A sin ω 0 t
(1)
x = ω 0 A cos ω 0 t
(2)
where ω 0 = k m
The time average of the kinetic energy is
T =
where τ =
2π
is the period of oscillation.
ω0
1
τ
t +τ
∫
t
1
mx 2 dt
2
(3)
82
CHAPTER 3
By inserting (2) into (3), we obtain
T =
t+τ
1
mA2ω 02
2τ
∫ cos
2
ω 0 t dt
(4)
t
or,
T =
mA2ω 02
4
(5)
In the same way, the time average of the potential energy is
U =
=
=
1
τ
t +τ
∫
t
1 2
kx dt
2
1
kA2
2τ
t +τ
∫ sin
2
ω 0 t dt
t
kA2
4
(6)
and since ω 02 = k m , (6) reduces to
U =
mA2ω 02
4
(7)
From (5) and (7) we see that
T = U
(8)
The result stated in (8) is reasonable to expect from the conservation of the total energy.
E = T +U
(9)
This equality is valid instantaneously, as well as in the average. On the other hand, when T and
U are expressed by (1) and (2), we notice that they are described by exactly the same function,
displaced by a time τ 2 :

mA2ω 02
cos 2 ω 0 t 
2


2
mA ω 0 t
2
sin ω 0 t 
U=

2
T=
(10)
Therefore, the time averages of T and U must be equal. Then, by taking time average of (9), we
find
T = U =
E
2
b) Space average:
The space averages of the kinetic and potential energies are
(11)
83
OSCILLATIONS
A
T=
1 1
mx 2 dx
∫
A0 2
(12)
and
mω 02
1 1
U = ∫ kx 2 dx =
2A
A0 2
A
A
∫x
2
dx
(13)
0
(13) is readily integrated to give
U=
mω 02 A2
6
(14)
To integrate (12), we notice that from (1) and (2) we can write
(
x 2 = ω 02 A2 cos 2 ω 0 t = ω 02 A2 1 − sin 2 ω 0 t
(
= ω 02 A2 − x 2
)
)
(15)
Then, substituting (15) into (12), we find
T=
=
mω 02
2A
∫
mω 02
2A
 3 A3 
A − 3 


(16)
mω 02 A2
6
(17)
A
 A2 − x 2  dx
0
or,
T=2
From the comparison of (14) and (17), we see that
T = 2U
(18)
To see that this result is reasonable, we plot T = T(x) and U = U(x):

1
x2  
mω 02 A2 1 − 2  
2
 A 

1

2 2
U = mω 0 x

2
T=
mA2ω 02
(19)
Energy
U = U(x)
E = const. =
1
mA2ω 02
2
T = T(x)
–A
O
A
x
And the area between T(x) and the x-axis is just twice that between U(x) and the x-axis.
84
3-5.
CHAPTER 3
Differentiating the equation of motion for a simple harmonic oscillator,
x = A sin ω 0 t
(1)
∆x = Aω 0 cos ω 0 t ∆t
(2)
we obtain
But from (1)
sin ω 0 t =
x
A
(3)
Therefore,
cos ω 0 t = 1 − ( x A)
2
(4)
and substitution into (2) yields
∆t =
∆x
(5)
ω 0 A2 − x 2
Then, the fraction of a complete period that a simple harmonic oscillator spends within a small
interval ∆x at position x is given by
∆t
τ
=
∆x
ω 0τ A2 − x 2
=
∆x
2π
A2 − x 2
(6)
∆t⁄τ
x
–A3
–A2
–A1
A1
A2
A3
This result implies that the harmonic oscillator spends most of its time near x = ±A, which is
obviously true. On the other hand, we obtain a singularity for ∆t τ at x = ±A. This occurs
because at these points x = 0, and (2) is not valid.
3-6.
k
m1
m2
x1
x2
x
Suppose the coordinates of m1 and m2 are x1 and x2 and the length of the spring at
equilibrium is . Then the equations of motion for m1 and m2 are
m1 x1 = − k ( x1 − x2 +
)
(1)
m2 x2 = − k ( x2 − x1 +
)
(2)
85
OSCILLATIONS
From (2), we have
x1 =
1
( m2 x2 + kx2 − k
k
)
(3)
Substituting this expression into (1), we find
d2
 m1 m2 x2 + ( m1 + m2 ) kx2  = 0
dt 2 
(4)
from which
x2 = −
m1 + m2
kx2
m1m2
(5)
m1 + m2
k
m1m2
(6)
Therefore, x2 oscillates with the frequency
ω=
We obtain the same result for x1 . If we notice that the reduced mass of the system is defined as
1
µ
=
1
1
+
m1 m2
(7)
we can rewrite (6) as
ω=
k
(8)
µ
k
µ
This means the system oscillates in the same way as a system consisting of a single mass µ.
Inserting the given values, we obtain µ
66.7 g and ω
2.74 rad ⋅ s −1 .
3-7.
A
hb
hs
Let A be the cross-sectional area of the floating body, hb its height, hs the height of its
submerged part; and let ρ and ρ0 denote the mass densities of the body and the fluid,
respectively.
The volume of displaced fluid is therefore V = Ahs . The mass of the body is M = ρ Ahb .
86
CHAPTER 3
There are two forces acting on the body: that due to gravity (Mg), and that due to the fluid,
pushing the body up ( − ρ0 gV = − ρ0 ghs A ).
The equilibrium situation occurs when the total force vanishes:
0 = Mg − ρ0 gV
= ρ gAhb − ρ0 ghs A
(1)
which gives the relation between hs and hb :
hs = hb
ρ
ρ0
(2)
For a small displacement about the equilibrium position ( hs → hs + x ), (1) becomes
Mx = ρ Ahb x = ρ gAhb − ρ0 g ( hs + x ) A
(3)
Upon substitution of (1) into (3), we have
ρ Ahb x = − ρ0 gxA
(4)
or,
x+g
ρ0
x=0
ρ hb
(5)
Thus, the motion is oscillatory, with an angular frequency
ω2 = g
g gA
ρ0
= =
ρhb hs V
(6)
where use has been made of (2), and in the last step we have multiplied and divided by A. The
period of the oscillations is, therefore,
τ=
Substituting the given values, τ
2π
ω
= 2π
V
gA
(7)
0.18 s .
3-8.
y
O
2a
m
s
2a
x
The force responsible for the motion of the pendulum bob is the component of the gravitational
force on m that acts perpendicular to the straight portion of the suspension string. This
component is seen, from the figure (a) below, to be
F = ma = mv = − mg cos α
(1)
87
OSCILLATIONS
where α is the angle between the vertical and the tangent to the cycloidal path at the position of
m. The cosine of α is expressed in terms of the differentials shown in the figure (b) as
cos α =
dy
ds
(2)
where
ds = dx 2 + dy 2
(3)
m
α
α
dy
F
ds
dx
S
mg
(b)
(a)
The differentials, dx and dy, can be computed from the defining equations for x(φ) and y(φ)
above:
dx = a (1 − cos φ ) dφ 


dy = − a sin φ dφ

(4)
Therefore,
ds2 = dx 2 + dy 2
2
= a 2 (1 − cos φ ) + sin 2 φ  dφ 2 = 2a 2 (1 − cos φ ) dφ 2


= 4 a 2 sin 2
φ
2
dφ 2
(5)
so that
ds = 2a sin
φ
2
dφ
(6)
Thus,
dy − a sin φ dφ
=
ds 2a sin φ dφ
2
= − cos
The velocity of the pendulum bob is
φ
2
= cos α
(7)
88
CHAPTER 3
v=
ds
φ dφ
= 2a sin
dt
2 dt
= −4 a
d
dt
φ

 cos 2 
(8)
from which
v = −4 a
Letting z ≡ cos
φ
2
d2
dt 2
φ

cos 2 
(9)
be the new variable, and substituting (7) and (9) into (1), we have
−4maz = mgz
(10)
or,
z+
g
z=0
4a
(11)
which is the standard equation for simple harmonic motion,
z + ω 02 z = 0
(12)
If we identify
ω0 =
where we have used the fact that
g
(13)
= 4a .
Thus, the motion is exactly isochronous, independent of the amplitude of the oscillations. This
fact was discovered by Christian Huygene (1673).
3-9.
The equation of motion for 0 ≤ t ≤ t0 is
mx = − k ( x − x0 ) + F = − kx + ( F + kx0 )
(1)
mx = − k ( x − x0 ) = − kx + kx0
(2)
while for t ≥ t0 , the equation is
It is convenient to define
ξ = x − x0
which transforms (1) and (2) into
mξ = − kξ + F ;
mξ = − kξ ;
0 ≤ t ≤ t0
t ≥ t0
(3)
(4)
89
OSCILLATIONS
The homogeneous solutions for both (3) and (4) are of familiar form ξ ( t ) = Ae iω t + Be − iω t , where
ω = k m . A particular solution for (3) is ξ = F k . Then the general solutions for (3) and (4) are
ξ− =
F
+ Ae iω t + Be − iω t ;
k
ξ+ = Ce iω t + De − iω t ;
0 ≤ t ≤ t0
t ≥ t0
(5)
(6)
To determine the constants, we use the initial conditions: x ( t = 0 ) = x0 and x(t = 0) = 0. Thus,
ξ− ( t = 0 ) = ξ− ( t = 0 ) = 0
(7)
The conditions give two equations for A and B:
F

+ A+B 
k

0 = iω ( A − B) 
0=
(8)
Then
A=B=−
F
2k
and, from (5), we have
ξ− = x − x 0 =
F
(1 − cos ω t ) ;
k
0 ≤ t ≤ t0
(9)
Since for any physical motion, x and x must be continuous, the values of ξ− ( t = t0 ) and
ξ− ( t = t0 ) are the initial conditions for ξ+ ( t ) which are needed to determine C and D:
ξ+ ( t = t0 ) =




− De − iω t0  

F
1 − cos ω t0 ) = Ce iω t0 + De − iω t0
(
k
F
ξ+ ( t = t0 ) = ω sin ω t0 = iω Ce iω t0
k
(10)
The equations in (10) can be rewritten as:
F

1 − cos ω t0 ) 
(
k


− iF

=
sin ω t0
k

Ce iω t0 + De − iω t0 =
Ce iω t0 − De − iω t0
(11)
Then, by adding and subtracting one from the other, we obtain
(
)
F − iω t0

e
1 − e iω t0 
2k


F iω t0
D=
e
1 − e − iω t0 
2k

C=
(
)
(12)
90
CHAPTER 3
Substitution of (12) into (6) yields
ξ+ =
(
)
(
)
F
 e − iω t0 − 1 e iω t + e iω t0 − 1 e − iω t 

2k 
=
F  iω (t − t0 ) iω t
− iω t − t
− e + e ( 0 ) − e − iω t 
e
2k 
=
F
 cos ω ( t − t0 ) − cos ω t 
k
(13)
Thus,
x − x0 =
3-10.
F
 cos ω ( t − t0 ) − cos ω t  ; t ≥ t0
k
(14)
The amplitude of a damped oscillator is expressed by
x ( t ) = Ae − βt cos (ω 1t + δ )
(1)
Since the amplitude decreases to 1 e after n periods, we have
β nT = β n
2π
ω1
=1
(2)
Substituting this relation into the equation connecting ω 1 and ω 0 (the frequency of undamped
oscillations), ω 12 = ω 02 − β 2 , we have
1 
 ω1 

= ω 12 1 + 2 2 

 2π n 
 4π n 
ω 02 = ω 12 + 
2
(3)
Therefore,
ω1 
1 
= 1 + 2 2 
ω 0  4π n 
−1 2
(4)
so that
ω1
1
≅ 1− 2 2
8π n
ω2
3-11.
The total energy of a damped oscillator is
1
1
2
2
mx ( t ) + kx ( t )
2
2
(1)
x ( t ) = Ae − βt cos (ω 1t − δ )
(2)
x ( t ) = Ae − βt  − β cos (ω 1t − δ ) − ω 1 sin (ω 1t − δ ) 
(3)
E (t) =
where
91
OSCILLATIONS
ω 1 = ω 02 − β 2 ,
ω0 =
k
m
Substituting (2) and (3) into (1), we have
E (t) =
A2 −2 βt
 mβ 2 + k cos 2 (ω 1t − δ ) + mω 12 sin 2 (ω 1t − δ )
e

2
(
)
+ 2mβω 1 sin (ω 1t − δ ) cos (ω 1t − δ ) 
(4)
Rewriting (4), we find the expression for E(t):
E (t) =
mA2 −2 βt  2
β cos 2 (ω 1t − δ ) + β ω 02 − β 2 sin 2 (ω 1t − δ ) + ω 02 
e


2
Taking the derivative of (5), we find the expression for
dE
:
dt
dE mA2 −2 βt
 2βω 02 − 4β 3 cos 2 (ω 1t − δ )
e
=

2
dt
− 4β 2 ω 02 − β 2 sin 2 (ω 1t − δ ) 0 − 2βω 2 

(
(5)
)
(6)
The above formulas for E and dE dt reproduce the curves shown in Figure 3-7 of the text. To
find the average rate of energy loss for a lightly damped oscillator, let us take β ω 0 . This
means that the oscillator has time to complete some number of periods before its amplitude
decreases considerably, i.e. the term e −2 βt does not change much in the time it takes to complete
one period. The cosine and sine terms will average to nearly zero compared to the constant term
in dE dt , and we obtain in this limit
dE
dt
− mβω 02 A2 e −2 βt
(7)
3-12.
θ
mg sin θ
mg
The equation of motion is
− m θ = mg sin θ
θ=−
g
sin θ
If θ is sufficiently small, we can approximate sin θ ≅ θ , and (2) becomes
(1)
(2)
92
CHAPTER 3
g
θ=− θ
(3)
θ ( t ) = θ 0 cos ω 0 t
(4)
which has the oscillatory solution
where ω 0 = g and where θ 0 is the amplitude. If there is the retarding force 2m g θ , the
equation of motion becomes
− m θ = mg sin θ + 2m g θ
(5)
or setting sin θ ≅ θ and rewriting, we have
θ + 2ω 0θ + ω 02θ = 0
(6)
Comparing this equation with the standard equation for damped motion [Eq. (3.35)],
x + 2β x + ω 02 x = 0
(7)
we identify ω 0 = β . This is just the case of critical damping, so the solution for θ ( t) is [see
Eq. (3.43)]
θ ( t ) = ( A + Bt ) e −ω0t
(8)
For the initial conditions θ ( 0 ) = θ 0 and θ(0) = 0, we find
θ ( t ) = θ 0 (1 + ω 0 t ) e −ω 0t
3-13.
For the case of critical damping, β = ω 0 . Therefore, the equation of motion becomes
x + 2β x + β 2 x = 0
(1)
x (t) = y (t) e −β t
(2)
If we assume a solution of the form
we have
x = ye − β t − β ye − β t
x = ye − β t − 2β ye − β t + β 2 ye − β t




(3)
Substituting (3) into (1), we find
ye − β t − 2β ye − β t + β 2 ye − β t + 2β ye − β t − 2β 2 ye − β t + β 2 ye − β t = 0
(4)
y=0
(5)
y ( t ) = A + Bt
(6)
or,
Therefore,
93
OSCILLATIONS
and
x ( t ) = ( A + Bt ) e − β t
(7)
which is just Eq. (3.43).
3-14.
For the case of overdamped oscillations, x(t) and x ( t ) are expressed by
x ( t ) = e − β t  A1eω 2t + A2 e −ω 2t 
(
) (
(1)
)
x ( t ) e − β t  −β A1eω 2t + + A2 e −ω 2t + A1ω 2 eω 2t − A2ω 2 e −ω 2t 
(2)
where ω 2 = β 2 − ω 02 . Hyperbolic functions are defined as
cosh y =
ey + e− y
,
2
sinh y =
ey − e−y
2
(3)
or,
e y = cosh y + sinh y 

−y
e = cosh y − sinh y 
(4)
Using (4) to rewrite (1) and (2), we have
x ( t ) = ( cosh β t − sinh β t ) ( A1 + A2 ) cosh ω 2t + ( A1 − A2 ) sinh ω 2t 
(5)
and
x ( t ) = ( cosh β t − sinh β t ) ( A1ω 2 − A1β ) ( cosh ω 2t + sinh ω 2t )
− ( A2 β + A2ω 2 ) ( cosh ω 2t − sinh ω 2t ) 
3-15.
(6)
We are asked to simply plot the following equations from Example 3.2:
x ( t ) = Ae − β t cos (ω 1t − δ )
(1)
v(t) = − Ae − β t β cos (ω 1t − δ ) + ω 1 sin (ω 1t − δ ) 
(2)
with the values A = 1 cm , ω 0 = 1 rad ⋅ s −1 , β = 0.1 s −1 , and δ = π rad. The position goes through
x = 0 a total of 15 times before dropping to 0.01 of its initial amplitude. An exploded (or
zoomed) view of figure (b), shown here as figure (B), is the best for determining this number, as
is easily shown.
94
CHAPTER 3
1
(b)
0.5
0
–0.5
x(t) (cm)
v(t) (cm/s)
–1
0
5
10
15
20
25
30
35
40
45
50
t (s)
1
(c)
v (cm/s)
0.5
0
–0.5
–1
–1
–0.5
0
0.5
1
x (cm)
0.01
x (cm)
(B)
0
–0.01
0
5
10
15
20
25
30
35
40
45
50
55
t (s)
3-16.
If the damping resistance b is negative, the equation of motion is
x − 2β x + ω 02 x = 0
(1)
where β ≡ − b 2m > 0 because b < 0. The general solution is just Eq. (3.40) with β changed to –β:
x ( t ) = e β t  A1 exp

(
)
(
)
β 2 − ω 02 t + A2 exp − β 2 − ω 02 t 

(2)
From this equation, we see that the motion is not bounded, irrespective of the relative values of
β 2 and ω 02 .
The three cases distinguished in Section 3.5 now become:
a) If ω 02 > β 2 , the motion consists of an oscillatory solution of frequency ω 1 = ω 02 − β 2 ,
multiplied by an ever-increasing exponential:
95
OSCILLATIONS
x ( t ) = e β t  A1e iω1t + A2 e − iω1t 
(3)
x ( t ) = ( A + Bt ) e β t
(4)
x ( t ) = e β t  A1eω 2t + A2 e −ω 2t 
(5)
ω 2 = β 2 − ω 02 ≤ β
(6)
b) If ω 02 = β 2 , the solution is
which again is ever-increasing.
c) If ω 02 < β 2 , the solution is:
where
This solution also increases continuously with time.
The tree cases describe motions in which the particle is either always moving away from its
initial position, as in cases b) or c), or it is oscillating around its initial position, but with an
amplitude that grows with the time, as in a).
Because b < 0, the medium in which the particle moves continually gives energy to the particle
and the motion grows without bound.
3-17.
For a damped, driven oscillator, the equation of motion is
x = 2β x + ω 02 x = A cos ω t
(1)
and the average kinetic energy is expressed as
T =
mA2
ω2
4 ω 02 − ω 2 2 + 4ω 2 β 2
(
)
(2)
Let the frequency n octaves above ω 0 be labeled ω 1 and let the frequency n octaves below ω 0
be labeled ω 2 ; that is
ω 1 = 2n ω 0
ω 2 = 2− n ω 0
(3)
The average kinetic energy for each case is
T
T
ω1
ω2
=
22 n ω 02
mA2
4 ω 02 − 22 n ω 02 2 + (4)22 n ω 02 β 2
(4)
=
2−2 n ω 02
mA2
4 ω 02 − 2−2 n ω 02 2 + (4)2−2 n ω 02 β 2
(5)
(
(
)
)
Multiplying the numerator and denominator of (5) by 24 n , we have
96
CHAPTER 3
T
ω2
22 n ω 02
mA2
4 ω 02 − 22 n ω 02 2 + (4)22 n ω 02 β 2
=
(
)
Hence, we find
T
ω1
= T
(6)
ω2
and the proposition is proven.
3-18. Since we are near resonance and there is only light damping, we have ω 0 ω R ω ,
where ω is the driving frequency. This gives Q ω 0 2β . To obtain the total energy, we use the
solution to the driven oscillator, neglecting the transients:
x ( t ) = D cos (ω t − δ )
(1)
We then have
E=
1
1
mD2 2
ω sin 2 (ω t − δ ) + ω 02 cos 2 (ω t − δ ) 
mx 2 + kx 2 =
2
2
2 
1
mω 02 D2
2
(2)
The energy lost over one period is
∫ ( 2mβ x ) ⋅ ( xdt ) = 2π mωβ D
T
2
(3)
0
where T = 2π ω . Since ω
ω 0 , we have
E
energy lost over one period
ω0
4πβ
Q
2π
(4)
which proves the assertion.
3-19.
The amplitude of a damped oscillator is [Eq. (3.59)]
D=
A
(1)
(ω 02 − ω 2 ) + 4ω 2β 2
2
At the resonance frequency, ω = ω R = ω 02 − β 2 , D becomes
DR =
A
(2)
2β ω 02 − β 2
Let us find the frequency, ω = ω ′ , at which the amplitude is
1
1
A
=
DR =
2
2 2β ω 02 − β 2
Solving this equation for ω ′ , we find
1
DR :
2
A
(ω
2
0
− ω′
)
2 2
+ 4ω ′ β
2
2
(3)
97
OSCILLATIONS
 β2 
ω ′ = ω − 2β ± 2βω 0 1 − 2 
 ω0 
2
2
0
12
2
(4)
For a lightly damped oscillator, β is small and the terms in β 2 can be neglected. Therefore,
ω ′ 2 ≅ ω 02 ± 2βω 0
(5)
or,

ω ′ ≅ ω 0 1 ±

β 
ω 0 
(6)
which gives
∆ω = (ω 0 + β ) − (ω 0 − β ) = 2β
(7)
We also can approximate ω R for a lightly damped oscillator:
ω R = ω 02 − 2β 2 ≅ ω 0
(8)
Therefore, Q for a lightly damped oscillator becomes
ω0 ω0
≅
2β ∆ω
Q≅
3-20.
(9)
From Eq. (3.66),
x=
− Aω
(ω 02 − ω 2 ) + 4ω 2β 2
2
sin (ω t − δ )
(1)
Therfore, the absolute value of the velocity amplitude v is given by
v0 =
Aω
(ω
2
0
−ω
)
2 2
+ 4ω β
2
(2)
2
The value of ω for v0 a maximum, which is labeled ω v , is obtained from
∂ v0
∂ω
=0
(3)
ω =ω v
and the value is ω v = ω 0 .
Since the Q of the oscillator is equal to 6, we can use Eqs. (3.63) and (3.64) to express β in terms
of ω 0 :
β2 =
ω 02
146
We need to find two frequencies, ω 1 and ω 2 , for which v0 = vmax
We find
(4)
2 , where vmax = v0 (ω = ω 0 ) .
98
CHAPTER 3
Aω
vmax
A
=
=
2 2 2β
(ω 02 − ω 2 ) + 4ω 2β 2
2
(5)
Substituting for β in terms of ω 0 from (4), and by squaring and rearranging terms in (5), we
obtain
(ω
2
0
2
− ω 1,2
)
(
)
2
2
ω 1,2
ω 02 = 0
73
(6)
2
1
ω 1,2ω 0 ≅ ± ω 1,2ω 0
73
6
(7)
2
−
from which
2
ω 02 − ω 1,2
=±
Solving for ω 1 , ω 2 we obtain
 ω 
ω 1,2 ≅  ± 0  ± ω 0
 12 
(8)
It is sufficient for our purposes to consider ω 1 , ω 2 positive: then
ω1 ≅
ω0
12
+ ω0 ;
ω2 ≅ −
ω0
12
+ ω0
(9)
so that
∆ω = ω 1 − ω 2 =
ω0
(10)
6
A graph of v0 vs. ω for Q = 6 is shown.
v0
A
vmax =
2β
A
2 2β
∆ω
ω0 ω0
12 12
1
ω0
6
ω0
3-21.
We want to plot Equation (3.43), and its derivative:
x ( t ) = ( A + Bt ) e − β t
(1)
v ( t ) = [ B − β ( A + Bt ) ] e − β t
(2)
where A and B can be found in terms of the initial conditions
A = x0
(3)
B = v0 + β x 0
(4)
99
OSCILLATIONS
The initial conditions used to produce figure (a) were ( x0 , v0 ) = ( −2, 4 ) , (1, 4) , (4,−1) , (1,−4) ,
(−1,−4) , and (−4, 0) , where we take all x to be in cm, all v in cm ⋅ s −1 , and β = 1 s −1 . Figure (b) is a
magnified view of figure (a). The dashed line is the path that all paths go to asymptotically as
t → ∞. This can be found by taking the limits.
lim v(t) = − β Bte − β t
(5)
lim x(t ) = Bte − β t
(6)
t →∞
t →∞
so that in this limit, v = –βx, as required.
4
(a)
3
2
v (cm/s)
1
0
–1
–2
–3
–4
–4
–2
0
x (cm)
2
–0.5
–0.25
0
x (cm)
0.25
4
(b)
0.4
v (cm/s)
0.2
0
–0.2
–0.4
3-22.
0.5
For overdamped motion, the position is given by Equation (3.44)
x ( t ) = A1e − β1t + A2 e − β2t
(1)
100
CHAPTER 3
The time derivative of the above equation is, of course, the velocity:
a)
v ( t ) = − A1β1 e − β1t − A2 β 2 e − β2t
(2)
x0 = A1 + A2
(3)
v0 = − A1β1 − A2 β 2
(4)
At t = 0:
The initial conditions x0 and v0 can now be used to solve for the integration constants A1 and
A2 .
b) When A1 = 0 , we have v0 = − β 2 x0 and v ( t ) = −β 2 x ( t ) quite easily. For A1 ≠ 0 , however, we
have v ( t ) → − β1 A1 e − β1t = − β1 x as t → ∞ since β1 < β 2 .
3-23. Firstly, we note that all the δ = π solutions are just the negative of the δ = 0 solutions.
The δ = π 2 solutions don’t make it all the way up to the initial “amplitude,” A , due to the
retarding force. Higher β means more damping, as one might expect. When damping is high,
less oscillation is observable. In particular, β 2 = 0.9 would be much better for a kitchen door
than a smaller β, e.g. the door closing (δ = 0), or the closed door being bumped by someone who
then changes his/her mind and does not go through the door ( δ = π 2 ).
101
OSCILLATIONS
1
β2 = 0.1, δ = 0
β2 = 0.5, δ = 0
β2 = 0.9, δ = 0
β2 = 0.1, δ = π/2
β2 = 0.5, δ = π/2
β2 = 0.9, δ = π/2
β2 = 0.1, δ = π
β2 = 0.5, δ = π
β2 = 0.9, δ = π
0.5
0
–0.5
–1
1
0.5
0
–0.5
–1
1
0.5
0
0.5
–1
0
5
10
15
0
5
10
15
0
5
10
15
3-24. As requested, we use Equations (3.40), (3.57), and (3.60) with the given values to
evaluate the complementary and particular solutions to the driven oscillator. The amplitude of
the complementary function is constant as we vary ω, but the amplitude of the particular
solution becomes larger as ω goes through the resonance near 0.96 rad ⋅ s −1 , and decreases as ω
is increased further. The plot closest to resonance here has ω ω 1 = 1.1 , which shows the least
distortion due to transients. These figures are shown in figure (a). In figure (b), the ω ω 1 = 6
plot from figure (a) is reproduced along with a new plot with Ap = 20 m ⋅ s −2 .
102
CHAPTER 3
ω/ω1 = 1/9
2
1
1
0
0
0
–1
ω/ω1 = 1.1
ω/ω1 = 1/3
–2
–1
0
10
20
30
0
10
ω/ω1 = 3
20
0
30
10
ω/ω1 = 6
0.5
0.5
0
0
–0.5
–0.5
20
xc
xp
x
Legend:
–1
0
10
20
30
–1
0
10
20
t (s)
30
t (s)
30
t (s)
(a)
0.5
0.5
0
0
–0.5
–0.5
Ap = 1
–1
0
5
10
15
20
25
Ap = 20
30
–1
0
5
10
15
20
25
30
(b)
3-25. This problem is nearly identical to the previous problem, with the exception that now
Equation (3.43) is used instead of (3.40) as the complementary solution. The distortion due to
the transient increases as ω increases, mostly because the complementary solution has a fixed
amplitude whereas the amplitude due to the particular solution only decreases as ω increases.
The latter fact is because there is no resonance in this case.
103
OSCILLATIONS
ω/ω1 = 1/9
ω/ω1 = 1/3
ω/ω1 = 1.1
1
1
0.5
0
0
0
–0.5
–0.5
–1
0
5
–1
10
0
5
–1
0
ω/ω1 = 6
ω/ω1 = 3
5
10
ω/ω1 = 6, Ap = 6
0
0
0
–0.5
–0.5
–1
10
0
5
Legend:
3-26.
–1
10
xc
xp
–0.5
0
5
10
–1
0
5
10
x
The equations of motion of this system are
m1 x1 = − kx1 − b1 ( x1 − x2 ) + F cos ω t 


m2 x2 = − b2 x2 − b1 ( x2 − x1 )

(1)
The electrical analog of this system can be constructed if we substitute in (1) the following
equivalent quantities:
1
;
C
m1 → L1 ;
k→
m2 → L2 ;
F → ε0 ;
b1 → R1 ; x → q
b2 → R2
Then the equations of the equivalent electrical circuit are given by
1

q1 = ε 0 cos ω t 
C


L2 q2 + R2 q2 + R1 ( q2 − q1 ) = 0

L1q1 + R1 ( q1 − q2 ) +
(2)
Using the mathematical device of writing exp(iω t) instead of cos ω t in (2), with the
understanding that in the results only the real part is to be considered, and differentiating with
respect to time, we have
104
CHAPTER 3
(
)
I1

= iωε 0 e iω t 
C


L2 I 2 + R2 I 2 + R1 I 2 − I1 = 0 
L1I1 + R1 I1 − I 2 +
( )
(
(3)
)
Then, the equivalent electrical circuit is as shown in the figure:
L1
ε0 cos ωt
I2(t)
R1
I1(t)
C
1
R2
L2
2
The impedance of the system Z is
1
+ Z1
ωC
(4)
1
1
1
=
+
Z1 R1 R2 + iω L2
(5)
Z = iω L1 − i
where Z1 is given by
Then,
Z1 =
R1  R2 ( R2 + R1 ) + ω 2 L22 + iω L2 R1 
(6)
( R1 + R2 )2 + ω 2 L22
and substituting (6) into (4), we obtain



1 
2
R1  R2 ( R2 + R1 ) + ω 2 L22  + i  R1ω L2 +  ω L1 −
R1 + R2 ) + ω 2 L22 
(

ω C



Z=
2
2 2
( R1 + R2 ) + ω L2
(
3-27.
)
(7)
From Eq. (3.89),
F (t) =
∞
1
a0 + ∑ ( an cos nω t + bn sin nω t )
2
n =1
(1)
∞
1
a0 + ∑ cn cos ( nω t − φ n )
2
n=1
(2)
We write
F (t) =
which can also be written using trigonometric relations as
F (t) =
∞
1
a0 + ∑ cn cos nω t cos φ n + sin nω t sin φ n 
2
n =1
Comparing (3) with (2), we notice that if there exists a set of coefficients cn such that
(3)
105
OSCILLATIONS
cn cos φ n = an 

cn sin φ n = bn 
(4)
then (2) is equivalent to (1). In fact, from (4),
cn2 = an2 + bn2 

b 
tan φ n = n 
an 
(5)
with an and bn as given by Eqs. (3.91).
3-28. Since F(t) is an odd function, F(–t) = –F(t), according to Eq. (3.91) all the coefficients an
vanish identically, and the bn are given by
bn =
ω
π
∫
π
ω
π
−
ω
F ( t ′ ) sin nω t ′ dt ′
=
π

ω 0
ω
 − ∫ − π sin nω t ′ dt ′ + ∫ 0 sin nω t ′ dt ′ 
π 
ω

=
ω
π
=
2
( cos 0 − cos nπ )
nπ
0
π 
 1

 1
 ω

cos
cos
ω
ω
n
t
n
t
+
−
′
′
 π  nω
 0 
  nω
−
ω


 4
for n odd

=  nπ
 0 for n even

(1)
Thus,
4

( 2n + 1) π  n = 0, 1, 2, …

=0

b( 2 n + 1) =
b( 2 n)
(2)
Then, we have
F (t) =
4
π
sin ω t +
4
4
sin 3ω t +
sin 5ω t + …
3π
5π
(3)
106
CHAPTER 3
F(t)
1
–π ⁄ω
π ⁄ω
t
–1
0.849
Terms 1 + 2
–π ⁄ω
π ⁄ω
–0.849
t
1.099
–π ⁄ω
π ⁄ω
Terms 1 + 2 + 3
t
–1.099
0.918
–π ⁄ω
π ⁄ω
Terms 1 + 2 + 3 + 4
t
–0.918
3-29. In order to Fourier analyze a function of arbitrary period, say τ = 2P ω instead of
2π ω , proportional change of scale is necessary. Analytically, such a change of scale can be
represented by the substitution
x=
πt
P
or
t=
Px
π
(1)
for when t = 0, then x = 0, and when t = τ = 2P ω , then x = 2π ω .
Thus, when the substitution t = Px π is made in a function F(t) of period 2P ω ′ , we obtain the
function
 Px 
F   = f ( x)
π 
(2)
and this, as a function of x, has a period of 2π ω . Now, f(x) can, of course, be expanded
according to the standard formula, Eq. (3.91):
f ( x) =
where
∞
1
a0 + ∑ ( an cos nω x + bn sin nω x )
2
n =1
(3)
107
OSCILLATIONS
an =
bn =
ω
π
ω
π
2π

∫ ω f ( x ′ ) cos nω x ′ dx ′ 
0
2π
∫ω
0


f ( x ′ ) sin nω x ′ dx ′ 

(4)
If, in the above expressions, we make the inverse substitutions
x=
πt
P
and
dx =
π
P
dt
(5)
the expansion becomes
∞
a

π t 
P π t
 nωπ t 
 nωπ t  
f   = F  ⋅  = F ( t ) = 0 + ∑  an cos 
+ bn sin 


2 n =1 
P
π P 
 P 
 P  
(6)
and the coefficients in (4) become

 nωπ t ′ 
dt
′

0
P 


ω 2ωP

 nωπ t ′ 
bn = ∫ F ( t ′ ) sin 
dt ′ 

0
P
 P 

an =
ω
2P
ω F t cos
( ′) 
P∫

For the case corresponding to this problem, the period of F(t) is
(7)
4π
ω
, so that P = 2π. Then,
substituting into (7) and replacing the integral limits 0 and τ by the limits −
τ
2
and +
τ
2
, we
obtain
2π

 nω t ′ 
dt ′ 

−
2 



ω 2ωπ
 nω t ′ 
bn =
dt ′ 
2π F ( t ′ ) sin 
∫

−
2π ω
 2 

an =
ω
2π
∫ ωωπ F (t ′ ) cos 
2
(8)
and substituting into (6), the expansion for F(t) is
F (t) =
a0 ∞ 
 nω t 
 nω t  
+ ∑  an cos 
+ bn sin 


2 n =1 
 2 
 2  
(9)
Substituting F(t) into (8) yields
2π

 nω t ′ 
dt ′ 

2 


ω 2ωπ

 nω t ′ 
bn =
dt ′ 
sin ω t ′ sin 
∫

2π 0
 2 

an =
Evaluation of the integrals gives
ω
2π
∫ ω sin ω t ′ cos 
0
(10)
108
CHAPTER 3
b2 =
1
; bn = 0
2
0


an ( n ≥ 2 ) = 
−4

2
 π n − 4
a0 = a1 = 0



n even 


n odd 


for n ≠ 2
(
)
(11)
and the resulting Fourier expansion is
F (t) =
3-30.
ωt 4
3ω t
5ω t
7ω t
1
4
4
4
−
−
−
+…
sin ω t +
cos
cos
cos
cos
2
3π
2 5π
2
21π
2
45π
2
(12)
The output of a full-wave rectifier is a periodic function F(t) of the form
π

 − sin ω t ; − ω < t ≤ 0
F (t) = 

π
 sin ω t ;
0<t<
ω

(1)
The coefficients in the Fourier representation are given by
an =
bn =
π

ω 0
ω sin ω t cos nω t dt
ω
ω
sin
t
cos
n
t
dt
−
+
′
′
′
′
′ ′ 
(
)
π
∫ −
∫
0
π ω

π
ω

ω 0
 ∫ − π ( − sin ω t ′ ) sin nω t ′ dt ′ + ∫ 0 sin ω t ′ sin nω t ′ dt ′ 
π ω





(2)
Performing the integrations, we obtain
4

 π 1 − n2 ; if n even ( or 0 )
an = 

0;
if n odd

(
bn = 0
)
for all n








(3)
The expansion for F(t) is
F (t) =
2
π
−
4
4
cos 2ω t −
cos 4ω t …
3π
15π
The exact function and the sum of the first three terms of (4) are shown below.
(4)
109
OSCILLATIONS
Sum of first
three terms
F(t)
1
sin ωt
.5
–π
3-31.
–π ⁄ 2
π⁄2
π
ωt
We can rewrite the forcing function so that it consists of two forcing functions for t > τ :

0

F (t) 
=  a (t τ )
m


a (t − τ )
 a (t τ ) −
τ

t<0
0<t<τ
(1)
t >τ
During the interval 0 < t < τ, the differential equation which describes the motion is
x + 2β x + ω 02 x =
at
(2)
τ
The particular solution is x p = Ct + D , and substituting this into (2), we find
2β C + ω 02Ct + ω 02 D =
at
(3)
τ
from which
2β C + ω 02 D = 0 


a
Cω 02 − = 0

τ

(4)
Therefore, we have
D=−
2β a
ω τ
4
0
C=
,
a
(5)
ω 02τ
which gives
xp =
a
ωτ
2
0
t−
2β a
(6)
ω 04τ
Thus, the general solution for 0 < t < τ is
x ( t ) = e − β t A cos ω 1t + B sin ω 1t  +
and then,
a
ωτ
2
0
t−
2β a
ω 04τ
(7)
110
CHAPTER 3
x ( t ) = − β e − β t A cos ω 1t + B sin ω 1t  + ω 1e − β t − A sin ω 1t + B cos ω 1t  +
a
ω 02τ
(8)
The initial conditions, x(0) = 0, x ( 0 ) = 0 , implies
2β a





a  2β 2
B=
− 1 
ω 1ω 02τ  ω 02

A=
ω 04τ
(9)
Therefore, the response function is
x (t) =
For the forcing function −
a  2β − βt
e−β t
e
cos
ω
t
+

1
ω 02τ  ω 02
ω1
a (t − τ )
τ
 2β 2

2β 
 2 − 1 sin ω 1t + t − 2 
ω0 
 ω0

(10)
in (1), we have a response similar to (10). Thus, we add these
two equations to obtain the total response function:
x (t) =

a  2β − β t
e − βt  2β 2
βt
−
−
+
e
cos
ω
t
e
cos
ω
t
τ
(
)

 2 − 1
1
1
2
2
ω0τ ω0
ω1  ω 0

(
)
(11)
×  sin ω 1t + − e sin ω 1 ( t − τ )  + τ 
βt
When τ → 0, we can approximate e βτ as 1 + βτ, and also sin ω 1τ ≅ ω 1τ , cos ω 1τ ≅ 1 . Then,
x ( t ) 
→
τ →0
a  2β − β t
e−β t


e
cos
ω
t
1
βτ
cos
ω
t
ω
τ
sin
ω
t
−
+
+
+
(
)
(

1
1
1
1 )

ω 02τ  ω 02
ω1
 2β 2

 2 − 1
 ω0

× sin ω 1t − (1 + βτ ) ( sin ω 1t − ω 1τ cos ω 1t )  + τ 
=

β
2β 3 
a 
−β t
− β t  2βω 1
−
−
−
+
1
cos
ω
sin ω 1t 
e
t
e
1
 2
2 
2
ω0 
ω 1 ω 1ω 0 
 ω0

(12)
If we use ω 12 = ω 02 − β 2 , the coefficient of e − β t sin ω 1t becomes β ω 1 . Therefore,
→
x ( t ) 
τ →0

a 
β
1 − e − β t cos ω 1t − e − β t
sin ω 1t 
2 
ω0 
ω1

This is just the response for a step function.
3-32.
a) Response to a Step Function:
From Eq. (3.100) H ( t0 ) is defined as
(13)
111
OSCILLATIONS
 0, t < t0
H ( t0 ) = 
 a1 , t > t0
(1)
With initial conditions x ( t0 = 0 ) and x ( t0 = 0 ) , the general solution to Eq. (3.102) (equation of
motion of a damped linear oscillator) is given by Eq. (3.105):


a 
β e − β (t − t0 )
− β ( t − t0 )
ω
1
−
cos
−
−
sin ω 1 ( t − t0 )  for t > t0 
e
t
t
1(
0)
2 
ω 0 
ω1



for t < t0 
x (t) = 0
x (t) =
(2)
where ω 1 = ω 02 − β 2 .
For the case of overdamping, ω 02 < β 2 , and consequently ω 1 = i β 2 − ω 02 is a pure imaginary
number. Hence, cos ω 1 ( t − t0 ) and sin ω 1 ( t − t0 ) are no longer oscillatory functions; instead,
they are transformed into hyperbolic functions. Thus, if we write ω 2 = β 2 − ω 02 (where ω 2 is
real),
cos ω 1 ( t − t0 ) = cos iω 2 ( t − t0 ) = cosh ω 2 ( t − t0 ) 


sin ω 1 ( t − t0 ) = sin iω 2 ( t − t0 ) = i sinh ω 2 ( t − t0 ) 
(3)
The response is given by [see Eq. (3.105)]


a 
β e − β (t − t0 )
− β ( t − t0 )
ω
−
−
−
1
cosh
sinh ω 1 ( t − t0 )  for t > t0 
e
t
t
2(
0)
2 
ω 0 
ω2



for t < t0 
x (t) = 0
x (t) =
(4)
For simplicity, we choose t0 = 0 , and the solution becomes
x (t) =

H ( 0) 
βe−β t
−β t
ω
−
−
e
t
1
cosh
sinh ω 2t 
2
2 
ω0 
ω2

(5)
This response is shown in (a) below for the case β = 5 ω 0 .
b) Response to an Impulse Function (in the limit τ → 0):
From Eq. (3.101) the impulse function I ( t0 , t1 ) is defined as
0

I ( t0 , t1 ) =  a


0
t < t0
t0 < t < t1
t > t1
For t1 − t2 = τ → 0 in such a way that aτ is constant = b, the response function is given by
Eq. (3.110):
(6)
112
CHAPTER 3
x (t) =
b
ω1
− β t −t
e ( 0 ) sin ω 1 ( t − t0 ) for t > t0
(7)
Again taking the “spike” to be at t = 0 for simplicity, we have
x (t) =
b
ω1
e − βt sin ω 1 ( t )
for t > 0
(8)
For ω 1 = iω 2 = i β 2 − ω 02 (overdamped oscillator), the solution is
x (t) =
b
ω2
e − βt sinh ω 2t ; t > 0
(9)
This response is shown in (b) below for the case β = 5 ω 0 .
1
x  H ( 0) ω 02 
(a)
0.5
0
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
ω0t
1
x (b ω 2 )
(b)
0.5
0
0
1
2
3
4
5
ω0t
3-33.
a) In order to find the maximum amplitude of the response function shown in Fig. 3-22, we
look for t1 such that x ( t ) given by Eq. (3.105) is maximum; that is,
(
∂ x (t)
)
∂t
=0
(1)
t = t1
From Eq. (3.106) we have
(
∂ x (t)
∂t
) = H ( 0) e
ω
2
0
−β t

β2 
ω 1 +  sin ω 1t
ω1 

(2)
113
OSCILLATIONS
For β = 0.2ω 0 , ω 1 = ω 02 − β 2 = 0.98ω 0 . Evidently, t1 = π ω 1 makes (2) vanish. (This is the
absolute maximum, as can be seen from Fig. 3-22.)
Then, substituting into Eq. (3.105), the maximum amplitude is given by
x ( t ) max
βπ
−

a 
ω1
= x ( t1 ) = 2 1 + e 
ω 0 

(3)
or,
x ( t1 ) ≅ 1.53
a
ω 02
(4)
b) In the same way we find the maximum amplitude of the response function shown in Fig.
3-24 by using x(t) given in Eq. (3.110); then,
(
∂ x (t)
∂t
)
t = t1

β
− β t −t 
sin ω 1 ( t − t0 ) 
= be ( 0 )  cos ω 1 ( t − t0 ) −
ω1

 t = t1
(5)
If (5) is to vanish, t1 is given by
t1 − t0 =
ω  1
1.37
tan −1  1  =
tan −1 ( 4.9) =
ω1
ω1
 β  ω1
1
(6)
Substituting (6) into Eq. (3.110), we obtain (for β = 0.2ω 0 )
x ( t ) max
−β
× 1.37
b
= x ( t1 ) =
sin ( 1.37 )
e ω1
0.98ω 0
(7)
or,
x ( t1 ) ≅ 0.76
aτ
ω0
(8)
The response function of an undamped (β = 0) linear oscillator for an impulse function
2π
I(0,τ ), with τ =
, can be obtained from Eqs. (3.105) and (3.108) if we make the following
3-34.
ω0
substitutions:
β=0;
t0 = 0 ;
ω1 = ω 0


2π 
t1 = τ =

ω0 
(For convenience we have assumed that the impulse forcing function is applied at t = 0.)
Hence, after substituting we have
(1)
114
CHAPTER 3
x (t) = 0
x (t) =
x (t) =
t<0
a
1 − cos ω 0 t 
ω 
0<t<
2
0
a
cos ( w0 t − 2π ) − cos w0 t  = 0
ω 
t >τ =
2
0
2π
ω0
2π
ω0









(2)
This response function is shown below. Since the oscillator is undamped, and since the impulse
lasts exactly one period of the oscillator, the oscillator is returned to its equilibrium condition at
the termination of the impulse.
2a
ω 02
a
ω 02
π 2π
ω0 ω0
3-35.
t
The equation for a driven linear oscillator is
x + 2β x + w20 x = f ( t )
where f(t) is the sinusoid shown in the diagram.
f(t)
a
I
II
III
t
Region I:
x=0
(1)
Region II:
x + 2β x + ω 02 x = a sin ω t
(2)
Region III:
x + 2β x + ω 02 x = 0
(3)
The solution of (2) is
x = e − β t ( A sin ω 1t + B cos ω 1t ) + xP
(4)
xP = Da sin (ω t − δ )
(5)
in which
where
D=
1
(ω − ω ) + 4β 2ω 2
2
0
2
(6)
115
OSCILLATIONS
δ = tan −1
2βω
ω 02 − ω 2
(7)
Thus,
x = e − βt ( A sin ω 1t + B cos ω 1t ) + D a sin (ω t − δ )
(8)
The initial condition x(0) = 0 gives
B = aD sin δ
(9)
and x ( 0 ) = 0
− β B + ω 1 A + Daω cos δ = 0
or,
A = ( β sin δ − ω cos δ )
aD
ω1
(10)
The solution of (3) is
x ( t ) = e − β t A′ sin ω 1t + B′ cos ω 1t 
(11)
We require that x ( t ) and x ( t ) for regions II and III match at t = π ω . The condition that
π
xII   = xIII
ω
π
  gives
ω
e − βπ ω ( A sin φ + B cos φ ) + Da sin (π − δ ) = e − βπ ω ( A′ sin φ + B′ cos φ )
where φ =
ω1
π or,
ω
A′ + B′ cot φ = A + B cot φ +
π
The condition that xII   = xIII
ω
aD sin δ βπ ω
e
sin φ
(12)
π
  gives
ω
− β e − βπ ω ( A sin φ + B cos φ ) + aDω cos (π − δ ) + e − βπ ω ( Aω 1 cos φ − Bω 1 sin φ )
= − β e − βπ ω ( A′ sin φ + B′ cos φ ) + e − βπ ω ( A′ω 1 cos φ − B′ω 1 sin φ )
or,
A′ (ω 1 cos φ − β sin φ ) − B′ (ω 1 sin φ + β cos φ )
= A ( − β sin φ + ω 1 cos φ ) − B (ω 1 sin φ + β cos φ ) − e βπ ω aDω cos δ
or,
 ω sin φ + β cos φ 
 ω 1 sin φ + β cos φ  βπ ω 

ω Da cos δ
A′ − B′  1
 = A − B
−e


 ω 1 cos φ − β sin φ 
 ω 1 cos φ − β sin φ 
 ω 1 cos φ − β sin φ 
(13)
116
CHAPTER 3
Substituting into (13) from (12), we have
B′
(ω1 cos φ − β sin φ ) cos φ + (ω1 sin φ + β cos φ ) sin φ
(ω1 cos φ − β sin φ ) sin φ

(ω1 cos φ − β sin φ ) cos φ + (ω1 sin φ + β cos φ ) sin φ + aDe βπ ω  sin δ +
ω cos δ
=


(ω1 cos φ − β sin φ ) sin φ
 sin φ ω 1 cos φ − β sin φ 
(14)
from which
B′ = aD sin δ +
a
ω1
De βπ ω  sin δ (ω 1 cos φ − β sin φ ) + ω cos δ sin φ 
(15)
aD sin δ e βπ ω
− B′ cot φ
sin φ
(16)
Using (12), we can find A′:
A′ = A + B cot φ +
Substituting for A, B, and B′ from (10), (9), and (15), we have
β
 β
  aDω cos δ
A′ = aD sin δ  + e βπ ω 
cos φ + sin φ   −
1 + e βπ ω cos φ
ω1
 ω1

 ω1
(
)
(17)
Thus, we obtained all constants giving us the response functions explicitly.
3-36. With the initial conditions, x ( t0 ) = x0 and x ( t0 ) = x0 , the solution for a step function for
t > t0 given by Eq. (3.103) yields
A1 = x0 −
a
ω 02
; A2 =
x0
ω1
+
β x0
βa
−
ω 1 ω 02ω 1
(1)
Therefore, the response to H ( t0 ) for the initial conditions above can be expressed as

x
βx 
− β t −t 
x ( t ) = e ( 0 )  x0 cos ω 1 ( t − t0 ) +  0 + 0  sin ω 1 ( t − t0 ) 
 ω1 ω1 



a 
β − β (t − t0 )
−β t−t
+ 2 1 − e ( 0 ) cos ω 1 ( t − t0 ) −
sin ω 1 ( t − t0 ) 
e
ω0 
ω1

(2)
for t > t0
The response to an impulse function I ( t0 , t1 ) = H ( t1 ) , for the above initial conditions will then be
given by (2) for t0 < t < t1 and by a superposition of solutions for H ( t0 ) and for H ( t1 ) taken
individually for t > t1 . We must be careful, however, because the solution for t > t1 must be
equal that given by (2) for t = t1 . This can be insured by using as a solution for H ( t1 ) Eq. (3.103)
with initial conditions x ( 0 ) = 0 , x ( 0 ) = 0 , and using t1 instead of t0 in the expression.
The solution for t > t1 is then
117
OSCILLATIONS

x
βx 
− β t −t 
x ( t ) = e ( 0 )  x0 cos ω 1 ( t − t0 ) +  0 + 0  sin ω 1 ( t − t0 )  + x1 ( t )
 ω1 ω1 


(3)
where
x1 ( t ) =
− β t−t
ae ( 0 )  βτ
β e βτ
e
t
t
t
t
−
−
−
−
+
cos
ω
τ
cos
ω
(
)
(
)
1
0
1
0
ω 02 
ω1

β e βτ
β
+
sin ω 1 ( t − t0 − τ ) −
sin ω 1 ( t − t0 ) 
ω1
ω1

(4)
for t > t1
We now allow a → ∞ as τ → 0 in such a way that aτ = b = constant; expanding (3) for this
particular case, we obtain

x
βx
b 
−β t−t 
x ( t ) = e ( 0 )  x0 cos ω 1 ( t − t0 ) +  0 + 0 +  sin ω 1 ( t − t0 ) 
 ω1 ω1 ω1 


t > t0
(5)
which is analogous to Eq. (3.119) but for initial conditions given above.
3-37. Any function F ( t ) m can be expanded in terms of step functions, as shown in the figure
below where the curve is the sum of the various (positive and negative) step functions.
In general, we have
x + 2β x + ω 02 x =
=
Fn ( t )
n =− ∞ m
∞
∑
∞
∑ H (t)
n =− ∞
n
(1)
where
 an ( t )
Hn (t) = 
0

t > tn = nτ
t < tn = nτ
(2)
Then, since (1) is a linear equation, the solution to a superposition of functions of the form given
by (2) is the superposition of the solutions for each of those functions.
According to Eq. (3.105), the solution for H n ( t ) for t > tn is
xn ( t ) =

an 
β e − β ( t − tn )
− β ( t − tn )
1
cos
sin ω 1 ( t − tn ) 
−
e
t
−
t
−
ω
n)
1(
2 
ω 0 
ω1

(3)
then, for
∞
F (t)
= ∑ Hn (t)
m
n =− ∞
the solution is
(4)
118
CHAPTER 3
x (t) =
=


β e − β ( t − tn )
− β ( t − tn )
−
−
−
H
t
1
e
cos
ω
t
t
sin ω 1 ( t − tn ) 
(
)
(
)

n
n
1
2 ∑
ω 0 n =− ∞
ω1


1
∞
∞
∞
∑ mH (t ) G (t ) = ∑ F ( t ) G (t )
n
n =− ∞
n
n =− ∞
n
(5)
n
where
 1

2
Gn ( t ) =  mω 0

 0


β e − β ( t − tn )
− β ( t − tn )
cos ω 1 ( t − tn ) −
sin ω 1 ( t − tn )  ;
1 − e
ω1


t ≥ tn
(6)
t < tn
or, comparing with (3)
 xn ( t ) man ,
Gn ( t ) = 

0
t ≥ tn
t < tn
(7)
Therefore, the Green’s function is the response to the unit step.
F (t )
m
tn tn+1
t
Hn(t)
3-38.
The solution for x(t) according to Green’s method is
x (t) = ∫
=
t
−∞
F ( t ′ ) G ( t , t ′ ) dt ′
F0
mω 1
∫
t
0
e − γ t ′ sin ω t ′ e − β (t − t ′) sin ω 1 ( t − t ′ ) dt ′
(1)
Using the trigonometric identity,
sin ω t ′ sin ω 1 ( t − t ′ ) =
we have
1
cos (ω 1 + ω ) t ′ − ω 1t  − cos (ω − ω 1 ) t ′ + ω 1t  

2
(2)
119
OSCILLATIONS
x (t) =
t
t

F0 e − β t 
( β −γ ) t ′
cos (ω + ω 1 ) t ′ − ω 1t  − ∫ dt ′e ( β −γ )t ′ cos (ω − ω 1 ) t ′ + ω 1t  
 ∫ dt ′e
2mω 1  0
0

(3)
Making the change of variable, z = (ω + ω 1 ) t ′ − ω 1t , for the first integral and y = (ω − ω 1 ) t ′ + ω 1t
for the second integral, we find
 ( β −γ )ω1t
F0 e − β t  e ω +ω1
x (t) =
2mω 1  ω + ω 1

( β −γ ) z
ωt
∫
dz e
ω +ω1
cos z −
− ω1t
e
− ( β − γ )ω1t
ωt
ω −ω1
ω − ω1
∫ dy e
(β −γ ) y
ω −ω 1
ω1t


cos y 


(4)
After evaluating the integrals and rearranging terms, we obtain
x (t) =
F0
ω
2
2
m  ( β − γ ) + (ω + ω 1 )   ( β − γ ) 2 + (ω − ω 1 ) 2 



sin ω t 

2

×  e − γ t  2 (γ − β ) cos ω t + [ β − γ ] + ω 12 − ω 2
ω 


(
)
sin ω 1t  
2

+ e − β t  2 ( β − γ ) cos ω 1t + [ β − γ ] + ω 2 − ω 12
ω  

(
3-39.
 sin ω t
F (t) = 
 0
0<t<π ω
π ω < t < 2π ω
From Equations 3.89, 3.90, and 3.91, we have
F (t) =
2
∞
1
a0 + ∑ ( an cos nω t + bn sin nω t )
2
n=1
τ
an =
F ( t ′ ) cos nω t ′ dt′
τ∫
bn =
2
F ( t ′ ) sin nω t ′ dt′
τ∫
an =
ω πω
sin ω t ′ cos nω t ′ dt ′
π ∫0
a0 =
ω πω
2
sin ω t ′ dt ′ =
∫
0
π
π
a1 =
ω πω
sin ω t ′ cos ω t ′ dt ′ = 0
π ∫0
0
τ
0
)
(5)
120
CHAPTER 3
π ω
ω πω
ω  cos (1 − n) ω t ′ cos (1 + n) ω t 
an ( n ≥ 2) = ∫ sin ω t ′ cos nω t ′ dt ′ =  −
−

π 0
π
2 (1 − n) ω
2 (1 + n) ω  0
Upon evaluating and simplifying, the result is
2

 π 1 − n2
an = 

0

(
n even
)
n = 0,1,2,…
n odd
b0 = 0 by inspection
b1 =
ω
π
∫
π ω
0
sin 2 ω t ′ dt ′ =
1
2
π ω
ω πω
ω  sin (1 − n) ω t ′ sin (1 + n) ω t ′ 
bn ( n ≥ 2) = ∫ sin ω t ′ sin nω t ′ dt ′ = 
−

π 0
π  2 (1 − n) ω
2 (1 + n) ω  0
=0
So
F (t) =
1
π
+
∞
1
2
sin 2 ω t + ∑
cos nω t
2
2
n = 2,4,6,… π 1 − n
(
)
or, letting n → 2n
F (t) =
1
π
+
∞
1
2
sin ω t + ∑
cos 2nω t
2
2
n = 1,2,… π 1 − 4 n
(
)
The following plot shows how well the first four terms in the series approximate the function.
1.0
Sum of first
four terms
0.5
π
ω
3-40.
2π
ω
t
The equation describing the car’s motion is
m
d2 y
= − k ( y − a sin ω t )
dt 2
where y is the vertical displacement of the car from its equilibrium position on a flat road, a is
the amplitude of sine-curve road, and
k = elastic coefficient =
dm × g 100 × 9.8
=
= 98000 N/m
dy
0.01
121
OSCILLATIONS
ω=
2π v0
λ
= 174 rad/s with v0 and λ being the car’s speed and wavelength of sine-curve road.
The solution of the motion equation can be cast in the form
y ( t ) = B cos (ω 0 t + β ) +
aω 0 2
sin ω t
ω 02 − ω 2
with
k
= 9.9 rad/s
m
ω0 =
We see that the oscillation with angular frequency ω has amplitude
A=
aω 02
= −0.16 mm
ω 02 − ω 2
The minus sign just implies that the spring is compressed.
3-41.
a) The general solution of the given differential equation is (see Equation (3.37))
(
(
)
)
x ( t ) = exp ( − β t )  A1 exp t β 2 − ω 02 + A2 exp −t β 2 − ω 02 
and
(
(
)
)
v(t) = x ′ ( t ) = − β exp ( − β t )  A1 exp t β 2 − ω 02 + A2 exp −t β 2 − ω 02 
(
(
)
)
+ exp ( − β t ) β 2 − ω 02  A1 exp t β 2 − ω 02 − A2 exp −t β 2 − ω 02 
at t = 0, x(t) = x0 , v(t) = v0
A1 =
⇒
v + β x0 
1
x0 + 0

2
β 2 − ω 02 
A1 =
and
v + β x0 
1
x0 − 0

2
β 2 − ω 02 
(1)
b)
i) Underdamped, β =
ω0
2
In this case, instead of using above parameterization, it is more convenient to work with the
following parameterization
(
x(t ) = A exp ( − β t ) cos t ω 0 2 − β 2 − δ
(
)
)
(2)
(
)
v(t) = − A exp ( − β t ) β cos t ω 02 − β 2 − δ + ω 02 − β 2 sin t ω 02 − β 2 − δ 
Using initial conditions of x(t) and v(t), we find
A=
2
x0
ω −β
2
0
2
v0
+β
x0
 v0

2
2
 x + β  + ω 0 − β and tan (δ ) =
ω 02 − β 2
0
(3)
122
In the case β =
CHAPTER 3
ω0
2
, and using (6) below we have
tan δ =
A=
2 v0
1
1
+
=−
3 x 0ω 0
3
3
⇒ δ = −30°
v02 v0ω 0
2
+
+ ω 02 =
x0
2
x0
x0
3
2 x0
3 ω0
so finally
 3

2
 1

ω 0 + 30°
x0 exp − ω 0 t cos  t




2
2
3
x(t ) =
(4)
ii) Critically damped, β = ω 0 , using the same parameterization as in i) we have from (2) and
(3):
x(t ) = A exp ( − β t ) = x0 exp ( −ω 0 t )
(5)
and v(t) = x ′(t) = −ω 0 x0 exp(−ω 0 t ⇒ v0 = −ω 0 x0
(6)
iii) Overdamped, β = ω 0 , returning to the original parameterization (1) we have (always
using relation (6)),
(
(
)
)
x ( t ) = exp ( − β t )  A1 exp t β 2 − ω 02 + A2 exp −t β 2 − ω 02 
=
(
3 + 1) x0
exp
2 3
((
) (
3 − 2) ω 0 t +
3 − 1) x0
exp − ( 3 + 2) ω 0 t
2 3
(
)
(7)
Below we show sketches for equations (4), (5), (7)
x
Underdamped
Critically damped
t
Overdamped
3-42.
a)
m ( x ′′ + ω 0 2 x ) − F0 sin ω t = 0
The most general solution is
x(t ) = a sin ω 0 t + b cos ω 0 t + A sin ω t
where the last term is a particular solution.
(1)
123
OSCILLATIONS
To find A we put this particular solution (the last term) into (1) and find
A=
F0
m (ω − ω 2 )
2
0
At t = 0, x = 0, so we find b = 0, and then we have
x(t ) = a sin ω 0 t + A sin ω t ⇒ v(t) = aω 0 cos ω 0 t + Aω cos ω t
At t = 0, v = 0 ⇒
A=
− aω 0
ω
x(t ) =
⇒
F0
1
(ω 0 sin ωt − ω sin ω 0t)
mω 0 (ω 0 + ω )(ω 0 − ω )
b) In the limit ω → ω 0 one can see that
x(t ) →
F0 t 3ω 0
6m
The sketch of this function is shown below.
x
t
3-43.
a) Potential energy is the elastic energy:
U(r ) =
1
k(r − a)2 ,
2
where m is moving in a central force field. Then the effective potential is (see for example,
Chapter 2 and Equation (8.14)):
U eff ( r ) = U (r ) +
l2
1
l2
2
=
k
(
r
−
a
)
+
2mr 2 2
2mr 2
where l = mvr = mω r 2 is the angular momentum of m and is a conserved quantity in this
problem. The solid line below is U eff ( r ) ; at low values of r, the dashed line represents
1
l2
k(r − a)2 , and the solid line is dominated by
. At large values of r,
2
2mr 2
1
U eff ( r ) ≅ U (r ) = k( r − a)2 .
2
U(r ) =
124
Potential energy
CHAPTER 3
U (r ) =
l2
2mr 2
1
k ( r − a )2
2
U eff (r )
r
b) In equilibrium circular motion of radius r0 , we have
k ( r0 − a ) = mω 02 r0
⇒ ω0 =
k ( r0 − a )
mr0
c) For given (and fixed) angular momentum l, V(r) is minimal at r0, because V ′(r ) r = r = 0 , so
0
we make a Taylor expansion of V(r) about r0 ;
3mω 02 (r − r0 )2 K (r − r0 )2
1
V (r ) = V (r0 ) + (r − r0 )V ′(r0 ) + (r − r0 )2 V ′′( r0 ) + ... ≈
=
2
2
2
where K = 3mω 02 , so the frequency of oscillation is
ω=
K
= 3ω 0 =
m
3k(r0 − a)
mr0
3-44. This oscillation must be underdamped oscillation (otherwise no period is present).
From Equation (3.40) we have
x(t ) = A exp ( − β t ) cos (ω 1t − δ )
so the initial amplitude (at t = 0) is A.
Now at t = 4T =
8π
ω1
8π 

x(4T ) = A exp  − β  cos (8π − δ )
ω1 

8π 

The amplitude now is A exp  − β  , so we have
ω1 

125
OSCILLATIONS
8π 

A exp  − β 
ω1  1

=
A
e
and because β = ω 02 − ω 12 , we finally find
ω1
8π
=
ω0
64π 2 + 1
3-45.
Energy of a simple pendulum is
mgl 2
θ where θ is the amplitude.
2
For a slightly damped oscillation θ (t) ≈ θ exp( −β t) .
Initial energy of pendulum is
mgl 2
θ .
2
Energy of pendulum after one period, T = 2π
l
, is
g
mgl
mgl 2
θ (T )2 =
θ exp (−2β T )
2
2
So energy lost in one period is
mgl 2
mgl 2
θ ( 1 − exp ( −2β T ) ) ≈
θ 2β T = mglθ 2 β T
2
2
So energy lost after 7 days is
mglθ 2 β T
(7 days)
= mglθ 2 β (7 days)
T
This energy must be compensated by potential energy of the mass M as it falls h meters:
Mgh = mglθ 2 β (7 days) ⇒ β =
Mh
= 0.01 s −1
mlθ (7 days)
2
Knowing β we can easily find the coefficient Q (see Equation (3.64))
ω 0 2 − 2β 2
ω
=
Q= R =
2β
2β
g
− 2β 2
l
= 178
2β
126
CHAPTER 3
CHAPTER
4
Nonlinear Oscillations
and Chaos
4-1.
0
= 0 + d
s
x
m
d
d
0
= 0 + d
(b)
(a)
m
θ
(c)
The unextended length of each spring is 0 , as shown in (a). In order to attach the mass m, each
spring must be stretched a distance d, as indicated in (b). When the mass is moved a distance x,
as in (c), the force acting on the mass (neglecting gravity) is
F = −2k ( s −
0
) sin θ
s=
+ x2
(1)
where
2
(2)
and
sin θ =
x
2
(3)
+ x2
Then,
F ( x) = −
2kx
2

+x 

= −2kx 1 −

2
2
+ x2 −
0
=−

2kx
2

+x 
2
2
+ x2 − ( − d) 

−1 2

−d 
− d  x2  
 = −2kx 1 −
1 + 2  
2
+ x2 


 
(4)
127
128
CHAPTER 4
Expanding the radical in powers of x 2
2
and retaining only the first two terms, we have

− d  1 x2 
F ( x ) ≅ −2kx 1 −
1 − 2 2  



  d  1 − d x2 
= −2kx 1 − 1 −  +
2 
 2
 

=−
2kd
x−
k ( − d)
x3
(5)
U ( x ) = − ∫ F ( x ) dx
(6)
3
The potential is given by
so that
U ( x) =
kd
x2 +
k ( − d) 4
x
4 3
(7)
4-2. Using the general procedure explained in Section 4.3, the phase diagram is constructed
as follows:
U(x)
E
E2 1
E
E4 3
E
E6 5
x
x·
E1
E2
E4
E5
E6
E3
x
129
NONLINEAR OSCILLATIONS AND CHAOS
4-3.
The potential U ( x ) = − ( λ 3x ) has the form shown in (a) below. The corresponding phase
3
diagram is given in (b):
(a)
U(x)
E5
E
E3 4
E
E1 2
E1
x
E2
x·
(b)
x
4-4.
Differentiation of Rayleigh’s equation above yields
(
)
x − a − 3bx 2 x + ω 02 x = 0
(1)
The substitution,
y = y0
3b
x
a
(2)
implies that
x=
a y
3b y 0
x=
a y
3b y 0
x=
a y
3b y 0











(3)
When these expressions are substituted in (1), we find
3ba y 2  y
a y
a 
a y
−
+ ω 02
=0
a −
2 
3b y 0
3b 
3b y 0
b y0  y0
Multiplying by y0
(4)
3b
and rearranging, we arrive at van der Pol’s equation:
a
y−
(
)
a 2
y0 − y 2 y + ω 02 y = 0
2
y0
(5)
130
CHAPTER 4
4-5.
a) A graph of the functions f1 ( x ) = x 2 + x + 1 and f 2 ( x ) = tan x in the region 0 ≤ x ≤ π 2 shows
that there is an intersection (i.e., a solution) for x1 ≅ 3π 8 .
tan x
x2 + x +1
O
π⁄2
x
The procedure is to use this approximate solution as a starting point and to substitute x1 = 3π 8
into f1 ( x ) and then solve for x = tan −1  f1 ( x1 )  . If the result is within some specified amount,
say 10 −4 , of 3π 8 , then this is our solution. If the result is not within this amount of the starting
value, then use the result as a new starting point and repeat the calculation. This procedure
leads to the following values:
x1
f1 ( x1 ) = x12 + x1 + 1
tan −1  f1 ( x1 ) 
Difference
1.1781
1.2974
1.3247
1.3304
1.3316
1.3318
3.5660
3.9806
4.0794
4.1004
4.1047
4.1056
1.2974
1.3247
1.3304
1.3316
1.3318
1.3319
0.11930
0.02728
0.00573
0.00118
0.00024
0.00005
Thus, the solution is x = 1.3319.
Parts b) and c) are solved in exactly the same way with the results:
b) x = 1.9151
c) x = 0.9271
4-6.
For the plane pendulum, the potential energy is
u = mg 1 − cos θ 
(1)
If the total energy is larger than 2mg , all values of θ are allowed, and the pendulum revolves
continuously in a circular path. The potential energy as a function of θ is shown in (a) below.
U
2mg
mg
θ
–π
–π⁄2
O
(a)
Since T = E – U(θ ), we can write
π⁄2
π
131
NONLINEAR OSCILLATIONS AND CHAOS
T=
1
1
mv 2 = m 2θ 2 = E − mg (1 − cos θ )
2
2
(2)
and, therefore, the phase paths are constructed by plotting
θ=
12
2
 E − mg (1 − cos θ ) 
2 
m
(3)
versus θ. The phase diagram is shown in (b) below.
·
θ
E = 2mg
3
E = mg
2
E = 3mg
5
E = mg
2
E = mg
–π
−
π
2
π
2
π
θ
(b)
4-7.
Let us start with the equation of motion for the simple pendulum:
θ = −ω 02 sin θ
where ω 2 ≡ g
(1)
. Put this in terms of the horizontal component by setting y ≡ x
= sin θ .
Solving for θ and taking time derivatives, we obtain
θ=
yy 2
y
+
2 32
(1 − y )
1 − y2
(2)
2
Since we are keeping terms to third order, we need to get a better handle on the y term. Help
comes from the conservation of energy:
1 2 2
m θ − mg cos θ = − mg cos θ 0
2
(3)
where θ 0 is the maximum angle the pendulum makes, and serves as a convenient parameter
that describes the total energy. When written in terms of y , the above equation becomes (with
the obvious definition for y0 )
y2
= 2ω 02
2
1− y
(
1 − y 2 − 1 − y02
Substituting (4) into (2), and the result into (1) gives
)
(4)
132
CHAPTER 4
(
)
y + ω 02 y 3 1 − y 2 − 2 1 − y02 = 0
(5)
Using the binomial expansion of the square roots and keeping terms up to third order, we can
obtain for the x equation of motion
 x2  3g
x + ω 02 x 1 + 20  − 3 x 3 = 0

 2
4-8.
(6)
For x > 0, the equation of motion is
mx = − F0
(1)
If the initial conditions are x ( 0 ) = A , x ( 0 ) = 0 , the solution is
x (t) = A −
F0 2
t
2m
(2)
For the phase path we need x = x ( x ) , so we calculate
x ( x) = ±
2F0
( A − x)
m
(3)
Thus, the phase path is a parabola with a vertex on the x-axis at x = A and symmetrical about
both axes as shown below.
x·
A
x
t=0
t=
1
τ
4
Because of the symmetry, the period τ is equal to 4 times the time required to move from x = A
to x = 0 (see diagram). Therefore, from (2) we have
τ=4
4-9.
2mA
F0
(4)
The proposed force derives from a potential of the form
1 2

kx

2
U ( x) = 
 1 ( x + δ ) x − δ ax 2
 2
which is plotted in (a) below.

x < a

x > a

133
NONLINEAR OSCILLATIONS AND CHAOS
U(x)
E7
E6
E5
E4
E3
E2
E1
–a
O
x
a
(a)
For small deviations from the equilibrium position (x = 0), the motion is just that of a harmonic
oscillator.
For energies E < E6 , the particle cannot reach regions with x < –a, but it can reach regions of
x > a if E > E4 . For E2 < E < E4 the possibility exists that the particle can be trapped near x = a.
A phase diagram for the system is shown in (b) below.
x·
…
E2
E1
E3
E4
E7
x
(b)
4-10.
The system of equations that we need to solve are
y
x 

 y  =  −0.05 y − sin x + 0.7 cos ω t 
  

(1)
The values of ω that give chaotic orbits are 0.6 and 0.7. Although we may appear to have chaos
for other values, construction of a Poincaré plot that samples at the forcing frequency show that
they all settle on a one period per drive cycle orbit. This occurs faster for some values of ω than
others. In particular, when ω = 0.8 the plot looks chaotic until it locks on to the point
(−2.50150, 0.236439) . The phase plot for ω = 0.3 shown in the figure was produced by numerical
integration of the system of equations (1) with 100 points per drive cycle. The box encloses the
point on the trajectory of the system at the start of a drive cycle. In addition, we also show
Poincaré plot for the case ω = 0.6 in figure, integrated over 8000 drive cycles with 100 points per
cycle.
134
CHAPTER 4
1
0.5
0
-0.5
–1
–1.5
–1
-0.5
0
0.5
1
1.5
3
2
1
0
–1
–2
–3
–4
–3
–2
–1
0
1
2
3
4
4-11. The three-cycle does indeed occur where indicated in the problem, and does turn
chaotic near the 80th iteration. This value is approximate, however, and depends on the
precision at which the calculations are performed. The behavior returns to a three-cycle near the
200th iteration, and stays that way until approximately the 270th iteration, although some may
see it continue past the 300th.
1
0.8
x
0.6
0.4
0.2
0
100
200
300
iteration
400
500
135
NONLINEAR OSCILLATIONS AND CHAOS
4-12.
x1 = 0.4
0.25
0.2
0.15
0.1
0.05
0
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
x1 = 0.75
0.25
0.2
0.15
0.1
0.05
0
0
0.2
0.4
These plots are created in the manner described in the text. They are created with the logistic
equation
xn + 1 = 0.9 ⋅ xn (1 − xn )
The first plot has the seed value x1 = 0.4 as asked for in the text. Only one additional seed has
been done here ( x1 = 0.75 ) as it is assumed that the reader could easily produce more of these
plots after this small amount of practice.
(1)
136
CHAPTER 4
4-13.
1
0.8
0.6
0.4
0.2
0
30
32
34
36
38
40
42
iteration
44
46
48
50
38
40
42
iteration
44
46
48
50
x1 = 0.7
x1 = 0.700000001
1
0.8
0.6
0.4
0.2
0
30
32
34
36
x1 = 0.7
x1 = 0.7000000001
The plots are created by iteration on the initial values of (i) 0.7, (ii) 0.700000001, and (iii)
0.7000000001, using the equation
(
xn + 1 = 2.5 ⋅ xn 1 − xn2
)
(1)
A subset of the iterates from (i) and (ii) are plotted together, and clearly diverge by n = 39. The
plot of (i) and (iii) clearly diverge by n = 43.
4-14.
1
0.8
0.6
0.4
0.2
0
20
22
24
26
28
iteration
x1 = 0.9
x1 = 0.9000001
fractional difference
30
32
34
137
NONLINEAR OSCILLATIONS AND CHAOS
The given function with the given initial values are plotted in the figure. Here we use the
notation x1 = 0.9 and y1 = 0.9000001 , with xn + 1 = f ( xn ) and yn + 1 = f ( yn ) where the function is
(
f ( x ) = 2.5 ⋅ x 1 − x 2
)
(1)
The fractional difference is defined as x − y x , and clearly exceeds 30% when n = 30.
4-15. A good way to start finding the bifurcations of the function f(α,x) = α sin πx is to plot its
bifurcation diagram.
1
0.8
x
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
α
One can expand regions of the diagram to give a rough estimate of the location of a bifurcation.
Its accuracy is limited by the fact that the map does not converge very rapidly near the
bifurcation point, or more precisely, the Lyapunov exponent approaches zero. One may
continue undaunted, however, with the help of a graphical fractal generating software
application, to estimate quite a few of the period doublings α n .Using Fractint for Windows, and
Equation (4.47) to compute the Feigenbaum constant, we can obtain the following:
n
1
2
3
4
5
6
7
8
α
δ
0.71978
0.83324
0.85859
0.86409
0.86526
0.86551
0.865564
0.8655761
4.475
4.611
4.699
4.680
4.630
4.463
One can see that although we should obtain a better value of δ as n increases, numerical
precision and human error quickly degrade the quality of the calculation. This is a perfectly
acceptable answer to this question.
One may compute the α n to higher accuracy by other means, all of which are a great deal more
complicated. See, for example, Exploring Mathematics with Mathematica, which exploits the
vanishing Lyapunov exponent. Using their algorithm, one obtains the following:
138
CHAPTER 4
n
1
2
3
4
5
6
7
α
δ
0.719962
0.833266
0.858609
0.864084
0.865259
0.865511
0.865560
4.47089
4.62871
4.66198
4.65633
5.13450
Note that these are shown here only as reference, and the student may not necessarily be
expected to perform to this degree of sophistication. The above values are only good to about
10 −6 , but this time only limited by machine precision. Another alternative in computing the
Feigenbaum constant, which is not requested in the problem, is to use the so-called
”supercycles,” or super-stable points Rn , which are defined by
f
( 2n−1 )  R , 1  = 1
 n 

2
2
The values Rn obey the same scaling as the bifurcation points, and are much easier to compute
since these points converge faster than for other α (the Lyapunov exponent goes to – ∞). See, for
example, Deterministic Chaos: An Introduction by Heinz Georg Schuster or Chaos and Fractals: New
Frontiers of Science by Peitgen, Jürgens and Saupe. As a result, the estimates for δ obtained in
this way are more accurate than those obtained by calculating the bifurcation points.
4-16.
The function y = f(x) intersects the line y = x at x = x0 , i.e. x0 is defined as the point
where x0 = f ( x0 ) . Now expand f(x) in a Taylor series, so that near x0 we have
f ( x)
f ( x0 ) + β ( x − x0 ) = x 0 + β ( x − x0 )
(1)
where
β≡
df
dx
(2)
x0
Now define ε n ≡ xn − x0 . If we have x1 very close to x0 , then ε 1 should be very small, and we
may use the Taylor expansion. The equation of iteration xn + 1 = f ( xn ) becomes
ε n+1
βε n
If the approximation (1) remains valid from the initial value, we have ε n +1
(3)
β nε1 .
a) The values xn − x0 = ε n form the geometric sequence ε 1 , βε 1 , β 2 ε 1 , … .
b) Clearly, when β < 1 we have stability since
lim ε n = 0
n →∞
Similarly we have a divergent sequence when β > 1 , although it will not really be exponentially
divergent since the approximation (1) becomes invalid after some number of iterations, and
normally the range of allowable xn is restricted to some subset of the real numbers.
139
NONLINEAR OSCILLATIONS AND CHAOS
4-17.
0.6
0.4
0.2
0
0
2
4
6
8
10
12
14
16
18
20
iteration
α = 0.4
α = 0.7
The first plot (with α = 0.4) converges rather rapidly to zero, but the second (with α = 0.7) does
appear to be chaotic.
4-18.
1
0.8
x
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
α
The tent map always converges to zero for α < 0.5. Near α = 0.5 it takes longer to converge, and
that is the artifact seen in the figure. There exists a “hole” in the region 0.5 < α < 0.7 (0.7 is
approximate), where the iterations are chaotic but oscillate between an upper and lower range
of values. For α > 0.7, there is only a single range of chaos, which becomes larger until it fills the
range (0,1) at α = 1.
4-19.
From the definition in Equation (4.52) the Lyapunov exponent is given by
λ = lim
n→∞
df
1 n −1
ln
∑
n i=0
dx
(1)
xi
The tent map is defined as
for 0 < x < 1 2
 2α x
f ( x) = 
 2α (1 − x ) for 1 2 < x < 1

This gives df dx = 2α , so we have
(2)
140
CHAPTER 4
n − 1
 ln ( 2α ) = ln ( 2α )
n→∞  n 
λ = lim 
(3)
As indicated in the discussion below Equation (4.52), chaos occurs when λ is positive: α > 1 2
for the tent map.
4-20.
0.4
y
0.2
0
–0.2
–0.4
–1.5
–1
–0.5
0
0.5
1
1.5
0.5
1
1.5
x
4-21.
0.4
y
0.2
0
–0.2
–0.4
–1.5
–1
–0.5
0
x
The shape of this plot (the attractor) is nearly identical to that obtained in the previous problem.
In Problem 4-20, however, we can clearly see the first few iterations (0,0), (1,0), (–0.4,0.3),
NONLINEAR OSCILLATIONS AND CHAOS
141
whereas the next iteration (1.076,–0.12) is almost on the attractor. In this problem the initial
value is taken to be on the attractor already, so we do not see any transient points.
4-22.
The following. system of differential equations were integrated numerically
y
x 

 y  =  −0.1 y − x 3 + B cos t 
  

(1)
using different values of B in the range [9.8,13.4], and with a variety of initial conditions. The
integration range is over a large number of drive cycles, throwing away the first several before
starting to store the data in order to reduce the effects of the transient response. For the case
B = 9.8, we have a one period per three drive cycle orbit. The phase space plot (line) and
Poincaré section (boxes) for this case are overlaid and shown in figure (a). All integrations are
done here with 100 points per drive cycle. One can experiment with B and determine that the
system becomes chaotic somewhere between 9.8 and 9.9. The section for B = 10.0, created by
integrating over 8000 drive cycles, is shown in figure (b). If one further experiments with
different values of B, and one is also lucky enough to have the right initial conditions, (0,0) is
one that works, then a transition will be found for B in the range (11.6,11.7). As an example of
the different results one can get depending on the initial conditions, we show two plots in
figure (c). One is a phase plot, overlaid with its section, for B = 12.0 and the initial condition
(0,0). Examination of the time evolution reveals that it has one period per cycle. The second plot
is a Poincaré section for the same B but with the initial condition (10,0), clearly showing chaotic
motion. Note that the section looks quite similar to the one for B = 10.0. Another transition is in
the range (13.3,13.4), where the orbits become regular again, with one period per drive cycle,
regardless of initial conditions. The phase plot for B = 13.4 looks similar to the one with B = 12.0
and initial condition (0,0).
To summarize, we may enumerate the above transition points by B1 , B2 , and B3 .
Circumventing the actual task of computing where these transition points are, we do know that
9.8 < B1 < 9.9 , 11.6 < B2 < 11.7 , and 13.3 < B3 < 13.4 . We can then describe the behavior of the
system by region.
•
B < B1 : one period per three drive cycles
•
B1 < B < B2 : chaotic
•
B2 < B < B3 : mixed chaotic/one period per drive cycle (depending on initial conditions)
•
B3 < B : one period per drive cycle
We should remind ourselves, though, that the above list only applies for B in the range we have
examined here. We do not know the behavior when B < 9.8 and B > 13.4, without going beyond
the scope of this problem.
142
CHAPTER 4
(a)
y
5
0
–5
–3
–2
–1
0
1
2
3
x
10
(b)
y
5
0
–5
2.4
2.6
2.8
3
3.2
3.4
1
2
3.6
x
5
y
(c)
0
–5
–3
–2
–1
0
3
x
10
(d)
y
5
0
–5
–10
2.4
2.6
2.8
3
3.2
x
3.4
3.6
3.8
143
NONLINEAR OSCILLATIONS AND CHAOS
4-23.
The Chirikov map is defined by
pn + 1 = pn − K sin qn
(1)
qn + 1 = qn − pn + 1
(2)
The results one should get from doing this problem should be some subset of the results shown
in figures (a), (b), and (c) (for K = 0.8, 3.2, and 6.4, respectively). These were actually generated
using some not-so-random initial points so that a reasonably complete picture could be made.
What look to be phase paths in the figures are actually just different points that come from
iterating on a single initial condition. For example, in figure (a), an ellipse about the origin (just
pick one) comes from iterating on any one of the points on it. Above the ellipses is chaotic orbit,
then a five ellipse orbit (all five come from a single initial condition), etc. The case for K = 3.2 is
similar except that there is an orbit outside of which the system is always undergoing chaotic
motion. Finally, for K = 6.4 the entire space is filled with chaotic orbits, with the exception of
two small lobes. Inside of these lobes are regular orbits (the ones in the left are separate from the
ones in the right).
1
(a)
p⁄π
0.5
0
–0.5
–1
–1
–0.5
0
0.5
1
0.5
1
q⁄π
1
(b)
p⁄π
0.5
0
–0.5
–1
–1
–0.5
0
q⁄π
144
CHAPTER 4
1
(c)
p⁄π
0.5
0
–0.5
–1
–1
–0.5
0
0.5
1
q⁄π
4-24.
a) The Van de Pol equation is
d2 x
dx
+ ω 0 2 x = µ ( a2 − x 2 )
2
dt
dt
Now look for solution in the form
x(t ) = b cos ω 0 t + u(t)
(1)
we have
dx
du
= − bω 0 sin ω 0 t +
dt
dt
and
d2 x
d2u
2
=
−
b
ω
cos
ω
t
+
0
0
dt 2
dt 2
Putting these into the Van de Pol equation, we obtain
d 2 u(t)
+ ω 0 u(t) = − µ b 2 cos 2 ω 0 t + u2 (t) + 2bu(t ) cos ω 0 t − a 2
dt 2
{
}
{
− bω 0 sin ω 0 t +
du(t )
dt
}
From this one can see that u(t) is of order µ (i.e. u ~ O(µ ) ), which is assumed to be small here.
Keeping only terms up to order µ , the above equation reads
d 2 u(t)
+ ω 0 u(t ) = − µ − b 3ω 0 sin ω 0 t cos 2 ω 0 t + a 2 bω 0 sin ω 0 t
2
dt
{
}

b2 
b2

= − µbω 0  a 2 −  sin ω 0 t − sin 3ω 0 t 


4
4


(where we have used the identity 4 sin ω 0 t cos 2 ω 0 t = sin ω 0 t + sin 3ω 0 t )
This equation has 2 frequencies ( ω 0 and 3ω 0 ), and is complicated. However, if b = 2 a then the
term sin ω 0 t disappears and the above equation becomes
145
NONLINEAR OSCILLATIONS AND CHAOS
d 2 u(t)
b3
(
)
sin 3ω 0 t
+
ω
=
µω
u
t
0
0
4
dt 2
We let b = 2 a , and the solution for this equation is
u(t ) = −
µb 3
µ a3
sin 3ω 0 t = −
sin 3ω 0 t
32ω 0
4ω 0
So, finally putting this form of u(t) into (1), we obtain one of the exact solutions of Van de Pol
equation:
u(t ) = 2 a cos ω 0 t −
µ a3
sin 3ω 0 t
4ω 0
b) See phase diagram below. Since µ = 0.05 is very small, then actually the second term in the
expression of u(t) is negligible, and the phase diagram is very close to a circle of radius
b = 2a = 2.
.
x
2
1
–2
–1
1
2
x
–1
–2
4-25. We have used Mathematica to numerically solve and plot the phase diagram for the van
de Pol equation. Because µ = 0.07 is a very small value, the limit cycle is very close to a circle of
radius b = 2a = 2.
a) In this case, see figure a), the phase diagram starts at the point (x = 1, x′ = 0) inside the limit
cycle, so the phase diagram spirals outward to ultimately approach the stable solution
presented by the limit cycle (see problem 4-24 for exact expression of stable solution).
146
CHAPTER 4
.
x
2
1
–2
–1
1
2
x
–1
–2
b) In this case, see figure b), the phase diagram starts at the point (x = 3, x′ = 0) outside the
limit cycle, so the phase diagram spirals inward to ultimately approaches the stable solution
presented by the limit cycle (see problem 4-24 for exact expression of stable solution).
.
x
2
1
–2
–1
1
2
3
x
–1
–2
4-26. We have used Mathematica to numerically solve and plot the phase diagram for the van
de Pol equation. Because µ = 0.5 is not a small value, the limit cycle is NOT close to a circle (see
problem 4-24 above).
a) In this case, see figure a), the phase diagram starts at the point (x = 1, x′ = 0) inside the limit
cycle, so the phase diagram spirals outward to ultimately approach the stable solution
presented by the limit cycle (see problem 4-24 for exact expression of stable solution).
147
NONLINEAR OSCILLATIONS AND CHAOS
.
x
2
1
–2
–1
1
2
x
–1
–2
b) In this case (see figure below), the phase diagram starts at the point (x = 3, x′ = 0) outside
the limit cycle, so the phase diagram spirals inward to ultimately approaches the stable solution
presented by the limit cycle (see problem 4-24 for exact expression of stable solution).
.
x
2
1
–2
–1
1
–1
–2
2
3
x
148
CHAPTER 4
CHAPTER
5
Gravitation
5-1.
a) Two identical masses:
The lines of force (dashed lines) and the equipotential surfaces (solid lines) are as follows:
b) Two masses, +M and –M:
In this case the lines of force do not continue outward to infinity, as in a), but originate on the
“negative” mass and terminate on the positive mass. This situation is similar to that for two
electrical charges, +q and –q; the difference is that the electrical lines of force run from +q to –q.
149
150
5-2.
CHAPTER 5
Inside the sphere the gravitational potential satisfies
∇ 2 φ = 4π G ρ ( r )
(1)
Since ρ(r) is spherically symmetric, φ is also spherically symmetric. Thus,
1 ∂  2 ∂φ 
= 4π G ρ ( r )
r
r 2 ∂r  ∂r 
(2)
The field vector is independent of the radial distance. This fact implies
∂φ
= constant ≡ C
∂r
(3)
2C
= 4π Gρ
r
(4)
Therefore, (2) becomes
or,
ρ=
C
2π Gr
(5)
5-3. In order to remove a particle from the surface of the Earth and transport it infinitely far
away, the initial kinetic energy must equal the work required to move the particle from r = Re
to r = ∞ against the attractive gravitational force:
∫
∞
Re
G
Me m
1
dr = mv02
2
r
2
(1)
where Me and Re are the mass and the radius of the Earth, respectively, and v0 is the initial
velocity of the particle at r = Re .
Solving (1), we have the expression for v0 :
v0 =
2G Me
Re
(2)
Substituting G = 6.67 × 10 −11 m 3/kg ⋅ s 2 , Me = 5.98 × 10 24 kg , Re = 6.38 × 10 6 m , we have
v0 ≅ 11.2 km/sec
5-4.
(3)
The potential energy corresponding to the force is
U = − ∫ F dx = mk 2
dx
mk 2
=
−
∫ x 3 2x 2
The central force is conservative and so the total energy is constant and equal to the potential
energy at the initial position, x = d:
(1)
151
GRAVITATION
E = constant =
1
1 k2
1 k2
mx 2 − m 2 = − m 2
2
2 x
2 d
(2)
Rewriting this equation in integrable form,
0
dt = − ∫
d
dx
1
1
k  2 − 2
d 
x
=−
2
0
d
x
dx
k ∫d d 2 − x 2
(3)
where the choice of the negative sign for the radical insures that x decreases as t increases.
Using Eq. (E.9), Appendix E, we find
t=
d
d2 − x2
k
0
(4)
d
or
t=
5-5.
d2
k
The equation of motion is
mx = −G
Mm
x2
(1)
Using conservation of energy, we find
1 2
1
1
x − G M = E = −G M
2
x
x∞
(2)
1 1 
dx
= − 2GM  − 
dt
 x x∞ 
(3)
where x∞ is some fixed large distance. Therefore, the time for the particle to travel from x∞ to x
is
x
t =−∫
x∞
1
=−
GM
1 1 
2GM  − 
 x x∞ 
dx
x
∫
x∞
xx∞
dx
2 ( x∞ − x )
Making the change of variable, x → y 2 , and using Eq. (E.7), Appendix E, we obtain
x
t=
x∞ 
x 
−1
 x ( x∞ − x ) − x∞ sin

2GM 
x∞ 
x∞
If we set x = 0 and x = x∞ 2 in (4), we can obtain the time for the particle to travel the total
distance and the first half of the distance.
(4)
152
CHAPTER 5
0
∫
T0 =
1  x∞ 
GM  2 
dt =
x∞
x∞ 2
T1 2 =
∫
1
GM
dt =
x∞
 x∞ 
 2 
32
32
(5)
 π
1 + 2 
(6)
Hence,
T1 2
T0
1+
=
π
2
π
Evaluating the expression,
T1 2
T0
= 0.818
(7)
or
T1 2
≅
T0
9
11
(8)
5-6.
z
P
α
s
θ
r2drd(cos θ)dφ
r
y
φ
x
Since the problem has symmetry around the z-axis, the force at the point P has only a
z-component. The contribution to the force from a small volume element is
dg z = −G
where ρ is the density. Using cos α =
a
ρ
s2
r 2 dr d ( cos θ ) dφ cos α
z − r cos θ
and integrating over the entire sphere, we have
s
+1
2π
g z = −Gρ ∫ r 2 dr ∫ d ( cos θ ) ∫ dφ
0
(1)
−1
0
(r
Now, we can obtain the integral of cos θ as follows:
z − r cos θ
2
+ z 2 − 2rz cos θ
)
32
(2)
153
GRAVITATION
I=
+1
∫
−1
=−
(
z − r cos θ
r + z 2 − 2rz cos θ
2
∂
∂z
+1
∫ (r
2
)
32
d ( cos θ )
+ z 2 − 2rz cos θ
)
12
d ( cos θ )
−1
Using Eq. (E.5), Appendix E, we find
I=−
=−
∂
∂z
 1 2
r + z 2 − 2rz cos θ
−
rz

(
)
12
+1


−1 
∂ 2 2
=
∂z  z  z 2
(3)
Therefore, substituting (3) into (2) and performing the integral with respect to r and φ, we have
g z = −Gρ
= −G
But
a3 2
⋅ 2π
3 z2
4π 3 1
a ρ 2
z
3
(4)
4π 3
a ρ is equal to the mass of the sphere. Thus,
3
g z = −GM
1
z2
(5)
Thus, as we expect, the force is the same as that due to a point mass M located at the center of
the sphere.
5-7.
dx
s
x
P
R
The contribution to the potential at P from a small line element is
dΦ = −G
where ρ =
M
ρ
s
dx
(1)
is the linear mass density. Integrating over the whole rod, we find the potential
Φ = −G
M
∫
2
− 2
1
x 2 + R2
dx
(2)
154
CHAPTER 5
Using Eq. (E.6), Appendix E, we have
Φ = −G
M
ln  x + x 2 + R2 

−
Φ=−
2

ln 

GM
2
2

+
+ R2

GM  2
4
=−
ln
2

+ R2
− +
4
 2






+ 4 R2 + 

2
+ 4 R2 − 
2
(3)
5-8.
z
z0
r 2 + ( z0 − z )
α
2
a
z
x
rdrdθdz
y
θ r
Since the system is symmetric about the z-axis, the x and y components of the force vanish and
we need to consider only the z-component of the force. The contribution to the force from a
small element of volume at the point (r,θ,z) for a unit mass at (0,0, z0 ) is
rdrdθ dz
dg z = −Gρ
r + ( z0 − z )
2
2
cos α
( z0 − z) rdrdθ dz
32
 r 2 + ( z0 − z ) 2 


= −Gρ
(1)
where ρ is the density of the cylinder and where we have used cos α =
( z0 − z )
2
r 2 + ( z0 − z )
. We can
find the net gravitational force by integrating (1) over the entire volume of the cylinder. We find
a
2π
0
0
g z = −Gρ ∫ rdr
z0 − z
∫ dθ ∫ dz 
2
r 2 + ( z0 − z ) 


0
32
Changing the variable to x = z0 − z , we have
a
g z = 2π Gρ ∫ rdr
0
Using the standard integral,
z0 −
∫
z0
xdx
 r + x 2 
2
32
(2)
155
GRAVITATION
∫
xdx
(
a2 ± x
=
)
2 3
−1
(3)
a ± x2
2
we obtain

a
r
g z = −2π Gρ ∫ dr 

2
0
 r + ( z0 −
)
2
−


r 2 + z02 

r
(4)
Next, using Eq. (E.9), Appendix E, we obtain
g z = −2π Gρ  a 2 + ( z0 −

)2 −
a 2 + z02 + 

(5)
Now, let us find the force by first computing the potential. The contribution from a small
element of volume is
dΦ = −Gρ
rdrdθ dz
r + ( z0 − z )
2
2
2π
a
r
0
0
(6)
Integrating over the entire volume, we have
dΦ = −Gρ ∫ dz
0
∫ dθ ∫ dr
r 2 + ( z0 − z )
(7)
2
Using Eq. (E.9), Appendix E, again, we find
2
dΦ = −2π Gρ ∫ dz  a 2 + ( z0 − z ) − ( z0 − z ) 


0
(8)
Now, we use Eqs. (E.11) and (E.8a), Appendix E, and obtain
 (z −
Φ = −2π Gρ  − 0
2

+
Thus, the force is
z0
2
)
a 2 + ( z0 −
a 2 + z02 −
)2 +
a2
ln  −2 ( z0 −

2
) + 2 ( z0 − ) 2 + a2 
a2 
1
ln −2z0 + 2 z02 + a 2  − z0 +


2
2
2



156
CHAPTER 5


∂Φ
1 2
= −2π Gρ 
gz = −
a + ( z0 −
∂z 0
2


−
1 2
1
a + z02 −
2
2
)
2
1
+
2
1−
( z0 − ) 2
2
a 2 + ( z0 − )
a2
+
2 −(z −
0
z0 −
( z0 − ) 2 + a 2
) + ( z0 − ) 2 + a 2


z +a
z
a
−
+ 

2 − z0 + z02 + a 2
a 2 + z02


2
0
2
z0
1−
2
0
2
(9)
or,
g z = −2π Gρ  a 2 + ( z0 −

)2 −
a 2 + z02 + 

(10
and we obtain the same result as in (5).
In this case, it is clear that it is considerably easier to compute the force directly. (See the
remarks in Section 5.4.)
5-9.
a
r
θ
P
R
The contribution to the potential at the point P from a small line element d is
Φ = −G ∫
ρd
(1)
r
where ρ is the linear mass density which is expressed as ρ =
M
. Using
2π a
r = R2 + a 2 − 2aR cos θ and d = adθ, we can write (1) as
Φ=−
GM
2π
2π
∫
0
dθ
R + a − 2aR cos θ
2
2
(2)
This is the general expression for the potential.
If R is much greater than a, we can expand the integrand in (2) using the binomial expansion:
a2  
1  a
= 1 −  2 cos θ − 2  
R 
R2 + a 2 − 2aR cos θ R   R
1
−1 2
2
 1 a
a2  3  a
a2   1
= 1 +  2 cos θ − 2  +  2 cos θ − 2   + …
R  8 R
R   R
 2  R
(3)
157
GRAVITATION
3
 a
If we neglect terms of order   and higher in (3), the potential becomes
 R
Φ=−
=−
2π
GM
2π
∫
0


3 a2
a
a2
θ
1
+
cos
−
+
cos 2 θ  dθ
 R
2
2
2R
2R



a2 3 a2 
π
π
−
+ π
2

R2 2 R 2 

GM
2π
(4)
or,
Φ ( R) ≅ −
GM  1 a 2 
1+
R  4 R 2 
(5)
We notice that the first term in (5) is the potential when mass M is concentrated in the center of
GMa 2
.
the ring. Of course this is a very rough approximation and the first correction term is −
4 R3
5-10.
P
φ
R
r
θ
a
x
R cos θ
R sin θ
Using the relations
( R sin θ ) 2 + a2 − 2aR sin θ cos φ
x=
r = x 2 + R2 cos 2 θ = R 2 + a 2 − 2 aR sin θ cos φ
ρ =
M
(the linear mass density),
2π a
(1)
(2)
(3)
the potential is expressed by
Φ = −G ∫
ρd
r
=
−GM
2π R
2π
∫
0
dφ
 a
a2 
1 −  2 sin θ cos φ − 2 
R 
 R
(4)
If we expand the integrand and neglect terms of order ( a R) and higher, we have
3
  a
a2  
1
2
sin
cos
θ
φ
−
−
 

R 2  
  R
Then, (4) becomes
−1 2
≅ 1+
a
1 a2 3 a2
sin θ cos φ −
sin 2 θ cos 2 φ
+
R
2 R2 2 R2
(5)
158
CHAPTER 5
Φ≅−

GM 
1 a2
3 a2
−
+
2
2
π
π
π sin 2 θ 

2
2
2π R 
2R
2R

Thus,
Φ ( R) ≅ −
GM
R
 1 a2  3

2
1 − 2 R 2 1 − 2 sin θ  



(6)
5-11.
P
a
r
θ
z
dm
The potential at P due to a small mass element dm inside the body is
dΦ = −G
dm
dm
= −G
2
2
r
z + a − 2 za cos θ
(1)
Integrating (1) over the entire volume and dividing the result by the surface area of the sphere,
we can find the average field on the surface of the sphere due to dm:
dΦ ave =
1
4π a 2
π


2π a 2 sin θ dθ
−
G
dm

∫ z 2 + a2 − 2za cos θ 
0


(2)
Making the variable change cos θ = x, we have
+1
dΦ ave
G
= − dm ∫
2
−1
(z
dx
2
(3)
)
+ a 2 − 2 zax
Using Eq. (E.5), Appendix E, we find
dΦ ave = −
G
 1
dm  −
2
 za
(z
2
)
+ a 2 − 2za +
=−
G
 − ( z − a) + ( z + a) 
dm 

2
za


=−
G
dm
z
1
za
(z
2

+ a 2 + 2 za 

)
(4)
This is the same potential as at the center of the sphere. Since the average value of the potential
is equal to the value at the center of the sphere at any arbitrary element dm, we have the same
relation even if we integrate over the entire body.
159
GRAVITATION
5-12.
dm
r
r'
O
R
P
Let P be a point on the spherical surface. The potential dΦ due to a small amount of mass dm
inside the surface at P is
dΦ = −
Gdm
r
(1)
The average value over the entire surface due to dm is the integral of (1) over dΩ divided by 4π.
Writing this out with the help of the figure, we have
dΦ ave = −
2π sin θ dθ
Gdm π
∫
2
0
4π
r ′ 2 + R − 2r ′R cos θ
(2)
Making the obvious change of variable and performing the integration, we obtain
dΦ ave = −
Gdm 1
du
Gdm
=−
∫
2
4π −1 r ′ 2 + R − 2r ′Ru
R
(3)
We can now integrate over all of the mass and get Φ ave = − Gm R . This is a mathematical
statement equivalent to the problem’s assertion.
5-13.
R1
R0
ρ1
R2
ρ2
R0 = position of particle. For R1 < R0 < R2 , we calculate the force by assuming that all mass for
which r < R0 is at r = 0, and neglect mass for which r > R0 . The force is in the radially inward
direction ( −er ).
The magnitude of the force is
F=
where M = mass for which r < R0
GMm
R02
160
CHAPTER 5
M=
So F = −
(
)
4
4
π R13 ρ1 + π R03 − R13 ρ2
3
3
4π Gm
ρ1R13 + ρ2 R03 − ρ2 R13 er
3R02
(
)
 ( ρ − ρ ) R3

4
F = − π Gm  1 22 1 + ρ2 R0  er
3
R0


5-14.
Think of assembling the sphere a shell at a time (r = 0 to r = R).
For a shell of radius r, the incremental energy is dU = dm φ where φ is the potential due to the
mass already assembled, and dm is the mass of the shell.
So
3 Mr 2 dr
 3M 
2
dm = ρ 4π r 2 dr = 
4
r
dr
π
=
3
R3
 4π R 
φ=−
Gm
r3
where m = M 3
r
R
So
U = ∫ du
 3 Mr 2 dr   GMr 2 
∫  R3   − R3 
r =0 
R
=
R
=−
3GM 2 4
r dr
R6 ∫0
U=−
3 GM 2
5 R
5-15. When the mass is at a distance r from the center of the Earth, the force is in the inward
radial direction and has magnitude Fr :
m
r
Fr =
Gm  4 3 
π r ρ  where ρ is the mass density of the Earth. The equation of motion is
r 2  3

161
GRAVITATION
Fr = mr = −
Gm  4 3 
π r ρ
r 2  3

or
r + ω 2 r = 0 where ω 2 =
4π Gρ
3
This is the equation for simple harmonic motion. The period is
T=
2π
3π
Gρ
=
ω
Substituting in values gives a period of about 84 minutes.
5-16.
M
z
θ
y
x
r 2 + h2
h
r
For points external to the sphere, we may consider the sphere to be a point mass of mass M. Put
the sheet in the x-y plane.
Consider force on M due to the sheet. By symmetry, Fx = Fy = 0
Fz = ∫ dFz =
With dm = ρs 2π rdr and cos θ =
∞
GMdm
cos θ
2
+ h2
∫ (r
r =0
)
h
r + h2
2
we have
Fz = 2πρsGMh
∞
∫
r =0
(r
rdr
2
+ h2

1
Fz = −2πρsGMh 
2
 r + h2

(
)
32
∞
)


12

0
Fz = 2πρsGM
The sphere attracts the sheet in the z -direction
with a force of magnitude 2πρsGM
162
CHAPTER 5
5-17.
y
water
x
moon
(not to scale)
Earth
Start with the hint given to us. The expression for g x and g y are given by
gx =
GMm y GMe y
2GMm x GMe x
−
−
; gy = −
3
3
D3
R3
D
R
(1)
where the first terms come from Equations (5.54) and the second terms come from the standard
assumption of an Earth of uniform density. The origin of the coordinate system is at the center
of the Earth. Evaluating the integrals:
∫
xmax
0
2
 2GMm GMe  xmax
;
g x dx = 
−
3
R3  2
 D
∫
ymax
0
2
 GMm GMe  ymax
−
g y dy =  −
3
R 3  2
 D
(2)
To connect this result with Example 5.5, let us write (1) in the following way
2
 GMe 2
ymax
GMm  2
2
xmax − ymax
 xmax +
=
3
3
2  2R
D 
(
)
(3)
The right-hand side can be factored as
GMe
GMe
x + ymax )( xmax − ymax ) =
( 2 R ) ( h ) = gh
3 ( max
2R
2R 3
2
If we make the approximation on the left-hand side of (3) that xmax
Equation (5.55). Turning to the exact solution of (3), we obtain
h = 2R
2
ymax
(4)
R 2 , we get exactly
Me M m
Me 2 M m
+ 3 −
−
3
R
D
R3
D3
Me Mm
Me 2 M m
+ 3 +
−
3
R
D
R3
D3
Upon substitution of the proper values, the answer is 0.54 m, the same as for Example 5.5.
Inclusion of the centrifugal term in g x does not change this answer significantly.
(5)
163
GRAVITATION
5-18.
From Equation (5.55), we have with the appropriate substitutions
hmoon
hsun
3GMm r 2
3
2 gD3
Mm  Res 
=
=
3GMs r 2
Ms  D 
3
2 gRes
(1)
Substitution of the known values gives
hmoon 7.350 × 10 22 kg  1.495 × 1011 m 
=
hsun
1.993 × 10 30 kg  3.84 × 10 8 m 
3
2. 2
(2)
5-19.
ωearth
ωmoon
Because the moon’s orbit about the Earth is in the same sense as the Earth’s rotation, the
difference of their frequencies will be half the observed frequency at which we see high tides.
Thus
1
1
1
=
−
2Ttides Tearth Tmoon
which gives Ttides
5-20.
dΦ( z) =
12 hours, 27 minutes.
The differential potential created by a thin loop of thickness dr at the point (0,0,z) is
2π rdrM −GM d ( r 2 )
−2GM
( z 2 + R2 − z )
=
⇒ Φ( z) = ∫ dΦ( z) =
2
2
2
2
2
2
2
R
R
R
π
z +r
z +r
−G
Then one can find the gravity acceleration,
dΦ
2GM  z 2 + R 2 − z 
g( z) = − kˆ
= − kˆ 2 

dz
R 
z 2 + R2 
where k̂ is the unit vector in the z-direction.
(1)
164
5-21.
CHAPTER 5
(We assume the convention that D > 0 means m is not sitting on the rod.)
The differential force dF acting on point mass m from the element of thickness dx of the rod,
which is situated at a distance x from m, is
dF =
G ( M L) mdx
x
2
GMm
⇒ F = ∫ dF =
L
L+ D
∫
D
dx
GMm
=
2
x
D(L + D)
And that is the total gravitational force acting on m by the rod.
CHAPTER
6
Some Methods in the
Calculus of Variations
6-1. If we use the varied function
y (α , x ) = x + α sin π (1 − x )
(1)
dy
= 1 − απ cos π (1 − x )
dx
(2)
Then
Thus, the total length of the path is
1
S=∫
0
2
 dy 
1 +   dx
 dx 
1
= ∫  2 − 2απ cos π (1 − x ) + α 2π 2 cos 2 π (1 − x ) 
12
dx
(3)
0
Setting π (1 − x ) ≡ u , the expression for S becomes
S=
1
π
π
∫
0
1

2 1 − απ cos u + α 2π 2 cos 2
2


u

12
du
(4)
The integral cannot be performed directly since it is, in fact, an elliptic integral. Because α is a
small quantity, we can expand the integrand and obtain
S=
2
π
π
∫
0
2
 1

1 2 2
1
1 2 2
2 
2 
1 −  απ cos u − α π cos u −  απ cos u − α π cos u + … du
2
8
2
 2

(5)
If we keep the terms up to cos 2 u and perform the integration, we find
S= 2+
2 2 2
π α
16
(6)
which gives
165
166
CHAPTER 6
∂S
2 2
π α
=
∂α
8
(7)
Therefore
∂S
∂α
=0
(8)
a=0
and S is a minimum when α = 0.
6-2.
The element of length on a plane is
dS = dx 2 + dy 2
(1)
from which the total length is
( x2 , y2 )
S=
∫
dx 2 + dy 2 =
( x1 , y1 )
x2
∫
x1
2
 dy 
1 +   dx
 dx 
(2)
If S is to be minimum, f is identified as
 dy 
f = 1+  
 dx 
2
(3)
Then, the Euler equation becomes
d d 
1 + y′2  = 0

dx dy ′ 
where y ′ =
(4)
dy
. (4) becomes
dx
d  y′

dx  1 + y ′ 2


=0

(5)
or,
y′
1 + y′2
= constant ≡ C
(6)
from which we have
y′ =
C2
= constant ≡ a
1 − C2
(7)
Then,
y = ax + b
This is the equation of a straight line.
(8)
167
SOME METHODS IN THE CALCULUS OF VARIATIONS
6-3.
The element of distance in three-dimensional space is
dS = dx 2 + dy 2 + dz 2
(1)
Suppose x, y, z depends on the parameter t and that the end points are expressed by
( x1 (t1 ) , y1 (t1 ) , z1 (t1 )) , ( x2 (t2 ) , y2 (t2 ) , z2 (t2 )) . Then the total distance is
t2
S=∫
t1
2
2
2
 dx   dy   dz 
 dt  +  dt  +  dt  dt
(2)
The function f is identified as
f = x2 + y2 + z2
Since
(3)
∂ f ∂f ∂ f
=
=
= 0 , the Euler equations become
∂x ∂y ∂z

d ∂f
=0 
dt ∂x


d ∂f
=0 

dt ∂y


d ∂f
=0 
dt ∂z

(4)
from which we have

= constant ≡ C1 

x2 + y2 + z2


y
= constant ≡ C2 

x2 + y2 + z2


z
= constant ≡ C3 

x2 + y2 + z2

x
(5)
From the combination of these equations, we have
y 
x
=
C1 C2 


y
z 
=
C2 C3 
If we integrate (6) from t1 to the arbitrary t, we have
(6)
168
CHAPTER 6
x − x1 y − y1 
=
C1
C2 


y − y1 z − z1 
=
C2
C3 
(7)
On the other hand, the integration of (6) from t1 to t2 gives
x2 − x1 y2 − y1 
=
C1
C2 


y2 − y1 z2 − z1 
=
C2
C3 
(8)
from which we find the constants C1 , C2 , and C3 . Substituting these constants into (7), we find
y − y1
x − x1
z − z1
=
=
x2 − x1 y2 − y1 z2 − z1
(9)
This is the equation expressing a straight line in three-dimensional space passing through the
two points ( x1 , y1 , z1 ) , ( x2 , y2 , z2 ) .
6-4.
z
ρ
dS
2
x
1
y
φ
The element of distance along the surface is
dS = dx 2 + dy 2 + dz 2
(1)
In cylindrical coordinates (x,y,z) are related to (ρ,φ,z) by
x = ρ cos φ 

y = ρ sin φ 


z=z

(2)
dx = − ρ sin φ dφ 

dy = ρ cos φ dφ 


dz = dz

(3)
from which
169
SOME METHODS IN THE CALCULUS OF VARIATIONS
Substituting (3) into (1) and integrating along the entire path, we find
2
S = ∫ ρ dφ + dz =
2
2
2
1
where z =
φ2
∫
ρ 2 + z 2 dφ
dz
. If S is to be minimum, f ≡ ρ 2 + z 2 must satisfy the Euler equation:
dφ
∂f
∂ ∂f
−
=0
∂ z ∂φ ∂ z
Since
(4)
φ1
(5)
∂f
= 0 , the Euler equation becomes
∂z
∂
∂φ
z
ρ + z2
2
=0
(6)
from which
z
ρ + z2
2
= constant ≡ C
(7)
or,
z=
C2
ρ
1 − C2
(8)
Since ρ is constant, (8) means
dz
= constant
dφ
and for any point along the path, z and φ change at the same rate. The curve described by this
condition is a helix.
6-5.
z
(x2,y2)
ds
(x1,y1)
y
x
The area of a strip of a surface of revolution is
dA = 2π × ds = 2π × dx 2 + dy 2
Thus, the total area is
(1)
170
CHAPTER 6
A = 2π
x2
∫x
1 + y 2 dx
(2)
x1
where y =
dy
. In order to make A a minimum, f ≡ x 1 + y 2 must satisfy equation (6.39). Now
dx
∂f
= 1 + y2
∂x
xy
∂f
=
∂y
1 + y2
Substituting into equation (6.39) gives
1 + y2 =
xy 2
d 
x 1 + y2 −
dx 
1 + y2

=
Multiplying by
 d  x
=

 dx  1 + y 2
(
1 + y 2 − xy ( dy dx ) 1 + y 2
)
1 + y2
1 + y 2 and rearranging gives
−
dy
dx
=
x y 1 + y2
(
)
Integration gives
− ln x + ln a =
y2
1
ln
2 1 + y2
where ln a is a constant of integration. Rearranging gives
y2 =
(
1
x a2 − 1
2
)
Integrating gives
y = b + a cosh −1
x
a
or
x = a cosh
which is the equation of a catenary.
y−b
a
−1 2



171
SOME METHODS IN THE CALCULUS OF VARIATIONS
6-6.
θ=π
0
y
(x1,y1)
θ=0
2a
(x2,y2)
x
If we use coordinates with the same orientation as in Example 6.2 and if we place the minimum
point of the cycloid at (2a,0) the parametric equations are
x = a (1 + cos θ ) 


y = a (θ + sin θ ) 
(1)
Since the particle starts from rest at the point ( x1 , y1 ) , the velocity at any elevation x is [cf. Eq.
6.19]
v = 2 g ( x − x1 )
(2)
Then, the time required to reach the point ( x2 , y2 ) is [cf. Eq. 6.20]
t=
x2
∫
x1
 1 + y′2 


 2 g ( x − x1 ) 
12
(3)
dx
Using (1) and the derivatives obtained therefrom, (3) can be written as
a
g
t=
θ1
 1 + cos θ 
∫  cos θ − cosθ1 
θ2 = 0
12
dθ
(4)
Now, using the trigonometric identity, 1 + cos θ = 2 cos 2 θ 2 , we have
t=
=
a
g
a
g
cos
θ1
∫
0
cos 2
θ
2
θ1
cos
0
2
∫
sin
θ1
2
θ
2
dθ
− cos 2
θ
2
θ1
2
dθ
− sin
2
θ
(5)
2
Making the change of variable, z = sin θ 2 , the expression for t becomes
sin
t=2
The integral is now in standard form:
a
g
θ1
∫
0
2
dz
sin 2
θ1
2
(6)
− z2
172
CHAPTER 6
x
= sin −1  
a
a =x
dx
∫
2
2
(7)
Evaluating, we find
t=π
a
g
(8)
Thus, the time of transit from ( x1 , y1 ) to the minimum point does not depend on the position of
the starting point.
6-7.
a
θ1
θ1
n1
(n2 > n1)
n2
c
n1
c
v2 =
n2
v1 =
θ2
b
x
d
The time to travel the path shown is (cf. Example 6.2)
1 + y′2
ds
t=∫ =∫
dx
v
v
(1)
Although we have v = v(y), we only have dv dy ≠ 0 when y = 0. The Euler equation tells us
y′
d 

dx  v 1 + y ′ 2


=0

(2)
Now use v = c n and y′ = –tan θ to obtain
n sin θ = const.
This proves the assertion. Alternatively, Fermat’s principle can be proven by the method
introduced in the solution of Problem 6-8.
6-8.
To find the extremum of the following integral (cf. Equation 6.1)
J = ∫ f ( y , x ) dx
we know that we must have from Euler’s equation
∂f
=0
∂y
This implies that we also have
(3)
173
SOME METHODS IN THE CALCULUS OF VARIATIONS
∂f
∂J
=∫
dx = 0
∂y
∂y
giving us a modified form of Euler’s equation. This may be extended to several variables and to
include the imposition of auxiliary conditions similar to the derivation in Sections 6.5 and 6.6.
The result is
∂g j
∂J
+ ∑ λ j ( x)
=0
∂y i
∂y i
j
when there are constraint equations of the form
g j ( yi , x ) = 0
a) The volume of a parallelepiped with sides of lengths a1 , b1 , c1 is given by
V = a1b1c1
(1)
We wish to maximize such a volume under the condition that the parallelepiped is
circumscribed by a sphere of radius R; that is,
a12 + b12 + c12 = 4 R 2
(2)
We consider a1 , b1 , c1 as variables and V is the function that we want to maximize; (2) is the
constraint condition:
g { a1 , b1 , c1 } = 0
(3)

∂g
∂V
+λ
=0 
∂a1
∂a1


∂g
∂V

+λ
=0 
∂b1
∂b1


∂g
∂V
+λ
=0 
∂c1
∂c1

(4)
b1c1 + 2λ a1 = 0 

a1c1 + 2λ b1 = 0 


a1b1 + 2λ c1 = 0 
(5)
Then, the equations for the solution are
from which we obtain
Together with (2), these equations yield
a1 = b1 = c1 =
2
R
3
Thus, the inscribed parallelepiped is a cube with side
2
R.
3
(6)
174
CHAPTER 6
b) In the same way, if the parallelepiped is now circumscribed by an ellipsoid with semiaxes
a, b, c, the constraint condition is given by
a12
b12
c12
=
=
=1
4 a 2 4b 2 4 z 2
(7)
where a1 , b1 , c1 are the lengths of the sides of the parallelepiped. Combining (7) with (1) and
(4) gives
a12 b12 c12
=
=
a2 b 2 c2
(8)
Then,
a1 = a
6-9.
2
2
2
, b1 = b
, c1 = c
3
3
3
(9)
The average value of the square of the gradient of φ ( x1 , x2 , x3 ) within a certain volume V
is expressed as
I=
1
V
1
v=
V
∫∫∫ ( ∇φ )
∫∫∫
2
dx1 dx2 dx3
  ∂φ  2  ∂ φ  2  ∂ φ  2 

  dx1dx2 dx3
 +
 +
∂x
∂x
∂x
  1   2   3  
(1)
In order to make I a minimum,
2
2
 ∂φ   ∂φ   ∂φ 
f =

 +
 +
 ∂x1   ∂x2   ∂x3 
2
must satisfy the Euler equation:
3
∂f
∂  ∂f 
−∑
=0
∂ φ i = 1 ∂x i   ∂ φ  
∂


  ∂xi  
(2)
If we substitute f into (2), we have
3
∂ ∂φ
=0
i ∂x i
∑ ∂x
i =1
(3)
which is just Laplace’sequation:
∇2 φ = 0
Therefore, φ must satisfy Laplace’s equation in order that I have a minimum value.
(4)
175
SOME METHODS IN THE CALCULUS OF VARIATIONS
6-10. This problem lends itself to the method of solution suggested in the solution of Problem
6-8. The volume of a right cylinder is given by
V = π R2 H
(1)
The total surface area A of the cylinder is given by
A = Abases + Aside = 2π R 2 + 2π RH = 2π R ( R + H )
(2)
We wish A to be a minimum. (1) is the constraint condition, and the other equations are
∂g
∂A

+λ
=0 
∂R
∂R


∂g
∂A
+λ
=0 
∂H
∂H

(3)
where g = V − π R 2 H = 0 .
The solution of these equations is
R=
1
H
2
(4)
6-11.
y
1
2a
θ
R
ds
}
The constraint condition can be found from the relation ds = Rdθ (see the diagram), where ds is
the differential arc length of the path:
(
ds = dx 2 + dy 2
)
12
= Rdθ
(1)
which, using y = ax 2 , yields
1 + 4a 2 x 2 dx = Rdθ
(2)
If we want the equation of constraint in other than a differential form, (2) can be integrated to
yield
A + Rθ =
x
2
4 ax 2 + 1 +
(
1
ln 2ax + 4 a 2 x 2 + 1
4a
)
(3)
where A is a constant obtained from the initial conditions. The radius of curvature of a parabola,
y = ax 2 , is given at any point (x,y) by r0 ≥ 1 2a . The condition for the disk to roll with one and
only one point of contact with the parabola is R < r0 ; that is,
176
CHAPTER 6
R<
6-12.
1
2a
(4)
The path length is given by
s = ∫ ds = ∫ 1 + y ′ 2 + z ′ 2 dx
(1)
g ( x, y, z) = x2 + y 2 + z2 − ρ2 = 0
(2)
and our equation of constraint is
The Euler equations with undetermined multipliers (6.69) tell us that
y′
d 

dx  1 + y ′ 2 + z ′ 2


dg
=λ
= 2λ y
dy

(3)
with a similar equation for z. Eliminating the factor λ, we obtain
y′
1 d 

y dx  1 + y ′ 2 + z ′ 2

 1 d 
z′
−

 z dx  1 + y ′ 2 + z ′ 2

=0

(4)
This simplifies to
(
(
)
)
z  y ′′ 1 + y ′ 2 + z ′ 2 − y ′ ( y ′y ′′ + z ′z ′′ )  − y  z ′′ 1 + y ′ 2 + z ′ 2 − z ′ ( y ′y ′′ + z ′z ′′ )  = 0
zy ′′ + ( yy ′ + zz ′ ) z ′y ′′ − yz ′′ − ( yy ′ + zz ′ ) y ′z ′′ = 0
(5)
(6)
and using the derivative of (2),
( z − xz ′ ) y ′′ = ( y − xy ′ ) z ′′
(7)
This looks to be in the simplest form we can make it, but is it a plane? Take the equation of a
plane passing through the origin:
Ax + By = z
(8)
and make it a differential equation by taking derivatives (giving A + By′ = z′ and By″ = z″) and
eliminating the constants. The substitution yields (7) exactly. This confirms that the path must
be the intersection of the sphere with a plane passing through the origin, as required.
6-13. For the reason of convenience, without lost of generality, suppose that the closed curve
passes through fixed points A(-a,0) and B(a,0) (which have been chosen to be on axis Ox). We
denote the part of the closed curve above and below the Ox axis as y1 ( x ) and y2 ( x ) respectively.
(note that y1 > 0 and y2 < 0 )
The enclosed area is
a
J ( y1 , y2 ) =
∫
−a
a
y1 ( x)dx −
∫
−a
a
y2 ( x)dx =
∫ ( y1 (x) − y2 (x)) dx =
−a
a
∫
−a
f ( y1 , y2 )dx
177
SOME METHODS IN THE CALCULUS OF VARIATIONS
The total length of closed curve is
K ( y1′ , y2′ ) =
a
∫
1 + ( y1′ ) dx +
2
−a
a
∫
1 + ( y2′ ) dx =
2
−a
∫{
a
1 + ( y1′ ) + 1 + ( y1′ )
2
−a
2
}
a
dx =
∫ g ( y1′ , y2′ ) dx
−a
Then the generalized versions of Eq. (6.78) (see textbook) for this case are
 ∂g
y1′
∂f
d ∂f
d ∂g 
d 

−
+λ
−
=0
 = 0 ⇒ 1− λ 
2
∂y1 dx ∂y1′
dx  1 + ( y1′ ) 
 ∂y1 dx ∂y1′ 
(1)
 ∂g
y2′
∂f
d ∂f
d ∂g 
d 

−
+λ
−
=0
 = 0 ⇒ 1− λ 
2
∂y2 dx ∂y2′
dx  1 + ( y2′ ) 
 ∂y2 dx ∂y2′ 
(2)
Analogously to Eq. (6.85);
from (1) we obtain
( x − A1 ) 2 + ( y1 − A2 ) 2 = λ 2
(3)
from (2) we obtain
( x − B1 ) 2 + ( y2 − B2 ) 2 = λ 2
(4)
where constants A’s, B’s can be determined from 4 initial conditions
( x = ± a, y1 = 0)
( x = ± a, y2 = 0)
and
We note that y1 < 0 and y2 > 0 , so actually (3) and (4) altogether describe a circular path of
radius λ . And this is the sought configuration that renders maximum enclosed area for a given
path length.
6-14. It is more convenient to work with cylindrical coordinates (r, φ ,z) in this problem. The
constraint here is z = 1 – r , then dz = –dr
ds2 = dr 2 + r 2 dφ 2 + dz 2 = 2 ( dr 2 + r 2 dβ 2 )
where we have introduced a new angular coordinate β =
φ
2
In this form of ds2 , we clearly see that the space is 2-dimensional Euclidean flat, so the shortest
line connecting two given points is a straight line given by:
r=
r0
cos ( β − β 0 )
=
this line passes through the endpoints (r = 1, φ = ±
r0
 φ − φ0 
cos 
 2 
π
2
) , then we can determine unambiguously
the shortest path equation
r(φ ) =
cos
π
2 2
 φ 
cos 
 2 
and z = 1 – r
178
CHAPTER 6
Accordingly, the shortest connecting length is
π 2
l=
π
 dr 
2
∫ dφ 2  dφ  + r = 2 2 sin 2 2
−π 2
2
6-15.
 dy  2

I[ y] = ∫   − y 2  dx
 

0
 dx
1
a) Treating I[y] as a mechanical action, we find the corresponding Euler-Lagrange equation
y( x ) = −
d2 y
dx 2
Combining with the boundary conditions (x = 0, y = 0) and (x = 1, y = 1), we can determine
unambiguously the functional form of y( x ) = (sin x) (sin 1) .
b) The corresponding minimum value of the integral is
1
 dy  2

1
I[ y] = ∫   − y 2  dx =
dx cos 2 x = cot (1) = 0.642
 
sin 2 1 ∫0

0
 dx
1
c) If x = y then I[y] = ( 2 3 ) = 0.667.
6-16.
a) S is arc length
2
2
2
9
 dy   dz 
 dy 
S = ∫ dx 2 + dy 2 + dz 2 = ∫ dx 1 +   +   = ∫ dx 1 +   + x = ∫ L dx
 dx   dx 
 dx 
4
Treating S and L like a mechanical action and Lagrangian respectively, we find the canonical
momentum associated with coordinate y
p=
δL
 dy 
δ 
 dx 
 dy 
 
dx
=
1+
9
 dy 
x+ 
 dx 
4
2
Because L does not depend on y explicitly, then E-L equation implies that p is constant
(i.e. dp dx = 0 ), then the above equation becomes
9
32
1+ x
dy
9
9 

4 ⇒y = p
=p
+
=
+
+B
1
1
dx
x
A
x

dx
1 − p2
1 − p2 ∫
4
4 
where A and B are constants. Using boundary conditions (x = 0, y = 0) and (x = 1, y = 1) one can
determine the arc equation unambiguously
179
SOME METHODS IN THE CALCULUS OF VARIATIONS
32
9 


y( x ) = 3 2
 1 + x − 1
13 − 8 
4

8
and
z = x3 2
b)
z
1
0.75
x
0.5
0.25
y
1
0.
0.75
0.5
0.25
0
1
0.75
0.5
0.25
00
6-17.
a) Equation of a ellipse
x2 y2
+
=1
a2 b2
which implies
xy ≤
ab
2
because
2xy x 2 y 2
≤ 2 + 2
ab
a
b
so the maximal area of the rectangle, whose corners lie on that ellipse, is
Max[A] = Max[4xy] = 2 iab.
This happens when
x=
a
2
and
y=
b
2
b) The area of the ellipse is A0 = π ab ; so the fraction of rectangle area to ellipse area is then
Max[ A] 2
=
A0
π
6-18. One can see that the surface xy = z is “locally” symmetric with respect to the line
x = − y = −z where x > 0, y < 0, z < 0. This line is a parabola. This implies that if the particle
starts from point (1,-1,-1) (which belongs to the symmetry line) under gravity ideally will move
downward along this line. Its velocity at altitude z (z < –1) can be found from the conservation
of energy.
v( z) = −2 g( z + 1)
180
CHAPTER 6
CHAPTER
7
Hamilton’s Principle—
Lagrangian and Hamiltonian Dynamics
7-1. Four coordinates are necessary to completely describe the disk. These are the x and y
coordinates, the angle θ that measures the rolling, and the angle φ that describes the spinning
(see figure).
φ
θ
y
x
Since the disk may only roll in one direction, we must have the following conditions:
dx cos φ + dy sin φ = R dθ
(1)
dy
= tan φ
dx
(2)
These equations are not integrable, and because we cannot obtain an equation relating the
coordinates, the constraints are nonholonomic. This means that although the constraints relate
the infinitesimal displacements, they do not dictate the relations between the coordinates
themselves, e.g. the values of x and y (position) in no way determine θ or φ (pitch and yaw),
and vice versa.
7-2.
Start with the Lagrangian
L=
=
(
m
v0 + at + θ cos θ
2 
) + ( θ sin θ )  + mg
2
2
cos θ
m
2
v0 + at ) + 2 ( v0 + at ) θ cos θ + 2θ 2  + mg cos θ
(

2 
(1)
(2)
181
182
CHAPTER 7
Now let us just compute
d  ∂L  d
2
  =  m ( v0 + at ) cos θ + m θ 
dt  ∂θ  dt 
= ma cos θ − m ( v0 − at ) θ sin θ + m 2θ
∂L
= − m ( v0 + at ) θ sin θ − mg sin θ
∂θ
(3)
(4)
(5)
According to Lagrange’s equations, (4) is equal to (5). This gives Equation (7.36)
θ=
g
sin θ +
a
cos θ = 0
(6)
To get Equation (7.41), start with Equation (7.40)
η=−
g cos θ e − a sin θ e
η
(7)
and use Equation (7.38)
a
g
tan θ e = −
(8)
to obtain, either through a trigonometric identity or a figure such as the one shown here,
g 2 + a2
θe
g
a
cos θ e =
g
g +a
2
2
sin θ e =
a
g + a2
2
(9)
Inserting this into (7), we obtain
η=−
a2 + g 2
η
(10)
as desired.
We know intuitively that the period of the pendulum cannot depend on whether the train is
accelerating to the left or to the right, which implies that the sign of a cannot affect the
frequency. From a Newtonian point of view, the pendulum will be in equilibrium when it is in
line with the effective acceleration. Since the acceleration is sideways and gravity is down, and
the period can only depend on the magnitude of the effective acceleration, the correct form is
clearly
a2 + g 2 .
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
183
7-3.
θ
R
ρ
φ θ
If we take angles θ and φ as our generalized coordinates, the kinetic energy and the potential
energy of the system are
T=
2
1
1
m ( R − ρ ) θ  + I φ 2
2
2
U =  R − ( R − ρ ) cos θ  mg
(1)
(2)
where m is the mass of the sphere and where U = 0 at the lowest position of the sphere. I is the
moment of inertia of sphere with respect to any diameter. Since I = ( 2 5) mρ 2 , the Lagrangian
becomes
L = T −U =
1
1
2
m ( R − ρ ) θ 2 + mρ 2φ 2 −  R − ( R − ρ ) cos θ  mg
2
5
(3)
When the sphere is at its lowest position, the points A and B coincide. The condition A0 = B0
gives the equation of constraint:
f (θ , φ ) = ( R − ρ ) θ − ρφ = 0
(4)
Therefore, we have two Lagrange’s equations with one undetermined multiplier:
∂f

 ∂L 
 ∂θ  + λ ∂θ = 0 


∂f
∂ L d  ∂L 

−  +λ
=0 
∂φ dt  ∂φ 
∂φ

∂L d
−
∂θ dt
(5)
After substituting (3) and ∂f ∂θ = R − ρ and ∂f ∂φ = − ρ into (5), we find
− ( R − ρ ) mg sin θ − m ( R − ρ ) θ + λ ( R − ρ ) = 0
2
(6)
2
mρ 2φ − λρ = 0
5
(7)
2
5
(8)
λ = − m ( R − ρ) θ
(9)
−
From (7) we find λ:
λ = − mρφ
or, if we use (4), we have
2
5
Substituting (9) into (6), we find the equation of motion with respect to θ :
184
CHAPTER 7
θ = −ω 2 sin θ
(10)
where ω is the frequency of small oscillations, defined by
5g
7 ( R − ρ)
ω=
(11)
7-4.
y
m
r
θ
x
If we choose (r,θ ) as the generalized coordinates, the kinetic energy of the particle is
T=
(
(
)
1
1
m x 2 + y 2 = m r 2 + r 2θ 2
2
2
)
(1)
Since the force is related to the potential by
f =−
∂U
∂r
(2)
we find
U=
A
α
rα
(3)
where we let U(r = 0) = 0. Therefore, the Lagrangian becomes
L=
(
)
1
A
m r 2 + r 2θ 2 − r α
2
α
(4)
Lagrange’s equation for the coordinate r leads to
mr − mrθ 2 + Ar α −1 = 0
(5)
Lagrange’s equation for the coordinate θ leads to
(
)
d
mr 2θ = 0
dt
(6)
Since mr 2θ = is identified as the angular momentum, (6) implies that angular momentum is
conserved. Now, if we use , we can write (5) as
mr −
2
mr
3
+ Ar α −1 = 0
(7)
Multiplying (7) by r , we have
mrr −
r 2
+ Ar r α −1 = 0
mr 3
(8)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
185
which is equivalent to
d 1
d  2  d A α 
2
mr
+
 dt  2mr 2  + dt  α r  = 0
dt  2


(9)
d
(T + U ) = 0
dt
(10)
Therefore,
and the total energy is conserved.
7-5.
y
r
m
φ
x
Let us choose the coordinate system so that the x-y plane lies on the vertical plane in a
gravitational field and let the gravitational potential be zero along the x axis. Then the kinetic
energy and the potential energy are expressed in terms of the generalized coordinates (r,φ) as
T=
A
U=
(
1
m r 2 + r 2φ 2
2
α
)
(1)
r α + mgr sin φ
(2)
)
(3)
from which the Lagrangian is
L = T −U =
(
1
A
m r 2 + r 2φ 2 − r α − mgr sin φ
α
2
Therefore, Lagrange’s equation for the coordinate r is
mr − mrφ 2 + Ar α −1 + mg sin φ = 0
(4)
Lagrange’s equation for the coordinate φ is
(
)
d
mr 2φ + mgr cos φ = 0
dt
(5)
Since mr 2φ is the angular momentum along the z axis, (5) shows that the angular momentum is
not conserved. The reason, of course, is that the particle is subject to a torque due to the
gravitational force.
186
CHAPTER 7
7-6.
y
M
φ
m
S
α
α
x
ξ
Let us choose ξ,S as our generalized coordinates. The x,y coordinates of the center of the hoop
are expressed by
x = ξ + S cos α + r sin α


− S) sin α 
y = r cos α + (
(1)
Therefore, the kinetic energy of the hoop is
Thoop =
=
(
)
1
1
m x 2 + y 2 + Iφ 2
2
2
(
1 
m ξ + S cos α
2 
) + ( −S sin α )  + 12 Iφ
2
2
2
(2)
Using I = mr 2 and S = rφ , (2) becomes
Thoop =
1
m  2S 2 + ξ 2 + 2ξS cos α 
2 
(3)
In order to find the total kinetic energy, we need to add the kinetic energy of the translational
motion of the plane along the x-axis which is
Tplane =
1
Mξ2
2
(4)
Therefore, the total kinetic energy becomes
T = mS2 +
1
( m + M ) ξ 2 + mξS cos α
2
(5)
The potential energy is
U = mgy = mg r cos α + ( − S) sin α 
(6)
Hence, the Lagrangian is
l = mS 2 +
1
( m + M ) ξ 2 + mξS cos α − mg r cos α + ( − S) sin α 
2
(7)
from which the Lagrange equations for ξ and S are easily found to be
2mS + mξ cos α − mg sin α = 0
(8)
( m + M ) ξ + mS cos α = 0
(9)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
187
or, if we rewrite these equations in the form of uncoupled equations by substituting for ξ and
S , we have


m cos 2 α 
2 −
 S − g sin α = 0 
m+ M 



mg sin α cos α

ξ=−
2
2 ( m + M ) − m cos α 
(10)
d
( m + M ) ξ + mS cos α  = 0

dt 
(11)
Now, we can rewrite (9) as
where we can interpret ( m + M ) ξ as the x component of the linear momentum of the total
system and mS cos α as the x component of the linear momentum of the hoop with respect to
the plane. Therefore, (11) means that the x component of the total linear momentum is a
constant of motion. This is the expected result because no external force is applied along the
x-axis.
7-7.
y1
y2
y
φ1
m
x1
φ2
x2
m
x
If we take (φ1 , φ 2 ) as our generalized coordinates, the x,y coordinates of the two masses are
x1 = cos φ1 

y1 = sin φ1 
(1)
x2 = cos φ1 + cos φ 2 

y2 = sin φ1 + sin φ2 
(2)
Using (1) and (2), we find the kinetic energy of the system to be
T=
(
)
(
m 2
m 2
x1 + y12 +
x2 + y22
2
2
=
m
2
=
m
2
)
2
φ12 + φ12 + φ 22 + 2φ1φ2 ( sin φ1 sin φ 2 + cos φ1 cos φ 2 ) 
2
 2φ12 + φ22 + 2φ1φ 2 cos (φ1 − φ 2 ) 
(3)
188
CHAPTER 7
The potential energy is
U = − mgx1 − mgx2 = − mg 2 cos φ1 + cos φ2 
(4)
Therefore, the Lagrangian is
1


L = m 2 φ12 + φ22 + φ1φ2 cos (φ1 − φ2 )  + mg 2 cos φ1 + cos φ 2 
2


(5)
∂L

= m 2φ1φ 2 sin (φ1 − φ 2 ) − 2mg sin φ1 
∂φ1


∂L
2
2

= 2m φ1 + m φ 2 cos (φ1 − φ 2 )

∂φ1


∂L
= − m 2φ1φ 2 sin (φ1 − φ2 ) − mg sin φ 2 
∂φ 2



∂L
2
2
= m φ2 + m φ1 cos (φ1 − φ2 )

∂φ 2

(6)
from which
The Lagrange equations for φ1 and φ 2 are
2φ1 + φ 2 cos (φ1 − φ 2 ) + φ 22 sin (φ1 − φ 2 ) + 2
φ2 + φ1 cos (φ1 − φ 2 ) − φ12 sin (φ1 − φ2 ) +
g
g
sin φ1 = 0
sin φ 2 = 0
(7)
(8)
7-8.
y
U1
v2
θ2
θ1
U2
x
v1
Let us choose the x,y coordinates so that the two regions are divided by the y axis:
x<0
 U1
U ( x) = 
 U 2
x>0
If we consider the potential energy as a function of x as above, the Lagrangian of the particle is
L=
(
)
1
m x2 + y2 − U ( x)
2
Therefore, Lagrange’s equations for the coordinates x and y are
(1)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
mx +
dU ( x )
=0
dx
189
(2)
my = 0
(3)
Using the relation
mx =
dP dP dx Px dpx
d
mx = x = x
=
dt
dt
dx dt m dx
(4)
(2) becomes
Px dPx dU ( x )
+
=0
m dx
dx
(5)
Integrating (5) from any point in the region 1 to any point in the region 2, we find
2
∫
1
dU ( x )
Px dPx
dx + ∫
dx = 0
m dx
dx
1
2
Px22
2m
−
Px21
2m
+ U 2 − U1 = 0
(6)
(7)
or, equivalently,
1
1
mx12 + U1 = mx22 + U 2
2
2
(8)
Now, from (3) we have
d
my = 0
dt
and my is constant. Therefore,
my1 = my2
(9)
1
1
my12 = my22
2
2
(10)
1
1
mv12 + U1 = mv22 + U 2
2
2
(11)
mv1 sin θ 1 = mv2 sin θ 2
(12)
From (9) we have
Adding (8) and (10), we have
From (9) we also have
Substituting (11) into (12), we find
sin θ 1 v2  U1 − U 2 
=
= 1 +

T1 
sin θ 2 v1 
12
(13)
190
CHAPTER 7
This problem is the mechanical analog of the refraction of light upon passing from a medium of
a certain optical density into a medium with a different optical density.
7-9.
y
x
O
M
ξ
φ
m
α
Using the generalized coordinates given in the figure, the Cartesian coordinates for the disk are
(ξ cos α, –ξ sin α), and for the bob they are ( sin φ + ξ cos α, – cos φ – ξ sin α). The kinetic
energy is given by
1
1
 1
2
2
T = Tdisk + Tbob =  Mξ 2 + Iθ 2  + m x bob
+ y bob
2
2
 2
(
)
(1)
1
1
1
( M + m) ξ 2 + Iθ 2 + m 2φ 2 + m φξ cos (φ + a)
2
2
2
(2)
Substituting the coordinates for the bob, we obtain
T=
The potential energy is given by
U = Udisk + U bob = Mgydisk + mgy bob = − ( M + m) gξ sin α − mg cos φ
(3)
Now let us use the relation ξ = Rθ to reduce the degrees of freedom to two, and in addition
substitute I = MR2 2 for the disk. The Lagrangian becomes
1 
1
3
L = T − U =  M + m ξ 2 + m 2φ 2 + m φξ cos ( φ + a) + ( M + m) gξ sin α + mg cos φ
4

2
2
(4)
The resulting equations of motion for our two generalized coordinates are
3

2
 M + m ξ − ( M + m) g sin α + m φ cos ( φ + α ) − φ sin ( φ + α )  = 0
2
1
φ + ξ cos ( φ + α ) +
7-10.
y
x
M
x
–y
S
M
g
sin φ = 0
(5)
(6)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
191
Let the length of the string be so that
(S − x) − y =
(1)
Then,
x = −y
(2)
a) The Lagrangian of the system is
L=
1
1
Mx 2 + My 2 − Mgy = My 2 − Mgy
2
2
(3)
Therefore, Lagrange’s equation for y is
d ∂L ∂L
−
= 2 My + Mg = 0
dt ∂y ∂y
(4)
from which
y=−
g
2
(5)
Then, the general solution for y becomes
y (t) = −
g 2
t + C1t + C2
4
(6)
If we assign the initial conditions y ( t = 0 ) = 0 and y ( t = 0 ) = 0 , we find
y (t) = −
g 2
t
4
(7)
b) If the string has a mass m, we must consider its kinetic energy and potential energy. These
are
Tstring =
U string = −
m
1
my 2
2
yg
y
mg 2
y
=−
2
2
(8)
(9)
Adding (8) and (9) to (3), the total Lagrangian becomes
L = My 2 − Mgy +
mg 2
1
my 2 +
y
2
2
(10)
Therefore, Lagrange’s equation for y now becomes
( 2 M + m) y −
mg
y + Mg = 0
(11)
In order to solve (11), we arrange this equation into the form
( 2 M + m) y =
mg 
M 
 y − m 
(12)
192
Since
CHAPTER 7
d2
dt 2
M  d2

−
=
y
y , (12) is equivalent to

m  dt 2
d2
dt 2
M 

 y − m  =
mg
( 2 M + m)
M 

 y − m 
(13)
which is solved to give
y−
M
= Aeγ t + Be − γ t
m
(14)
where
mg
( 2 M + m)
γ=
(15)
If we assign the initial condition y ( t = 0 ) = 0 ; y ( t = 0 ) = 0 , we have
A = +B = −
M
2m
Then,
y (t) =
M
(1 − cosh γ t )
m
(16)
7-11.
x
y′
er′
m
φ
x′
y
The x,y coordinates of the particle are
x = R cos ω t + R cos ( φ + ω t ) 

y = R sin ω t + R sin ( φ + ω t ) 
(1)
Then,
(
)
x = − Rω sin ω t − R φ + ω sin ( φ + ω t ) 


y = Rω cos ω t + R φ + ω cos ( φ + ω t ) 
(
)
(2)
Since there is no external force, the potential energy is constant and can be set equal to zero. The
Lagrangian becomes
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
L=
=
(
1
m x2 + y2
2
193
)
m 2 2
2
R ω + R 2 ( φ + ω ) + 2R2ω ( φ + ω ) cos φ 

2
(3)
from which
∂L
= − mR2ω φ + ω sin φ
∂φ
(
)
(4)
d ∂L d
=  mR 2 φ + ω + ω cos φ 
dt ∂φ dt 
(
)
(5)
Therefore, Lagrange’s equation for φ becomes
φ + ω 2 sin φ = 0
(6)
which is also the equation of motion for a simple pendulum. To make the result appear
reasonable, note that we may write the acceleration felt by the particle in the rotating frame as
a = ω 2 R ( i ′ + e′r )
(7)
where the primed unit vectors are as indicated in the figure. The part proportional to e′r does
not affect the motion since it has no contribution to the torque, and the part proportional to i′ is
constant and does not contribute to the torque in the same way a constant gravitational field
provides a torque to the simple pendulum.
7-12.
m
r
θ
Put the origin at the bottom of the plane
L = T −U =
(
)
1
m r 2 + r 2 θ 2 − mgr sin θ
2
θ = αt; θ = α
L=
(
)
1
m r 2 + α 2 r 2 − mgr sin α t
2
Lagrange’s equation for r gives
mr = mα 2 r − mg sin α t
or
r − α 2 r = − g sin α t
(1)
194
CHAPTER 7
The general solution is of the form r = rp + rh where rh is the general solution of the
homogeneous equation r − α 2 r = 0 and rp is a particular solution of Eq. (1).
So
rh = Aeα t + Be −α t
For rp , try a solution of the form rp = C sin α t . Then rp = −C α 2 sin α t . Substituting into (1) gives
−C α 2 sin α t − C α 2 sin α t = − g sin α t
C=
g
2α 2
So
r ( t ) = Aeα t + Be −α t +
g
2α 2
sin α t
We can determine A and B from the initial conditions:
r ( 0 ) = r0
(2)
r ( 0) = 0
(3)
(2) implies r0 = A + B
(3) implies 0 = A − B +
g
2α 2
Solving for A and B gives:
A=
r (t) =
g 
1
r0 − 2 

2
2α 
B=
g 
1
r0 +

2
2α 2 
g 
g  −α t
g
1
1
r0 − 2  eα t +  r0 +
e +
sin α t
2

2
2α 
2
2α 
2α 2
or
r ( t ) = r0 cosh α t +
g
2α 2
( sin α t − sinh α t )
7-13.
a)
a
b θ
m
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
x=
1 2
at − b sin θ
2
y = − b cos θ
x = at − bθ cos θ
y = bθ sin θ
L=
=
(
)
1
m x 2 + y 2 − mgy
2
(
)
1
m a 2t 2 − 2 at bθ cos θ + b 2θ 2 + mgb cos θ
2
d ∂L ∂L
=
dt ∂θ ∂θ
gives
d
 − mat b cos θ + mb 2θ  = mat bθ sin θ − mgb sin θ
dt 
This gives the equation of motion
θ+
g
a
sin θ − cos θ = 0
b
b
b) To find the period for small oscillations, we must expand sin θ and cos θ about the
equilibrium point θ 0 . We find θ 0 by setting θ = 0 . For equilibrium,
g sin θ 0 = a cos θ 0
or
tan θ 0 =
a
g
a2 + g 2
a
θ0
g
Using the first two terms in a Taylor series expansion for sin θ and cos θ gives
f (θ 0 ) + f ′ (θ ) θ =θ (θ − θ 0 )
f (θ )
0
sin θ
sin θ 0 + (θ − θ 0 ) cos θ 0
cos θ
cos θ 0 − (θ − θ 0 ) sin θ 0
tan θ 0 =
a
implies sin θ 0 =
g
cos θ 0 =
g
a2 + g 2
a
a + g2
2
,
195
196
CHAPTER 7
Thus
1
sin θ
a + g2
2
1
cos θ
a2 + g 2
( a + gθ − gθ 0 )
( g − aθ + aθ 0 )
Substituting into the equation of motion gives
0 =θ +
g
b a2 + g
( a + gθ − gθ 0 ) −
2
a
b a2 + g 2
( g − aθ + aθ 0 )
This reduces to
g 2 + a2
θ+
g 2 + a2
θ=
b
b
θ0
The solution to this inhomogeneous differential equation is
θ = θ 0 + A cos ωθ + B sin ωθ
where
(g
ω=
2
+ a2
)
14
b1 2
Thus
T=
2π
=
ω
(g
2π b1 2
2
+ a2
)
14
7-14.
a
θ b
m
x = b sin θ
y=
1 2
at − b cos θ
2
x = bθ cos θ
y = at + bθ sin θ
T=
(
)
(
1
1
m x 2 + y 2 = m b 2θ 2 + a 2t 2 + 2abtθ sin θ
2
2
)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
1

U = mgy = mg  at 2 − b cos θ 
2

L = T −U =
(
)
1
1


m b 2θ 2 + a 2t 2 + 2 abtθ sin θ + mg  b cos θ − at 2 


2
2
Lagrange’s equation for θ gives
d
 mb 2θ + mabt sin θ  = mabtθ cos θ − mgb sin θ
dt
b 2θ + ab sin θ + abtθ cos θ = abtθ cos θ − gb sin θ
θ+
For small oscillations, sin θ
a+ g
sin θ = 0
b
θ
θ+
a+ g
θ =0.
b
Comparing with θ + ω 2θ = 0 gives
T=
2π
ω
b
a+ g
= 2π
7-15.
θ
k
m
b = unextended length of spring
= variable length of spring
(
θ2)
T=
1
m
2
U=
1
1
2
2
k ( − b ) + mgy = k ( − b ) − mg cos θ
2
2
2
L = T −U =
+
2
1
m
2
(
2
+
2
θ2) −
1
( − b ) 2 + mg cos θ
2
Taking Lagrange’s equations for and θ gives
:
d
 m  = m θ 2 − k ( − b ) + mg cos θ
dt  
197
198
CHAPTER 7
θ:
d
 m 2θ  = − mg sin θ
dt 
This reduces to
− θ2 +
θ+
2
k
( − b ) − g cos θ = 0
m
g
θ+
sin θ = 0
7-16.
x = a sin ωt
θ b
m
For mass m:
x = a sin ω t + b sin θ
y = − b cos θ
x = aω cos ω t + bθ cos θ
y = bθ sin θ
Substitute into
T=
(
1
m x2 + y2
2
)
U = mgy
and the result is
L = T −U =
(
)
1
m a 2ω 2 cos 2 ω t + 2abωθ cos ω t cos θ + b 2θ 2 + mgb cos θ
2
Lagrange’s equation for θ gives
(
)
d
mabω cos ω t cos θ + mb 2θ = − mabwθ cos ω t sin θ − mgb sin θ
dt
− abω 2 sin ω t cos θ − abωθ cos ω t sin θ + b 2θ = − abωθ cos ω t sin θ − gb sin θ
or
θ+
g
a
sin θ − ω 2 sin ω t cos θ = 0
b
b
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
199
7-17.
y
C
A
h
θ
qθ
mg
B
Using q and θ (= ω t since θ (0) = 0), the x,y coordinates of the particle are expressed as
x = h cos θ + q sin θ = h cos ω t + q ( t ) sin ω t 

y = h sin θ − q cos θ = h sin ω t − q ( t ) cos ω t 
(1)
x = − hω sin ω t + qω cos ω t + q sin ω t 

y = hω cos ω t + qω sin ω t − q cos ω t 
(2)
from which
Therefore, the kinetic energy of the particle is
T=
(
1
m x2 + y2
2
)
(
)
1
m h 2ω 2 + q2ω 2 + q2 − mhω q
2
(3)
U = mgy = mg ( h sin ω t − q cos ω t )
(4)
=
The potential energy is
Then, the Lagrangian for the particle is
L=
1
1
1
mh 2ω 2 + mq2ω 2 + mq2 − mgh sin ω t + mgq cos ω t − mhω q
2
2
2
(5)
Lagrange’s equation for the coordinate is
q − ω 2 q = g cos ω t
(6)
The complementary solution and the particular solution for (6) are written as
qc ( t ) = A cos ( iω t + δ ) 


g
qP ( t ) = − 2 cos ω t 
2ω

(7)
so that the general solution is
q ( t ) = A cos ( iω t + δ ) −
Using the initial conditions, we have
g
2ω 2
cos ω t
(8)
200
CHAPTER 7

=0 


q ( 0 ) = − iω A sin δ = 0

q ( 0 ) = A cos δ −
g
2ω 2
(9)
Therefore,
δ = 0, A =
g
(10)
2ω 2
and
g
q (t) =
2ω 2
( cos iωt − cos ωt )
(11)
( cosh ωt − cos ωt )
(12)
or,
q (t) =
g
2ω 2
q(t)
g
2ω 2
t
In order to compute the Hamiltonian, we first find the canonical momentum of q. This is
obtained by
p=
∂L
= mq − mω h
∂q
(13)
Therefore, the Hamiltonian becomes
H = pq − L
= mq2 − mω hq −
1
1
1
mω 2 h 2 − mω 2 q2 − mq2 + mgh sin ω t − mgq cos ω t + mω qh
2
2
2
so that
H=
1 2 1
1
mq − mω 2 h 2 − mω 2 q2 + mgh sin ω t − mgq cos ω t
2
2
2
(14)
Solving (13) for q and substituting gives
p2
1
H=
+ ω hp − mω 2 q2 + mgh sin ω t − mgq cos ω t
2m
2
(15)
The Hamiltonian is therefore different from the total energy, T + U. The energy is not conserved
in this problem since the Hamiltonian contains time explicitly. (The particle gains energy from
the gravitational field.)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
201
7-18.
y
S
R
θ
O
A
B
θ
x
x
–y
m C
From the figure, we have the following relation:
AC = − s = − Rθ
(1)
where θ is the generalized coordinate. In terms of θ, the x,y coordinates of the mass are
x = AC cos θ + R sin θ = ( − Rθ ) cos θ + R sin θ 


y = R cos θ − AC sin θ = R cos θ − ( − Rθ ) sin θ 
(2)
x = Rθθ sin θ − θ sin θ 


y = Rθθ cos θ − θ cos θ 
(3)
from which
Therefore, the kinetic energy becomes
T=
(
)
1
1
m x 2 + y 2 = m  2θ 2 + R 2θ 2θ 2 − 2R θθ 2 
2
2
(4)
The potential energy is
U = mgy = mg R cos θ − ( − Rθ ) sin θ 
(5)
Then, the Lagrangian is
L = T −U =
1
m  2θ 2 + R 2θ 2θ 2 − 2R θθ 2  − mg R cos θ − ( − Rθ ) sin θ 
2
(6)
Lagrange’s equation for θ is
( − Rθ ) θ − Rθ 2 − g cos θ = 0
(7)
Now let us expand about some angle θ 0 , and assume the deviations are small. Defining
ε ≡ θ − θ 0 , we obtain
ε+
g sin θ 0
g cos θ 0
ε=
− Rθ 0
− Rθ 0
(8)
The solution to this differential equation is
ε = A sin (ω t + δ ) +
where A and δ are constants of integration and
cos θ 0
sin θ 0
(9)
202
CHAPTER 7
g sin θ 0
− Rθ 0
ω≡
(10)
is the frequency of small oscillations. It is clear from (9) that θ extends equally about θ 0 when
θ0 = π 2 .
7-19.
P
φ
θ
d
m2
m1
m2 g
m1g
Because of the various constraints, only one generalized coordinate is needed to describe the
system. We will use φ, the angle between a plane through P perpendicular to the direction of the
gravitational force vector, and one of the extensionless strings, e.g., 2 , as our generalized
coordinate.
The, the kinetic energy of the system is
T=
1
m1
2
( φ)
1
2
+
1
m2
2
( φ)
2
(1)
2
The potential energy is given by
U = − m1 g
1
sin (π − ( φ + θ ) ) − m2 g
2
sin φ
(2)
from which the Lagrangian has the form
L = T −U =
(
1
m1
2
2
1
+ m2
2
2
)φ
2
+ m1 g
1
sin ( φ + θ ) + m2 g
2
sin φ
(3)
The Lagrangian equation for φ is
m2 g
2
cos φ + m1 g
1
(
cos ( φ + θ ) − m1
2
1
+ m2
2
2
)φ = 0
(4)
This is the equation which describes the motion in the plane m1 , m2 , P .
To find the frequency of small oscillations around the equilibrium position (defined by φ = φ0 ),
we expand the potential energy U about φ0 :
U ( φ ) = U ( φ0 ) + U ′ ( φ 0 ) φ +
=
1
U ′′ (φ0 ) φ 2 + …
2
1
U ′′ (φ0 ) φ 2
2
where the last equality follows because we can take U (φ0 ) = 0 and because U ′ (φ0 ) = 0 .
From (4) and (5), the frequency of small oscillations around the equilibrium position is
(5)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
ω2 =
203
U ′′ (φ0 )
2
1
m1
+ m2
(6)
2
2
The condition U ′ (φ0 ) = 0 gives
tan φ0 =
+ m1 1 cos θ
m1 1 sin θ
m2
2
(7)
or,
sin φ0 =
m2
(m
+ m22
2 2
1 1
+ m1
2
1
cos θ
+ 2m1m2
2
2
1 2
cos θ
)
(8)
12
Then from (2), (7), and (8), U ′′ (φ0 ) is found to be
U ′′ (φ0 ) = g sin φ0 ( m2
=
(m
2 2
1 1
(
= g m12
g ( m2
+ m22
2
1
+ m1
2
2
2
+ m22
2
1
+ m1
cos θ + m1
1
cos θ )
+ 2m2 m1
2
2
1
2 1
+ 2m2 m1
cos θ
2 1
)
sin θ cot φ0 )
12
cos θ
)

 m2

1 cos θ +
2 + m1
m12 21 sin 2 θ 

m2 2 + m1 1 cos θ 
12
(9)
Finally, from (6) and (9), we have
ω
2
(m
=g
2 2
1 1
+ m22
2
2
(m
+ 2m1m2
2
1 1
+ m2
1 2
2
2
)
cos θ
)
12
(10)
which, using the relation,
2
1
cos θ =
+
2
2
2
− d2
(11)
1 2
can be written as
ω =
2
(
(m
g ( m1 + m2 ) m1
Notice that ω 2 degenerates to the value g
(so that 1 = 2 ).
2
1
+ m2
2
2
2
1 1
+ m2
2
2
) − d m m 
)
2
1
2
12
(12)
appropriate for a simple pendulum when d → 0
204
CHAPTER 7
7-20. The x-y plane is horizontal, and A, B, C are the fixed points lying in a plane above the
hoop. The hoop rotates about the vertical through its center.
z
A
B
y
C
θ′
x
A′
R
B′
θ
C′
The kinetic energy of the system is given by
2
T=
1 2 1
MR2 2 1  ∂z  2
θ + M  θ
Iω + Mz 2 =
2
2
2
2  ∂θ 
For small θ, the second term can be neglected since ( ∂z ∂θ )
θ =0
(1)
=0
The potential energy is given by
U = Mgz
(2)
where we take U = 0 at z = – .
Since the system has only one degree of freedom we can write z in terms of θ. When θ = 0,
z = − . When the hoop is rotated thorough an angle θ, then
z2 =
2
− ( R − R cos θ ) − ( R sin θ )
2
2
(3)
so that
z = − 
2
+ 2R 2 ( cos θ − 1) 
12
(4)
and the potential energy is given by
U = − Mg 
2
+ 2R2 ( cos θ − 1) 
12
(5)
for small θ, cos θ − 1 ≅ −θ 2 2 ; then,
 R 2θ 2 
U ≅ − Mg 1 − 2 


12
 R 2θ 2 
≅ − Mg 1 −
2 2 

(6)
From (1) and (6), the Lagrangian is
L = T −U =
 R 2θ 2 
1
MR 2θ 2 + Mg 1 −
,
2
2 2 

(7)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
205
for small θ. The Lagrange equation for θ gives
θ+
g
θ =0
(8)
where
g
ω=
(9)
which is the frequency of small rotational oscillations about the vertical through the center of
the hoop and is the same as that for a simple pendulum of length .
7-21.
ω
θ
From the figure, we can easily write down the Lagrangian for this system.
T=
(
mR 2 2
θ + ω 2 sin 2 θ
2
)
U = − mgR cos θ
(1)
(2)
The resulting equation of motion for θ is
θ − ω 2 sin θ cos θ +
g
sin θ = 0
R
(3)
The equilibrium positions are found by finding the values of θ for which
0 =θ
θ =θ 0
g

=  ω 2 cos θ 0 −  sin θ 0

R
(4)
Note first that 0 and π are equilibrium, and a third is defined by the condition
cos θ 0 =
g
ω 2R
(5)
To investigate the stability of each of these, expand using ε = θ − θ 0


ε = ω 2  cos θ 0 −
g

− ε sin θ 0  ( sin θ 0 + ε cos θ 0 )

ω 2R
(6)
206
CHAPTER 7
For θ 0 = π , we have


ε = ω2 1 +
g 
ε
ω 2 R 
(7)
g 
ε
ω 2 R 
(8)
indicating that it is unstable. For θ 0 = 0 , we have


ε = ω2 1 −
which is stable if ω 2 < g R and unstable if ω 2 > g R . When stable, the frequency of small
oscillations is
ω 2 − g R . For the final candidate,
ε = −ω 2 sin 2 θ 0 ε
with a frequency of oscillations of
(9)
ω 2 − ( g ω R) , when it exists. Defining a critical frequency
2
ω c2 ≡ g R , we have a stable equilibrium at θ 0 = 0 when ω < ω c , and a stable equilibrium at
θ 0 = cos −1 (ω c2 ω 2 ) when ω ≥ ω c . The frequencies of small oscillations are then ω 1 − (ω c ω )
2
and ω 1 − (ω c ω ) , respectively.
4
To construct the phase diagram, we need the Hamiltonian
H ≡θ
∂L
−L
∂θ
(10)
which is not the total energy in this case. A convenient parameter that describes the trajectory
for a particular value of H is
H
1
K≡
=
2 2
mω c R
2
 θ  2  ω  2

  −   sin 2 θ  − cos θ
 ω c   ω c 

(11)
so that we’ll end up plotting
2
2
 θ 
ω
2
 ω  = 2 ( K + cos θ ) +  ω  sin θ
c
c
(12)
for a particular value of ω and for various values of K. The results for ω < ω c are shown in
figure (b), and those for ω > ω c are shown in figure (c). Note how the origin turns from an
attractor into a separatrix as ω increases through ω c . As such, the system could exhibit chaotic
behavior in the presence of damping.
207
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
2
1.5
K
1
0.5
0
3
2
1
0
1
2
3
1
2
3
θ
(b)
2
1.5
K
1
0.5
0
3
2
1
0
θ
(c)
7-22.
The potential energy U which gives the force
k −( t τ )
e
x2
(1)
∂U
∂x
(2)
k −t τ
e
x
(3)
F ( x, t) =
must satisfy the relation
F=−
we find
U=
208
CHAPTER 7
Therefore, the Lagrangian is
1
k
mx 2 − e − t τ
x
2
L = T −U =
(4)
The Hamiltonian is given by
H = px x − L = x
∂L
−L
∂x
(5)
so that
H=
px2 k − t τ
+ e
2m x
(6)
The Hamiltonian is equal to the total energy, T + U, because the potential does not depend on
velocity, but the total energy of the system is not conserved because H contains the time
explicitly.
7-23.
The Hamiltonian function can be written as [see Eq. (7.153)]
H = ∑ p j qj − L
(1)
j
For a particle which moves freely in a conservative field with potential U, the Lagrangian in
rectangular coordinates is
L=
(
)
1
m x2 + y2 + z2 − U
2
and the linear momentum components in rectangular coordinates are
∂L

= mx 
∂x


py = my


pz = mz


px =
(2)
1

H =  mx 2 + my 2 + mz 2  −  m x 2 + y 2 + z 2 − U 
2

(
=
(
)
)
(
1
1 2
m x2 + y2 + z2 + U =
px + py2 + pz2
2
2m
)
(3)
which is just the total energy of the particle. The canonical equations are [from Eqs. (7.160) and
(7.161)]
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
px = mx = −
∂U
= Fx
∂x
py = my = −
∂U
= Fy
∂y
pz = mz = −
∂U
= Fz
∂z
209
(4)
These are simply Newton’s equations.
7-24.
θ
m
The kinetic energy and the potential energy of the system are expressed as
T=
1
m
2
(
2
)
+ 2θ 2 =
U = − mg cos θ
(
)
1

m α 2 + 2θ 2 
2



(1)
so that the Lagrangian is
L = T −U =
(
)
1
m α 2 + 2θ 2 + mg cos θ
2
(2)
The Hamiltonian is
H = pθ θ − L =
=
∂L
θ−L
∂θ
pθ2
1
− mα 2 − mg cos θ
2
2m
2
(3)
which is different from the total energy, T + U. The total energy is not conserved in this system
because work is done on the system and we have
d
(T + U ) ≠ 0
dt
(4)
210
CHAPTER 7
7-25.
z
r
z
m
y
θ
x
In cylindrical coordinates the kinetic energy and the potential energy of the spiraling particle
are expressed by
1

m  r 2 + r 2θ 2 + z 2  
2


U = mgz

(1)
i.e., z = kθ 


r = const.

(2)
T=
Therefore, if we use the relations,
z = kθ
the Lagrangian becomes
L=

1  r2 2
m  2 z + z 2  − mgz
2 k

(3)
 r2

∂L
= m  2 + 1 z
∂z
k

(4)
Then the canonical momentum is
pz =
or,
z=
pz
(5)
 r2

m  2 + 1
k

The Hamiltonian is
H = pz z − L = p z
pz
 r2

m  2 + 1
k

−
pz2
 r2

2m  2 + 1
k

+ mgz
(6)
or,
H=
1
2
Now, Hamilton’s equations of motion are
pz2
 r2

m  2 + 1
k

+ mgz
(7)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
−
∂H
= pz ;
∂z
∂H
=z
∂p z
211
(8)
so that
−
∂H
= − mg = pz
∂z
∂H
=
∂p z
pz
 r2

m  2 + 1
k

=z
(9)
(10)
Taking the time derivative of (10) and substituting (9) into that equation, we find the equation
of motion of the particle:
z=
g
r

 k 2 + 1


2
(11)
It can be easily shown that Lagrange’s equation, computed from (3), gives the same result as
(11).
7-26.
a)
θ
m
L = T −U =
L=
1 2 2
m θ − mgy
2
1 2 2
m θ + mg cos θ
2
pθ =
∂L
= m 2θ
∂θ
so
θ=
pθ
m 2
Since U is velocity-independent and the coordinate transformations are time-independent, the
Hamiltonian is the total energy
H = T +U =
The equations of motion are
pθ2
− mg cos θ
2m 2
212
CHAPTER 7
θ=
p
∂H
∂H
= θ 2 and pθ = −
= − mg sin θ
∂pθ m
∂θ
b)
a
x
m2
m1
T=
1
1
1 x2
m1 x 2 + m2 x 2 + I 2
2
2
2 a
where I = moment of inertia of the pulley
U = − m1 gx − m2 g ( − x )
px =
∂L ∂T 
I 
=
=  m1 + m2 + 2  x
∂x ∂x 
a 
So
x=
px
I 

 m1 + m2 + a 2 
H=T+U
H=
px2
I 

2  m1 + m2 + 2 
a 

− m1 gx − m2 g ( − x )
The equations of motion are
x=
px
∂H
=
I 
∂p x

2  m1 + m2 + 2 
a 

px = −
∂H
= m1 g − m2 g = g ( m1 − m2 )
∂p x
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
213
7-27.
a)
m2
k,b
θ
m1
xi , yi = coordinates of mi
Using , θ as polar coordinates
x2 = x1 + cos θ
y2 = y1 + sin θ
x2 = x1 + cos θ − θ sin θ
(1)
y2 = y1 + sin θ + θ cos θ
(2)
If we substitute (1) and (2) into
L = T −U =
(
(
)
)
1
1
1
2
m1 x12 + y12 + m2 x22 + y22 − k ( − b )
2
2
2
the result is
L=
(
)
1
1
( m1 + m2 ) x12 + y12 + m2
2
2
(
2
+ 2θ 2
)
1
2
+ m2 ( x1 cos θ + y1 sin θ ) + m2 θ ( y1 cos θ − x1 sin θ ) − k ( − b )
2
The equations of motion are
x1 :
d
( m1 + m2 ) x1 + m2 cos θ − m2 θ sin θ  = 0
dt 
= m1 x1 + m2 x2 = px
So px = constant
y1 :
d
( m1 + m2 ) y1 + m2 sin θ + m2 θ cos θ  = 0
dt 
= m1 y1 + m2 y2 = py
So py = constant
:
d
 m2 + m2 ( x1 cos θ + y1 sin θ )  = m2 θ 2 − k ( − b ) + m2θ ( y1 cos θ − x1 sin θ )
dt 
which reduces to
214
CHAPTER 7
− θ 2 + x1 cos θ + y1 sin θ +
θ:
k
( − b) = 0
m2
d
 m2 2θ + m2 ( y1 cos θ − x1 sin θ ) 
dt 
= − m2 ( x1 sin θ − y1 cos θ ) + m2 θ ( − x1 cos θ − y1 sin θ )
which reduces to
θ+
b)
2
θ+
cos θ
y1 −
sin θ
x1 = 0
As was shown in (a)
∂L
= px = constant
∂x1
∂L
= py = constant
∂y1
c)
(total linear momentum)
Using L from part (a)
px1 =
∂L
= ( m1 + m2 ) x1 + m2 cos θ − m2 θ sin θ
∂x1
py1 =
∂L
= ( m1 + m2 ) y1 + m2 sin θ − m2 θ cos θ
∂y1
p =
∂L
= m2 x1 cos θ + m2 y1 sin θ + m2
∂
pθ = − m2 x1 sin θ + m2 y1 cos θ + m2 2θ
Inverting these equations gives (after much algebra)
x1 =
sin θ 
1 
px1 − p cos θ +
pθ 

m1 

y1 =
cos θ 
1 
py1 − p sin θ −
pθ 

m1 

=
1 
m1 + m2 
p 
 − px1 cos θ − py1 sin θ +
m1 
m2

θ=
1 
m1 + m2 
pθ 
 px1 sin θ − py1 cos θ +
m1 
m2

Since the coordinate transformations are time independent, and U is velocity independent,
H = T +U
Substituting using the above equations for qi in terms of pi gives
215
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
H=
m1 + m2
1  2
2
 px1 + py1 +
m2
2m1 
 2 pθ2 
 p + 2  − 2 p px1 cos θ + py1 sin θ


pθ
 1
2
+2
px1 sin θ − py1 cos θ  + k ( − b )
 2
(
)
(
)
The equations of motion are
x1 =
sin θ 
1 
∂H
=
px1 − p cos θ +
pθ 

∂px1 m1 

y1 =
cos θ 
1 
∂H
=
py1 − p sin θ −
pθ 

∂py1 m1 

=
θ=

1  m1 + m2
∂H
p − px1 cos θ − py1 sin θ 
=

m1  m2
∂p


1  m1 + m2
∂H
pθ + px1 sin θ − py1 cos θ 
=

∂pθ m1  m2

px1 = −
∂H
=0
∂x1
py1 = −
∂H
=0
∂y1
2
p
∂H ( m1 + m2 ) pθ
p =−
=
+ θ 2 px1 sin θ − py1 cos θ − k ( − b )
3
m1m2
m1
∂
(
pθ = −
)
p
∂H p
=
− px1 sin θ + py1 cos θ − θ  px1 cos θ + py1 sin θ 
m1
∂θ m1
(
)
Note: This solution chooses as its generalized coordinates what the student would most likely
choose at this point in the text. If one looks ahead to Section 8.2 and 8.3, however, it would
show another choice of generalized coordinates that lead to three cyclic coordinates ( xCM , yCM ,
and θ ), as shown in those sections.
7-28.
F = − kr −2
so
U = − kr −1
L = T −U =
pr =
pθ =
(
)
1
k
m r 2 + r 2θ 2 +
2
r
p
∂L
= mr so r = r
m
∂r
∂L
= mr 2θ
∂θ
so θ =
pθ
mr 2
Since the coordinate transformations are independent of t, and the potential energy is velocityindependent, the Hamiltonian is the total energy.
216
CHAPTER 7
H = T +U =
(
)
1
k
m r 2 + r 2θ 2 −
2
r
p2  k
1  pr2
m  2 + r 2 2θ 4  −
mr  r
2 m
=
pr2
p2
k
+ θ2−
r
2m 2mr
H=
Hamilton’s equations of motion are
∂H p r
=
∂p r m
r=
θ=
pr = −
p2
k
∂H
= θ3 − 2
r
∂r mr
pθ = −
∂H
=0
∂θ
p
∂H
= θ
∂pθ mr 2
7-29.
a
k
θ
m
b = unextended length of spring
= variable length of spring
a)
x = sin θ
y=
x = sin θ + θ cos θ
1 2
at − cos θ
2
Substituting into T =
(
1
m x2 + y2
2
U = mgy +
y = at − cos θ + θ sin θ
)
1
2
k ( − b)
2
gives
L = T −U =
1 
m
2 
2

at 2  k
2
+ 2θ 2 + a 2t 2 + 2at θ sin θ − cos θ  + mg  cos θ −
− ( − b)

2  2

Lagrange’s equations give:
(
)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
:
d
 m − amt cos θ  = m θ 2 + matθ sin θ + mg cos θ − k ( − b )
dt 
θ:
d
 m 2θ + mat sin θ  = mat sin θ − mg sin θ + mat θ cos θ
dt 
217
Upon simplifying, the equations of motion reduce to:
− θ 2 − ( a + g ) cos θ +
θ+
2
θ+
k
( − b) = 0
m
a+ g
p =
∂L
= m − mat cos θ
∂
pθ =
∂L
= m 2θ + mat sin θ
∂θ
or
θ=
b)
sin θ = 0
or
=
p
+ at cos θ
m
pθ
at sin θ
−
2
m
(2)
Since the transformation equations relating the generalized coordinates to rectangular
coordinates are not time-independent, the Hamiltonian is not the total energy.
H = ∑ pi qi − L = p + pθ θ − L
Substituting (1) and (2) for
H=
and θ and simplifying gives
p2
p2
1
1
at
2
+ θ 2 − pθ sin θ + atp cos θ + k ( − b ) + mgat 2 − mg cos θ
2 m 2m
2
2
The equations for θ and
are
θ=
=
p
at
∂H
= θ 2 − sin θ
∂pθ m
∂H p
= + at cos θ
m
∂p
agreeing with (1) and (2)
The equations for p and pθ are
pθ2
∂H
at
p =−
= − 2 pθ sin θ − k ( − b ) + mg cos θ +
∂
m 3
or
p +
pθ = −
at
2
pθ sin θ + k ( − b ) − mg cos θ +
(1)
pθ2
=0
m 3
at
∂H
= − pθ cos θ + at p sin θ − mg sin θ
∂θ
218
CHAPTER 7
or
pθ −
c)
sin θ
θ , cos θ 1 −
at
pθ cos θ − at p sin θ + mg sin θ = 0
θ2
2
Substitute into Lagrange’s equations of motion
 θ2  k
− θ 2 − ( a + g ) 1 −  + ( − b ) = 0
2 m

θ+
For small oscillations, θ
1, θ
1,
2
θ+
a+ g
θ−
at θ
=0
1 . Dropping all second-order terms gives
+
θ+
k
k
= a+ g+ b
m
m
a+ g
θ =0
For θ,
Tθ =
2π
ω
= 2π
a+ g
The solution to the equation for is
=
homogeneous
= A cos
+
particular
k
k
m
t + B sin
t + (a + g) + b
m
m
k
So for ,
T =
2π
ω
= 2π
m
k
7-30.
a) From the definition of a total derivative, we can write
 ∂g ∂qk ∂g ∂pk 
dg ∂g
=
+ ∑
+

dt ∂t
∂ p k ∂t 
k  ∂qk ∂t
Using the canonical equations
(1)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
∂qk
∂H
= qk =
∂t
∂p k
219




∂pk
∂H 
= pk = −
∂t
∂qk 
(2)
 ∂ g ∂H ∂ g ∂ H 
dg ∂g
=
+ ∑
−

dt ∂t
∂ p k ∂ qk 
k  ∂qk ∂pk
(3)
dg ∂g
=
+  g , H 
dt ∂t 
(4)
we can write (1) as
or
qj =
b)
∂q j
∂t
=
∂H
∂p j
(5)
According to the definition of the Poisson brackets,
 ∂q ∂ H ∂g j ∂H 
qj , H  = ∑  j
−



∂pk ∂qk 
k  ∂ qk ∂ p k
(6)
but
∂q j
∂qk
= δ jk and
∂q j
∂pk
= 0 for any j,k
(7)
then (6) can be expressed as
∂H


 q j , H  = ∂p = q j
j
(8)
In the same way, from the canonical equations,
pj = −
∂H
∂q j
(9)
so that
 ∂p ∂H ∂p j ∂ H 
 pj , H  = ∑  j
−



∂ p k ∂ qk 
k  ∂qk ∂pk
(10)
but
∂p j
∂pk
then,
= δ jk and
∂p j
∂ qk
= 0 for any j,k
(11)
220
CHAPTER 7
pj = −
c)
∂H
=  p j , H 
∂q j 
(12)
 ∂p ∂ p j ∂ p k ∂ p j 
 pk , p j  = ∑  k
−



∂p ∂ q 
 ∂ q ∂p
(13)
∂pk
= 0 for any k,
∂q
(14)
since,
the right-hand side of (13) vanishes, and


 pk , p j  = 0
(15)
 ∂ q ∂ q j ∂ qk ∂ q j 
 qk , q j  = ∑  k
−



 ∂ q ∂p ∂p ∂ q 
(16)
In the same way,
since
∂q j
∂p
= 0 for any j,
(17)


 qk , q j  = 0
(18)
the right-hand side of (16) vanishes and
d)
 ∂q ∂p j ∂qk ∂p j 
 qk , p j  = ∑  k
−



 ∂ q ∂p ∂p ∂q 
= ∑δk δ j
(19)
or,
 qk , p j  = δ kj


(20)
e) Let g ( pk , qk ) be a quantity that does not depend explicitly on the time. If g ( pk , qk )
commutes with the Hamiltonian, i.e., if
 g , H  = 0
(21)
dg
=0
dt
(22)
then, according to the result in a) above,
and g is a constant of motion.
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
221
7-31. A spherical pendulum can be described in terms of the motion of a point mass m on the
surface of a sphere of radius , where corresponds to the length of the pendulum support rod.
The coordinates are as indicated below.
z
θ
mg
y
φ
x
The kinetic energy of the pendulum is
T=
1
1
1
I1φ 2 + I 2θ 2 = m
2
2
2
2
(φ
2
sin 2 θ + θ 2
)
(1)
and the potential energy is
U = mg cos θ
(2)
The Lagrangian is
L=
1
m
2
2
(φ
2
)
sin 2 θ + θ 2 − mg cos θ
(3)
so that the momenta are
∂L
= m 2θ
∂θ
(4)
∂L
= m 2φ sin 2 θ
∂φ
(5)
pθ =
pφ =
The Hamiltonian then becomes
H = pθ θ + pφ φ −
1
m
2
p2
= θ 2 + V θ , pφ
2m
(
2
(φ
2
)
sin 2 θ + θ 2 + mg cos θ
)
(6)
which is just the total energy of the system and where the effective potential is
(
pθ2
V θ , pφ =
+ mg cos θ
2m 2 sin 2 θ
)
When pφ = 0 , V(θ,0) is finite for all θ, with a maximum at θ = 0 (top of the sphere) and a
minimum at θ = π (bottom of the sphere); this is just the case of the ordinary pendulum. For
different values of pφ , the V–θ diagram has the appearance below:
(7)
222
CHAPTER 7
V
π
2
0
π θ
Pφ = 0
When pφ > 0 , the pendulum never reaches θ = 0 or θ = π because V is infinite at these points.
The V–θ curve has a single minimum and the motion is oscillatory about this point. If the total
energy (and therefore V) is a minimum for a given pφ ,θ is a constant, and we have the case of a
conical pendulum.
For further details, see J. C. Slater and N. H. Frank, Mechanics, McGraw-Hill, New York, 1947,
pp. 79–86.
7-32.
The Lagrangian for this case is
L = T −U =
(
)
1
k
m r 2 + r 2θ 2 + r 2 sin 2 θφ 2 +
2
r
(1)
where spherical coordinates have been used due to the symmetry of U.
The generalized coordinates are r, θ, and φ, and the generalized momenta are
pr =
∂L
= mr
∂r
(2)
pθ =
∂L
= mr 2θ
∂θ
(3)
pφ =
∂L
= mr 2φ sin 2 θ
∂φ
(4)
The Hamiltonian can be constructed as in Eq. (7.155):
H = pr r + pθ θ + pφ φ − L
(
)
=
1
k
m r 2 + r 2θ 2 + r 2φ 2 sin 2 θ −
2
r
=
 k
pφ2
pθ2
1  pr2
+
 +
−
2
2
2
mr sin θ  r
2  m mr
(5)
Eqs. (7.160) applied to H as given in (5) reproduce equations (2), (3), and (4). The canonical
equations of motion are obtained applying Eq. (7.161) to H:
2
pr = −
pφ
p2
∂H
k
= − 2 + θ3 +
3
∂r
r
mr
mr sin 2 θ
(6)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
223
pφ2 cot θ
∂H
pθ = −
=
∂θ mr 2 sin 2 θ
pφ = −
(7)
∂H
=0
∂φ
(8)
The last equation implies that pφ = const , which reduces the number of variables on which H
depends to four: r , θ , pr , pθ :
H=
1
2m
 2 pθ2
const  k
 pr + 2 + 2
−
r
r sin 2 θ  r

(9)
For motion with constant energy, (9) fixes the value of any of the four variables when the other
three are given.
From (9), for a given constant value of H = E, we obtain

p 2 sin 2 θ + const 2mk 
+
pr =  2 mE − θ 2

r sin 2 θ
r 

12
and so the projection of the phase path on the r − pr plane are as shown below.
pr
θ3
2mE
θ2
θ1
θ1 < θ2 < θ3
7-33.
x
m1
x′
m2
m3
Neglect the masses of the pulleys
T=
1
1
1
2
2
m1 x 2 + m2 ( x ′ − x ) + m3 ( − x − x ′ )
2
2
2
U = − m1 gx − m2 g ( − x + x ′ ) − m3 g ( − x + ′ − x ′ )
(10)
224
CHAPTER 7
L=
1
1
m1 + m2 + m3 ) x 2 + ( m2 + m3 ) x ′ 2 + xx ′ ( m3 − m2 )
(
2
2
+ g ( m1 − m2 − m3 ) x + g ( m2 − m3 ) x ′ + constant
We redefine the zero in U such that the constant in L = 0.
px =
∂L
= ( m1 + m2 + m3 ) x + ( m3 − m2 ) x ′
∂x
(1)
px ′ =
∂L
= ( m3 − m2 ) x + ( m2 + m3 ) x ′
∂x ′
(2)
Solving (1) and (2) for px and px ′ gives
x = D−1 ( m2 + m3 ) px + ( m2 − m3 ) px ′ 
x ′ = D−1 ( m2 + m3 ) px + ( m1 + m2 + m3 ) px ′ 
where D = m1m3 + m1m2 + 4 m2 m3
H =T +U
=
1
1
m1 + m2 + m3 ) x 2 + ( m2 + m3 ) x ′ 2 + ( m3 − m2 ) xx
(
2
2
− g ( m1 − m2 − m3 ) x − g ( m2 − m3 ) x ′
Substituting for x and x′ and simplifying gives
H=
1
1
m2 + m3 ) D−1 px2 + ( m1 + m2 + m3 ) D−1 px2′
(
2
2
+ ( m2 − m3 ) D−1 px px ′ − g ( m1 − m2 + m3 ) x − g ( m2 − m3 ) x ′
where D = m1m3 + m1 m2 + 4 m2 m3
The equations of motion are
x=
x′ =
∂H
= ( m2 + m3 ) D−1 px + ( m2 − m3 ) D−1 px ′
∂p x
∂H
= ( m2 − m3 ) D−1 px ′ + ( m1 + m2 + m3 ) D−1 px ′
∂p x ′
px = −
∂H
= g ( m1 − m2 − m3 )
∂x
px ′ = −
∂H
= g ( m2 − m3 )
∂x ′
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
225
7-34.
x
θ
r
m
R
M
The coordinates of the wedge and the particle are
xM = x
xm = r cos θ + x
yM = 0
ym = − r sin θ
(1)
The Lagrangian is then
L=
M+m 2 m 2
x +
r + r 2θ 2 + 2 xr cos θ − 2 xrθ sin θ + mgr sin θ
2
r
(
)
(2)
Note that we do not take r to be constant since we want the reaction of the wedge on the
particle. The constraint equation is f ( x , θ , r ) = r − R = 0 .
a) Right now, however, we may take r = R and r = r = 0 to get the equations of motion for x
and θ. Using Lagrange’s equations,
(
x = aR θ sin θ + θ 2 cos θ
θ=
)
(3)
x sin θ + g cos θ
R
(4)
where a ≡ m ( M + m) .
b) We can get the reaction of the wedge from the Lagrange equation for r
λ = mx cos θ − mRθ 2 − mg sin θ
(5)
We can use equations (3) and (4) to express x in terms of θ and θ , and substitute the resulting
expression into (5) to obtain

a−1 
2
 Rθ + g sin θ
2
 1 − a sin θ 
(
λ=
)
(6)
To get an expression for θ , let us use the conservation of energy
H=
M+m 2 m 2 2
x +
R θ − 2xRθ sin θ − mgR sin θ = − mgR sin θ 0
2
2
(
)
(7)
where θ 0 is defined by the initial position of the particle, and − mgR sin θ 0 is the total energy of
the system (assuming we start at rest). We may integrate the expression (3) to obtain
x = aRθ sin θ , and substitute this into the energy equation to obtain an expression for θ
θ2 =
2 g ( sin θ − sin θ 0 )
(
R 1 − a sin 2 θ
)
(8)
226
CHAPTER 7
Finally, we can solve for the reaction in terms of only θ and θ 0
λ=−
(
mMg 3 sin θ − a sin 3 θ − 2 sin θ 0
( M + m) ( 1 − a sin θ )
2
)
2
(9)
7-35. We use zi and pi as our generalized coordinates, the subscript i corresponding to the
ith particle. For a uniform field in the z direction the trajectories z = z(t) and momenta p = p(t)
are given by
1 2 
gt 
2


pi = pi 0 − mgt

z i = z i 0 + vi 0 t −
(1)
where zi 0 , pi 0 , and vi 0 = pi 0 m are the initial displacement, momentum, and velocity of the ith
particle.
Using the initial conditions given, we have
z1 = z0 +
p0 t 1 2
− gt
m 2
(2a)
p1 = p0 − mgt
(2b)
z2 = z0 + ∆z0 +
p0 t 1 2
− gt
m 2
(2c)
p2 = p0 − mgt
z 3 = z0 +
(2d)
( p0 + ∆p0 ) t − 1 gt 2
(2e)
2
m
p3 = p0 + ∆p0 − mgt
z4 = z0 + ∆Z0 +
(2f)
( p0 + ∆p0 ) t − 1 gt 2
m
2
p4 = p0 + ∆p0 − mgt
(2g)
(2h)
The Hamiltonian function corresponding to the ith particle is
H i = Ti + Vi =
p2
mzi2
+ mgzi = i + mgzi = const.
2
2m
From (3) the phase space diagram for any of the four particles is a parabola as shown below.
(3)
227
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
Area at t = 0
p0 + ∆p0
p0
1
2
3
4
Area at t = t1
∆p0
p
∆z0
z0 + ∆z0
z0
z
From this diagram (as well as from 2b, 2d, 2f, and 2h) it can be seen that for any time t,
p1 = p2
(4)
p3 = p 4
(5)
Then for a certain time t the shape of the area described by the representative points will be of
the general form
p
(p3,z3)
3
(p4,z4)
4
1 (p1,z1)
2 (p2,z2)
z
where the base 1 2 must parallel to the top 3 4 . At time t = 0 the area is given by ∆z0 ∆p0 , since
it corresponds to a rectangle of base ∆z0 and height ∆p0 . At any other time the area will be
given by
{
A = base of parallelogram t = t = ( z2 − z1 ) t = t
1
1
= ( z 4 − z 3 ) t = t = ∆z 0
1
{
x height of parallelogram t = t = ( p3 − p1 )
1
= ( p 4 − p2 )
}
t = t1
t = t1
=∆p0 ∆z0
= ∆p0
}
(6)
Thus, the area occupied in the phase plane is constant in time.
7-36.
The initial volume of phase space accessible to the beam is
V0 = π R02π p02
(1)
After focusing, the volume in phase space is
V1 = π R12π p12
(2)
228
CHAPTER 7
where now p1 is the resulting radius of the distribution of transverse momentum components
of the beam with a circular cross section of radius R1 . From Liouville’s theorem the phase space
accessible to the ensemble is invariant; hence,
V0 = π R02π p02 = V1 = π R12π p12
(3)
from which
p1 =
R0 p0
R1
(4)
If R1 < R0 , then p1 > p0 , which means that the resulting spread in the momentum distribution
has increased.
This result means that when the beam is better focused, the transverse momentum components
are increased and there is a subsequent divergence of the beam past the point of focus.
7-37.
Let’s choose the coordinate system as shown:
x1
x2
x3
m1
m3
m2
The Lagrangian of the system is
2
2
2
1   dx 
 dx  
 dx 
L = T − U =  m1  1  + m2  2  + m3  3   + g ( m1 x1 + m2 x2 + m3 x3 )
 dt 
 dt  
2   dt 
with the constraints
x1 + y = l1
and
which imply 2x1 + x2 + x3 − (2l1 + l2 ) = 0 ⇒ 2
x2 − y + x3 − y = l2
d 2 x1 d 2 x2 d 2 x3
+ 2 + 2 =0
dt 2
dt
dt
The motion equations (with Lagrange multiplier λ ) are
(1)
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
m1 g − m1
d 2 x1
+ 2λ = 0
dt 2
(2)
m2 g − m2
d 2 x2
+λ =0
dt 2
(3)
m3 g − m3
d 2 x3
+λ =0
dt 2
(4)
Combining (1)–(4) we find
λ=
−4 g
4
1
1
+
+
m1 m2 m3
Finally, the string tension that acts on m1 is (see Eq. (2))
T1 = m1 g − m1
7-38.
8g
d 2 x1
= −2λ =
2
4
1
1
dt
+
+
m1 m2 m3
The Hamiltonian of the system is
2
H =T +U =
229
kx 2 bx 4 p 2 kx 2 bx 4
1  dx 
m  +
+
=
+
+
2  dt 
2
4
2m
2
4
The Hamiltonian motion equations that follow this Hamiltonian are
dx ∂H p
=
=
dt ∂p m
dp
∂H
=−
= −(kx + bx 3 )
dt
∂x
7-39.
z
The Lagrangian of the rope is
2
2
mgz 2
1  dz   mz z  1  dz 
L = T − U = m  −  −
g  = m  +
2  dt   b 2  2  dt 
2b
230
CHAPTER 7
from which follows the equation of motion
mgz
d2 z
∂L d ∂L
=
⇒
=m 2
∂z dt ∂z
b
dt
7-40.
θ1
m
θ2
m
2m
x
We choose the coordinates for the system as shown in the figure.
The kinetic energy is
2
2
2

dθ dx
1
1  2  dθ 1   dx 
 dx 
T = 2m   + m  b 
cos θ 1 
+   + 2b 1

 dt 
dt dt
2
2   dt   dt 

2
2
dθ 1
dθ 2
dθ 2
1  dx
  dθ 1
 
cos θ 1 + b
cos θ 2  +  b
sin θ 1 + b
sin θ 2  
+ m 
+b
  dt
 
dt
dt
dt
2  dt
The potential energy is
U = − mgb cos θ 1 − mg(b cos θ 1 + b cos θ 2 )
And the Lagrangian is
1
dx dθ 1
 dx 
 dθ 
 dθ 
cos θ 1 + mb 2  2 
L = T − U = 2m   + mb 2  1  + 2mb
 dt 
 dt 
 dt 
2
dt dt
2
+ mb
2
dx dθ 2
dθ dθ
cos θ 2 + mb 2 1 2 cos (θ 1 − θ 2 ) + 2mgb cos θ 1 + mgb cos θ 2
dt dt
dt dt
From this follow 3 equations of motion
∂L d ∂L
=
∂x dt ∂x
2
⇒ 0=4
 d 2θ

d2 x
d 2θ
+ b  2 21 cos θ 1 + 22 cos θ 2 
2
dt
dt
 dt

2
  dθ  2

 dθ 
− b  2  1  sin θ 1 +  2  sin θ 2 
 dt 
  dt 

231
HAMILTON’S PRINCIPLE—LAGRANGIAN AND HAMILTONIAN DYNAMICS
∂L
d ∂L
=
∂θ 1 dt ∂θ 1
⇒ − 2 g sin θ 1 = 2b
d 2θ 1
d 2θ 2
d2 x
+
+
b
2
cos
θ
cos(θ 1 − θ 2 )
1
dt 2
dt 2
dt 2
 dθ 
+ b  2  sin(θ 1 − θ 2 )
 dt 
2
∂L
d ∂L
=
∂θ 2 dt ∂θ 2
− g sin θ 2 = b
7-41.
⇒
d 2θ 2 d 2 x
d 2θ 1
 dθ 
b
+
+
cos
θ
cos (θ 1 − θ 2 ) − b  1  sin (θ 1 − θ 2 )
2
2
2
2
 dt 
dt
dt
dt
2
For small angle of oscillation θ we have
1 2  dθ 
1  db 
mb   + m  
 dt 
2
2  dt 
2
T=
2
and
U = − mgb cos θ
So the Lagrangian reads
1 2  dθ 
1  db 
mb   + m   + mgb cos θ


2
2  dt 
dt
2
L = T −U =
2
from which follow 2 equations of motion
∂L d ∂L
=
∂b dt ∂b
∂L d ∂ L
=
∂θ dt ∂θ
d2b
dα
 dθ 
⇒ b   + g cos θ = 2 = −
 dt 
dt
dt
2
⇒ − mgb sin θ = 2mb
db dθ
d 2θ
dθ
d 2θ
+ mb 2 2 = −2mbα
+ mb 2 2
dt dt
dt
dt
dt
232
CHAPTER 7
CHAPTER
8
Central-Force Motion
8-1.
x3
m1
r1
m2
r2
x2
x1
In a uniform gravitational field, the gravitational acceleration is everywhere constant. Suppose
the gravitational field vector is in the x1 direction; then the masses m1 and m2 have the
gravitational potential energies:
U g(1) = − F (1) x1(1) = − m1α x1(1) 

( 2)
( 2) ( 2 )
( 2) 
U g = − F x1 = − m2α x1 
(
(1)
)
where r1 = x1(1) , x2(1) , x3(1) and where α is the constant gravitational acceleration. Therefore,
introducting the relative coordinate r and the center of mass coordinate R according to
r = r1 − r2


m1r1 + m2 r2 = ( m1 + m2 ) R 
(2)
we can express r1 and r2 in terms of r and R by




m1
r2 = −
r+R 
m1 + m2

r1 =
m2
r+R
m1 + m2
(3)
233
234
CHAPTER 8
which differ from Eq. (8.3) in the text by R. The Lagrangian of the two-particle system can now
be expressed in terms of r and R:
L=
1
1
2
2
m1 r1 + m2 r2 − U ( r ) − U g(1) − U g( 2)
2
2
2
=
2
m2
m1
1
1
r + R + m2 −
r + R − U (r)
m1
2
2
m1 + m2
m1 + m2
 m2



m1
x + X  + m2α  −
x + X
+ m1α 
 m1 + m2

 m1 + m2

(4)
where x and X are the x1 components of r and R, respectively. Then, (4) becomes
2
2
 m2  2 1
 m1  2 1
1
L = m1 
 r + m2 
 r + ( m1 + m2 ) R
2
2
2
 m1 + m2 
 m1 + m2 
m1m2 − m1m2
x + ( m1 + m2 ) α X
− U (r) + α
m1 + m2
(5)
Hence, we can write the Lagrangian in the form
L=
2
1
1
2
µ r − U ( r ) + ( m1 + m2 ) R + ( m1 + m2 ) α X
2
2
(6)
where µ is the reduced mass:
µ=
m1m2
m1 + m2
(7)
Therefore, this case is reducible to an equivalent one-body problem.
8-2.
Setting u = 1 r , Eq. (8.38) can be rewritten as
θ = −∫
2µE
2
(
+
du
2µ k
2
(1)
u−u
2
)
where we have used the relation du = − 1 r 2 dr . Using the standard form of the integral [see
Eq. (E.8c), Appendix E]:
∫
dx
ax 2 + bx + c
=
 2ax + b 
−1
sin −1 
 + const.
2
−a
 b − 4 ac 
(2)
we have


2 2µ k


− + 2

−1 
r
θ + const. = sin 

2
  2µ k  + 8 µ E 
2
  2 

(3)
235
CENTRAL-FORCE MOTION
or, equivalently,
2 2µ k
− + 2
r
2
µE
 2µ k 
 2  + 8 2
sin (θ + const.) =
(4)
We can choose the point from which θ is measured so that the constant in (4) is − π 2 . Then,
2
cos θ =
1
−1
µk r
(5)
2E 2
1+
µk 2
which is the desired expression.
8-3. When k → k 2 , the potential energy will decrease to half its former value; but the kinetic
energy will remain the same. Since the original orbit is circular, the instantaneous values of T
and U are equal to the average values, T and U . For a 1 r 2 force, the virial theorem states
T =−
1
U
2
(1)
Hence,
1
1
E = T +U = − U +U = U
2
2
Now, consider the energy diagram
E
D
C
B
r
–k/r
A
where
CB = E
original total energy
CA = U
original potential energy
CD = UC
original centrifugal energy
The point B is obtained from CB = CA − CD . According to the virial theorem, E = (1 2) U or
CB = (1 2) CA. Therefore,
CD = CB = BA
(2)
236
CHAPTER 8
Hence, if U suddenly is halved, the total energy is raised from B by an amount equal to (1 2) CA
or by CB . Thus, the total energy point is raised from B to C; i.e., E(final) = 0 and the orbit is
parabolic.
8-4. Since the particle moves in a central, inverse-square law force field, the potential energy
is
U=−
k
r
(1)
so that the time average is
U =−
1
τ
τ∫
0
k
dt
r
(2)
Since this motion is a central motion, the angular momentum is a constant of motion. Then,
µ r 2 θ ≡ = const.
(3)
from which
dt =
µr 2
dθ
(4)
Therefore, (2) becomes
U =−
1
τ
2π
k µr 2
kµ
dθ = −
τ
r
∫
0
Now, substituting α r = 1 + ε cos θ and τ = ( 2µ
U =−
k α
2π a 3 2
)π
2π
∫ rdθ
(5)
0
α a 3 2 , (5) becomes
2π
1
∫ 1 + ε cos θ dθ
(6)
0
where a is the semimajor axis of the ellipse. Using the standard integral [see Eq. (E.15),
Appendix E],
2π
1
∫ 1 + ε cos θ dθ =
0
2π
1− ε2
(7)
and the relation,
α = a (1 − ε 2 )
(8)
(6) becomes
U =−
The kinetic energy is
k
a
(9)
237
CENTRAL-FORCE MOTION
2
1
T = − µr 2 +
2
2µ r 2
(10)
and the time average is
T =−
1
τ
τ
∫ T dt =
2π
kµ
a3
1
∫ Tr
dθ
(11)
2π
kµ  µ
π 2
2
θ
+
rr
d
(
)


a 3  2 ∫0
µ 
(12)
0
2π
2
0
Part of this integral is trivial,
T =
1
2π
To evaluate the integral above, substitute the expression for r and make a change of variable
1
2
2π
∫ ( rr )
0
2
1 ε
dθ =  
2 µ 
2 2π
sin 2 θ dθ
∫ (1 + ε cos θ )
 ε
= 
 µ
2
0
2 1
1 − x 2 dx
∫ (1 + ε x )
(13)
2
−1
The reader is invited to evaluate this integral in either form. The solution presented here is to
integrate by parts twice, which gives a third integral that can be looked up in a table:
1
1 − x 2 dx
∫ (1 + ε x )
−1
2
1 − x2
=−
ε (1 + ε x )
1
−
x dx
ε ∫
−1
−1
1  x sin −1 x
=− 
ε  1 + ε x
1
1
1
−1
sin −1 x dx 

2
−1 ( 1 + ε x ) 

1
−∫
1 
2
tan −1
= − 2  sin −1 x +
2
ε 
1− ε

=
(14)
1 − x 2 (1 + ε x )
(15)
(1 − ε )(1 − x ) 
(1 + ε )(1 + x ) −1 
1
π 
1 
−1 +

2 
ε 
1 − ε2 
(16)
(17)
Substituting this into (13) and then into (12), we obtain the desired answer,
T =
k
2a
(18)
This explicitly verifies the virial theorem, which states that for an inverse-square law force,
T =−
1
U
2
(19)
238
CHAPTER 8
8-5.
m2
a
m1
Suppose two particles with masses m1 and m2 move around one another in a circular orbit with
radius a. We can consider this motion as the motion of one particle with the reduced mass µ
moving under the influence of a central force G m1m2 a 2 . Therefore, the equation of motion
before the particles are stopped is
m1m2
a2
µ aω 2 = G
(1)
where
1
µ
=
1
1
+
,
m1 m2
ω=
2π
(2)
τ
The radius of circular motion is
 G m1m2 τ 2 
a=

2
 4π µ 
13
(3)
After the circular motion is stopped, the particle with reduced mass µ starts to move toward the
force center. We can find the equation of motion from the conservation of energy:
−G
m1m2 1 2
mm
= µx − G 1 2
a
2
x
(4)
or,
 2G m1m2
x=
µ

 1 1 
 −  
x a 
12
(5)
Therefore, the time elapsed before the collision is
0
t = ∫ dt = − ∫
a
dx
2G m1m2  1 1 
 x − a 
µ
(6)
where the negative sign is due to the fact that the time increases as the distance decreases.
Rearranging the integrand, we can write
t=−
aµ
2G m1m2
0
x
dx
a−x
∫
a
(7)
Setting x ≡ y 2 ( dx = 2 y dy ) , the integral in (7) becomes
0
I=∫
a
x
dx = 2
a−x
0
∫
a
y2
a − y2
dy
239
CENTRAL-FORCE MOTION
Using Eq. (E.7), Appendix E, we find
 y a − y2 a
 y 
+ sin −1   
I = 2 −
2
2
 a  

0
=−
πa
a
2
(8)
Therefore,
t=
aµ
aπ
2G m1m2 2
or,
t=
τ
(9)
4 2
8-6.
m1
x1
m2
O
x2
r
r = x2 − x1
When two particles are initially at rest separated by a distance r0 , the system has the total
energy
E0 = −G
m1m2
r0
(1)
The coordinates of the particles, x1 and x2 , are measured from the position of the center of
mass. At any time the total energy is
E=
1
1
mm
m1 x12 + m2 x22 − G 1 2
2
2
r
(2)
and the linear momentum, at any time, is
p = m1 x1 + m2 x2 = 0
(3)
From the conservation of energy we have E = E0 , or
−G
m1m2 1
1
mm
= m1 x12 + m2 x22 − G 1 2
r0
2
2
r
(4)
Using (3) in (4), we find
x1 = v1 = m2
2G  1 1 
 − 
M  r r0 
(5)
x1 = v2 = − m1
2G
M
1 1 
 − 
 r r0 
240
CHAPTER 8
8-7. Since F ( r ) = − kr is a central force, angular momentum is conserved and the areal velocity,
dA dt = 2µ , is trivially constant (see Section 8.3). In order to compute U , we start with
dr
dt =
(1)
2

2
−
−
E
U
µ 
2µ r 2 
and
U=
kr 2
2
(2)
The time average of the potential energy becomes
U =
=
1
τ
τ
∫ U dt
0
2
τ∫
rmax
rmin
kr 3
2
dr
(3)
2

2  2 kr 4
Er
−
−

2
2µ 
µ
Substituting
r2 = x
dr =
1
dx
2r
(4)
(4) becomes
U =
k
2τ
µ
2 ∫
x dx
2
rmax
2
rmin
(5)
2
k
−
+ Ex − x 2
2µ
2
Using the integrals in Eqs. (E.9) and (E.8c), Appendix E,
∫
x dx
ax 2 + bx + c
=
 2ax + b 
b
1
ax 2 + bx + c +
sin −1 

2
a
2a − a
 b − 4 ac 
(6)
(5) becomes
k
U =
2τ
12
µ  E  2
−  
2  k  k


rmax
2 
 E − kr 2  2
k 4
2
−
− r + Er −
sin 

2 12 
2
2µ 
 E2 − k   k

 
µ  

 rmin
−1
But rmax and rmin were originally defined as the roots of
vanishes at both limits of integration. On the other hand,
E−U −
(7)
2
2µ r 2
. Hence, the second term
241
CENTRAL-FORCE MOTION
τ =2∫
dr
rmax
2

2
−
−
E
U
2

µ
2µ r 
rmin
= 2µ
∫
rdr
rmax
(8)
2
k
− r 4 + Er 2 −
2
2µ
rmin
or, using (5),
τ=
µ
2 ∫
dx
2
rmax
2
rmin
−
2
k 2
x + Ex −
2
2µ
µ 2
−1 
=−
  sin 

2 12
  E2 − k  
 
µ  

12
2 k 
E − kr 2
rmax
(9)
rmin
Using (9) to substitute for τ in (7), we have
U =
E
2
(10)
Now,
T =E− U =
E
2
(11)
The virial theorem states:
T =
n+1
U
2
when U = kr n +1
(12)
In our case n = 1, therefore,
T = U =
8-8.
E
2
(13)
The general expression for θ (r) is [see Eq. (8.17)]
θ (r) = ∫
(
)
r 2 dr
2


2µ  E − U −
2 
2µ r 

where U = − ∫ kr dr = − kr 2 2 in the present case. Substituting x = r 2 and dx = 2r dr into (1), we
have
(1)
242
CHAPTER 8
θ (r) =
1
2∫
dx
µk
2µE
x 2 x2 + 2 x − 1
(2)
Using Eq. (E.10b), Appendix E,
∫x
dx
ax + bx + c
2
=
 bx + 2c 
1
sin −1 

−c
 x b 2 − 4 ac 
(3)
and expressing again in terms of r, we find
  µE 2 
  2 r − 1
1


−1 
θ ( r ) = sin
2
2
 2 µ E
2
µk
+ 2
r
4



 + θ0



(4)
or,
1
sin 2 (θ − θ 0 ) =
2
k
1+
µE2
−
1
r2
2
µE
2
k
1+
µE2
(5)
In order to interpret this result, we set

k
≡ ε′ 
2
µE


2

≡ α′ 
µE

(6)
= 1 + ε ′ cos 2θ
(7)
1+
2
and specifying θ 0 = π 4 , (5) becomes
α′
r2
or,
α ′ = r 2 + ε ′r 2 ( cos 2 θ − sin 2 θ )
(8)
Rewriting (8) in x-y coordinates, we find
α ′ = x2 + y2 + ε ′ ( x2 − y2 )
(9)
or,
y2
x2
1=
+
α′
α′
1 + ε′ 1 − ε′
Since a′ > 0, ε′ > 1 from the definition, (10) is equivalent to
(10)
243
CENTRAL-FORCE MOTION
1=
y2
x2
+
α′
α′
1 + ε′ 1− ε′
(11)
which is the equation of a hyperbola.
8-9.
(a) By the virial theorem, T = − U 2 for a circular orbit.
The firing of the rocket doesn’t change U, so U f = U i
But
Tf =
(
)
1
m v 2 + v 2 = 2Ti
2
So
E f = 2Ti + U i = −U i + U i = 0
Ef
=0
Ei
The firing of the rocket doesn’t change the angular momentum since it fires in a radial direction.
f
=1
i
(b) E = 0 means the orbit is parabolic. The satellite will be lost.
E(r) = 0
U (r) = −
GMe ms
r
T (r) = E − U =
V (r) = U (r) +
GMe ms
r
2
2µ r 2
=−
2
GMe ms
+
2µ r 2
r
Behavior of V(r) is determined by
 2 2µ r 2

 − GM m r
e s

for small r
for large r
244
CHAPTER 8
Energy
V(r)
T(r)
E(r) = 0
r0
U(r)
Minimum in V(r) is found by setting
dV
= 0 at r = r0
dr
2
GMe ms
0=−
+ 3
µ r0
r02
r0 = −
8-10.
2
µGMe ms
For circular motion
T=
1
meω 2 re2
2
U=−
GMs me
re
We can get ω 2 by equating the gravitational force to the centripetal force
GMs me
= meω 2 re
re2
or
ω2 =
GMs
re3
So
T=
GM GMs me
1
1
me re2 ⋅ 3 s =
=− U
re
2
2re
2
E=T +U =
If the sun’s mass suddenly goes to
1
U
2
1
its original value, T remains unchanged but U is halved.
2
245
CENTRAL-FORCE MOTION
E′ = T ′ + U ′ = T +
1
1
1
U=− U+ U=0
2
2
2
The energy is 0, so the orbit is a parabola. For a parabolic
orbic, the earth will escape the solar system.
8-11.
For central-force motion the equation of orbit is [Eq. (8.21)]
d2  1  1
µr 2
+ = − 2 F (r)
2  
dθ  r  r
r
a
θ
force
center
(1)
In our case the equation of orbit is
r = 2 a cos θ
(2)
2
1 d2 
−1
 + 1 ( cos θ ) −1 = − 4 a µ F ( r ) cos 2 θ
cos
θ
(
)
2
 2a
2 a dθ 2 
(3)
Therefore, (1) becomes
But we have
d2 
d  sin θ 
−1
cos θ )  =
2 (
 dθ  cos 2 θ 
dθ


=
2 sin 2 θ
1
+
cos θ cos 3 θ
(4)
Therefore, we have
2 sin 2 θ
8 a3 µ
1
1
+
+
=
−
F ( r ) cos 2 θ
3
2
cos θ cos θ cos θ
(5)
or,
F (r) = −
2
2
8 a2
=
−
µ
8 a 3 µ cos 5 θ
2
1
r5
(6)
so that
F (r) = −
k
r5
(7)
246
CHAPTER 8
8-12.
re
βre
The orbit of the comet is a parabola (ε = 1), so that the equation of the orbit is
α
r
= 1 + cos θ
(1)
We choose to measure θ from perihelion; hence
r (θ = 0 ) = β rE
(2)
Therefore,
2
α=
µk
= 2β rE
(3)
Since the total energy is zero (the orbit is parabolic) and the potential energy is U = − k r , the
time spent within the orbit of the Earth is
T = 2∫
=
dr
rE
ΒrE
2µ
k
2

2 k
−

2
µ  r 2µ r 
rE
∫β
rE
r dr
r − β rE
(4)

2µ  2 ( −2β rE − r )
=
r − β rE 
−
3
k 
 β rE
rE
from which
T=
2µ
k
2 3 2

 3 rE ( 2β + 1) 1 − β 
(5)
Now, the period and the radius of the Earth are related by
τ E2 =
or,
4π 2 µE 3
rE
k′
(6)
247
CENTRAL-FORCE MOTION
k′ τE
µE 2π
rE3 2 =
(7)
Substituting (7) into (5), we find
T=
2µ 2
k 3
k′ τE
( 2β + 1) 1 − β
µE 2π
(8)
where k = GMs µ and k ′ = GMs µE . Therefore,
T=
1
3π
2 (1 − β ) (1 + 2β ) τ E
(9)
where τ E = 1 year . Now, β = rMercury rE = 0.387 . Therefore,
T=
1
3π
2 (1 − 0.387 ) (1 + 2 × 0.387 ) × 365 days
so that
T = 76 days
8-13.
(10)
Setting u ≡ 1 r we can write the force as
F=−
k λ
− = − ku 2 − λ u3
r2 r3
(1)
Then, the equation of orbit becomes [cf. Eq. (8.20)]
d2u
µ 1
+ u = − 2 2 − ku2 − λ u3
2
dθ
u
(
)
(2)
from which
d2u
 µλ  µ k
+ u 1 − 2  = 2
2
dθ


(3)
d 2 u  µλ  
µk 1 
+ 1 − 2   u − 2
=0
2
µλ 
dθ


1− 2 



(4)
or,
If we make the change of variable,
v=u−
µk
2
1
1−
µλ
(5)
2
we have
d 2 v  µλ 
+ 1− 2  v = 0
dθ 2 

(6)
248
CHAPTER 8
or,
d2v
+ β 2v = 0
dθ 2
(7)
where β 2 = 1 − µλ 2 . This equation gives different solutions according to the value of λ. Let us
consider the following three cases:
i)
λ<
µ:
2
For this case β 2 > 0 and the solution of (7) is
v = A cos ( βθ − δ )
By proper choice of the position θ = 0, the integration constant δ can be made to equal zero.
Therefore, we can write
1
= A cos βθ +
r
2
µk
− µλ
(9)
When β = 1 (λ = 0), this equation describes a conic section. Since we do not know the value of
the constant A, we need to use what we have learned from Kepler’s problem to describe the
motion. We know that for λ = 0,
1 µk
= 2 (1 + ε cos θ )
r
and that we have an ellipse or circle (0 ≤ ε < 1) when E < 1, a parabola (ε = 1) when E = 0, and a
hyperbola otherwise. It is clear that for this problem, if E ≥ 0, we will have some sort of
parabolic or hyperbolic orbit. An ellipse should result when E < 0, this being the only bound
orbit. When β ≠ 1, the orbit, whatever it is, precesses. This is most easily seen in the case of the
ellipse, where the two turning points do not have an angular separation of π. One may obtain
most constants of integration (in particular A) by using Equation (8.17) as a starting point, a
more formal approach that confirms the statements made here.
ii)
λ=
2
µ
For this case β 2 = 0 and (3) becomes
d2u µk
= 2
dθ 2
(10)
so that
u=
1 µk 2
=
θ + Aθ + B
r 2 2
(11)
from which we see that r continuously decreases as θ increases; that is, the particle spirals in
toward the force center.
249
CENTRAL-FORCE MOTION
iii)
λ>
2
µ
For this case β 2 < 0 and the solution (7) is
v = A cosh
(
−β 2 θ − δ
)
(12)
δ may be set equal to zero by the proper choice of the position at which θ = 0. Then,
1
= A cosh
r
(
)
−β 2 θ +
2
µk
− µλ
(13)
Again, the particle spirals in toward the force center.
8-14.
The orbit equation for the central-force field is [see Eq. (8.17)]
2µ r 4
 dr 
=
2
 dθ 
2
2


E
U
−
−

2
2µ r 

(1)
But we are given the orbit equation:
r = kθ 2
(2)
from which
2
 dr 
2 2
 dθ  = 4 k θ
(3)
Substituting (2) into (3), we have
2
 dr 
2 r
 dθ  = 4 k k = 4 kr
(4)
From (1) and (4), we find the equation for the potential U:
4 kr =
2

2µ r 4 
E
U
−
−

2
2
2µ r 

(5)
from which
U = E−
2
2k
µ
2
1
1
−
3
r
2µ r 2
(6)
and F ( r ) = − ∂U ∂r . Therefore,
F (r) = −
 6k 1 
+
µ  r 4 r 3 
2
(7)
250
CHAPTER 8
8-15.
A
B
m
r
b 2
θ
P
Let us denote by v the velocity of the particle when it is infinitely far from P and traveling along
the line AB. The angular momentum is
=
k
= mvb 2 = vb 2
b
(1)
k
2 b2
(2)
where we have used m = 1. Therefore,
v=
The total energy E of the particle is equal to the initial kinetic energy:
E=
1 2
k
v = 4
2
4b
(3)
The general orbit equation for a force, F ( r ) = − k r 5 , is
dθ =
dr
r
2
2


k
2 E + 4 − 2 
4r
2r 

(4)
Substituting for and E from (1) and (3), we have
dθ =
k 1
b r2
=b 2
=b 2
dr
k
k
k
+ 4− 2 2
4
b r
2b
2r
dr
r − 2b 2 r 2 + b 4
4
dr
(r
2
− b2
)
2
or,
dθ = − b 2
dr
r − b2
2
(5)
where we have taken the negative square root because r decreases as θ increases (see the diagram).
We can now use the integral [see Eq. (E.4b), Appendix E]
251
CENTRAL-FORCE MOTION
dx
1
= − coth −1
2
2
−b
ab
∫a x
2
 ax 
 b 
(6)
from which we obtain
r
θ = 2 coth −1   + θ 0
(7)
θ − θ0 
r = b coth 
 2 
(8)
b
or,
Now, coth φ → ∞ as φ → 0, since r → ∞ as θ → 0, we must have θ 0 = 0 . Thus,
(
r = b coth θ
2
)
(9)
Notice that r is always greater than b (because coth φ → 1 as φ → ∞), so that the denominator in
(5) never equals zero nor changes sign. Thus, r always decreases as θ increases. This is, the
particle spirals in toward P but never approaches closer than a distance b.
8-16.
The total energy of the particle is
E = T +U
(1)
a principle that by no means pushes the philosophical envelope of physical interpretation. The
impulse that causes v → v + δv changes the kinetic energy, not the potential energy. We
therefore have
δ E = δ T = δ  mv 2  = mv δ v


1
2
(2)
By the virial theorem, for a nearly circular orbit we have
1
E = − mv 2
2
(3)
so that
δE
−E
=
2δ v
v
(4)
where we have written –E since E < 0. The major and minor axes of the orbit are given by
a=−
k
2E
b=
−2µE
(5)
Now let us compute the changes in these quantities. For a we have
 k  kδ E
 δE
= 2 = a 

 2E  2E
 −E 
δ a = −δ 
and for b we have
(6)
252
CHAPTER 8


δ
+
=
−2µE
−2µE3
 −2µE 
δb = δ 
Easily enough, we can show that δ
(7)
= δ v v and therefore
δa
a
8-17.
δE 
δ
 1 
 − 2 δ E  = b  − 2E 
=
δb
b
=
δE
−E
=
2δ v
v
(8)
The equation of the orbit is
α
r
= 1 + ε cos θ
(1)
from which
α
r=
(2)
1 + ε cos θ
2E 2
. Therefore, the radial distance r can vary from the
mk 2
maximum value α (1 − ε ) to the minimum value α (1 + ε ) . Now, the angular velocity of the
particle is given by
where α =
2
µ k and ε = 1 +
ω=
(3)
µr 2
Thus, the maximum and minimum values of ω become
ω max =
ω min =
µr
2
min
2
µ rmax
=
=
α 
 1 + ε 
2
α 
 1 − ε 
2
µ 
µ 









(4)
Thus,
ω max  1 + ε 
=
=n
ω min  1 − ε 
2
(5)
from which we find
ε=
n −1
n +1
(6)
8-18. Kepler’s second law states that the areal velocity is constant, and this implies that the
angular momentum L is conserved. If a body is acted upon by a force and if the angular
momentum of the body is not altered, then the force has imparted no torque to the body; thus,
253
CENTRAL-FORCE MOTION
the force must have acted only along the line connecting the force center and the body. That is,
the force is central.
Kepler’s first law states that planets move in elliptical orbits with the sun at one focus. This
means the orbit can be described by Eq. (8.41):
α
r
= 1 + ε cos θ
with 0 < ε < 1
(1)
On the other hand, for central forces, Eq. (8.21) holds:
d2
dθ 2
µr
1 1
 r  + r = − 2 F ( r )
2
(2)
Substituting 1 r from (1) into the left-hand side of (2), we find
1
µ
= − 2 r2 F (r)
a
(3)
which implies, that
F (r) = −
2
(4)
αµ r 2
8-19. The semimajor axis of an orbit is defined as one-half the sum of the two apsidal
distances, rmax and rmin [see Eq. (8.44)], so
1
1 α
α 
α
+
=
rmax + rmin ] = 
[

2
2 1 + ε 1 − ε  1 − ε 2
(1)
This is the same as the semimajor axis defined by Eq. (8.42). Therefore, by using Kepler’s Third
Law, we can find the semimajor axis of Ceres in astronomical units:
 kC

τ C2 
aC  4π 2 µC

=
aE  kE τ 2 
 4π 2 µE E 
(2)
where kc = γ Ms mc , and
1
µc
=
1
1
+
Ms mc
Here, Ms and mc are the masses of the sun and Ceres, respectively. Therefore, (2) becomes
aC  Ms + mc
=
aE  Ms + me

from which
 τc 
 
τE 
2 13



(3)
254
CHAPTER 8
1


333, 480 +

aC 
8, 000
=
( 4.6035) 2 
aE  333, 480 + 1



13
(4)
so that
aC
≅ 2.767
aE
(5)
The period of Jupiter can also be calculated using Kepler’s Third Law:
 4π 2 µ J 3 
aJ 

τ J  kJ

=
τ E  4π 2 µE 3 
aE 

 kE

12
 M + m  aJ  3 
E
= s
  
 Ms + mJ  aE  
12
(6)
from which
12
τ J  333, 480 + 1

=
( 5.2028 ) 3 
τ E  333, 480 + 318.35

(7)
τJ
≅ 11.862
τE
(8)
Therefore,
The mass of Saturn can also be calculated from Kepler’s Third law, with the result
ms
≅ 95.3
me
8-20.
(9)
Using Eqs. (8.42) and (8.41) for a and r, we have
τ
1
 1 + ε cos θ 
 a
cos θ
  cos θ = ∫ dt 
r
τ 0  1 − ε 2 
4
4
(1)
From Kepler’s Second Law, we can find the relation between t and θ:
dt =
τ
τ 1
α2
dA =
dθ
π ab
π ab 2 (1 + ε cos θ ) 2
(2)
since dA = (1 2) r 2 dθ . Therefore, (1) becomes
4
1
1
 a
  cos θ =
r
τ 1− ε2
(
)
4
τ a2
π ab 2
2π
∫ cos θ (1 + ε cos θ )
0
It is easily shown that the value of the integral is 2πε. Therefore,
2
dθ
(3)
255
CENTRAL-FORCE MOTION
4
1
 a
  cos θ =
r
1 − ε2
(
)
4
1 2
α ε
ab
(4)
After substituting a and b in terms of ε and α [see Eqs. (8.42) and (8.43)], we obtain
ε
 a
  cos θ =
r
1 − ε2
4
(
)
52
(5)
8-21. If we denote the total energy and the potential of the family of orbits by E and U(r), we
have the relation
2
1 2
µr +
+ U (r) = E
2
2µ r 2
(1)
1


= 2µ r 2  E − U ( r ) − µ r 2 


2
(2)
from which
2
Here, E and U(r) are same for all orbits, and the different values of result from different values
of (1 2) µ r 2 . For stable circular motion, r = 0 , but for all other motions, r ≠ 0 . Therefore, for noncircular motions, r 2 > 0 and is smaller than for the circular case. That is, the angular
momentum of the circular orbit is the largest among the family.
8-22.
For the given force, F ( r ) = − k r 3 , the potential is
k
2r 2
(1)
 2
 1
 µ − k r2


(2)
U (r) = −
and the effective potential is
V (r) =
1
2
The equation of the orbit is [cf. Eq. (8.20)]
d2u
µ
+ u = − 2 2 − ku3
2
dθ
u
(
)
(3)
or,
d2u  µk 
+ 1− 2 u = 0
dθ 2 

Let us consider the motion for various values of .
(4)
256
i)
CHAPTER 8
= µk :
2
In this case the effective potential V(r) vanishes and the orbit equation is
d2u
=0
dθ 2
(5)
1
= Aθ + B
r
(6)
with the solution
u=
and the particle spirals toward the force center.
ii)
> µk :
2
In this case the effective potential is positive and decreases monotonically with increasing r. For
any value of the total energy E, the particle will approach the force center and will undergo a
reversal of its motion at r = r0 ; the particle will then proceed again to an infinite distance.
V(r)
E
r
r0
Setting 1 − µ k
2
≡ β 2 > 0 , (4) becomes
d2u
+ β 2u = 0
dθ 2
(7)
1
= A cos ( βθ − δ )
r
(8)
with the solution
u=
Since the minimum value of u is zero, this solution corresponds to unbounded motion, as
expected from the form of the effective potential V(r).
iii)
2
< µk :
For this case we set µ k
2
− 1 ≡ G 2 > 0 , and the orbit equation becomes
d2u
− G2u = 0
2
dθ
(9)
with the solution
u=
1
= A cosh ( βθ − δ )
r
so that the particle spirals in toward the force center.
(10)
257
CENTRAL-FORCE MOTION
In order to investigate the stability of a circular orbit in a 1 r 3 force field, we return to Eq. (8.83)
and use g ( r ) = k µ r 3 . Then, we have
x−
2
µ ρ 1 + ( x ρ ) 
2
3
3
=−
k
µρ 1 + ( x ρ ) 
3
3
(11)
or,
2


1
=0
x + k −  ⋅
3
µ  µρ 1 + ( x ρ )  3



Since r r = p = 0 , Eq. (8.87) shows that k =
(12)
µ . Therefore, (12) reduces to
2
x=0
(13)
so that the perturbation x increases uniformly with the time. The circular orbit is therefore not
stable.
We can also reach the same conclusion by examining the basic criterion for stability, namely,
that
∂V
∂r
= 0 and
r=ρ
∂ 2V
∂r 2
>0
r=ρ
The first of these relations requires k = 2 µ while the second requires 2 µ > k . Since these
requirements cannot be met simultaneously, no stable circular orbits are allowed.
8-23.
Start with the equation of the orbit:
α
r
= 1 + ε cos θ
(1)
and take its time derivative
r ε
ε
= θ sin θ =
sin θ
2
r
α
αµ r 2
(2)
Now from Equation (8.45) and (8.43) we have
τ=
2µ
⋅ π ab =
2πµ aα
1− ε2
(3)
so that from (2)
r max =
as desired.
ε
2π aε
⋅ =
µ α τ 1 − ε2
(4)
258
CHAPTER 8
8-24.
(b)
(a)
r
b
ra
θ
rp
a) With the center of the earth as the origin, the equation for the orbit is
α
= 1 + ε cox θ
(1)
rmin = a (1 − ε )
(2)
r
Also we know
rmax = a (1 + ε )
rmin = rp = 300 km + re = 6.67 × 10 6 m
rmax = ra = 3500 km + re = 9.87 × 10 6 m
a=
(
)
1
ra + rp = 8.27 × 106 m
2
Substituting (2) gives ε = 0.193. When θ = 0,
a
= 1+ ε
rmin
which gives α = 7.96 × 106 m . So the equation of the orbit is
7.96 × 10 6 m
= 1 + 0.193 cos θ
r
When θ = 90°,
r = α = 7.96 × 10 6 m
The satellite is 1590 km above the earth.
b)
b
β
a – rmin
θ
259
CENTRAL-FORCE MOTION
θ =π −β
= π − tan −1
b
a − rmin
Using b = α a
θ = π − tan −1
αa
a − rmin
101°
Substituting into (1) gives
r = 8.27 × 10 6 m ; which is
1900 km above the earth
8-25. Let us obtain the major axis a by exploiting its relationship to the total energy. In the
following, let M be the mass of the Earth and m be the mass of the satellite.
E=−
GMm 1
GMm
= mvp2 =
2a
2
rp
(1)
where rp and vp are the radius and velocity of the satellite’s orbit at perigee. We can solve for a
and use it to determine the radius at apogee by
 2GM 
ra = 2 a − rp = rp 
− 1
2
 rp vp

−1
(2)
Inserting the values
we obtain ra
G = 6.67 × 10 −11 N ⋅ m 2 ⋅ kg −2
(3)
M = 5.976 × 10 24 kg
(4)
rp = 6.59 × 106 m
(5)
vp = 7.797 × 10 3 m ⋅ s −1
(6)
1.010 rp = 6.658 × 106 m , or 288 km above the earth’s surface. We may get the
speed at apogee from the conservation of angular momentum,
mra va = mrp vp
(7)
giving va = 27,780 km ⋅ hr −1 . The period can be found from Kepler’s third law
t2 =
4π 2 a 3
GM
Substitution of the value of a found from (1) gives τ = 1.49 hours.
(8)
260
CHAPTER 8
8-26.
r
ra
rp
First, consider a velocity kick ∆v applied along the direction of travel at an arbitrary place in
the orbit. We seek the optimum location to apply the kick.
E1 = initial energy
=
1
GMm
mv 2 −
2
r
E2 = final energy
=
1
GMm
2
m ( v + ∆v ) −
2
r
We seek to maximize the energy gain E2 − E1 :
E2 − E1 =
(
1
m 2v ∆v + ∆ v 2
2
)
For a given ∆v , this quantity is clearly a maximum when v is a maximum; i.e., at perigee.
Now consider a velocity kick ∆V applied at perigee in an arbitrary direction:
∆v
v1
v2
The final energy is
1
GMm
mv22 =
2
rp
This will be a maximum for a maximum v 2 ; which clearly occurs when v 1 and ∆v are along
the same direction.
Thus, the most efficient way to change the energy of
an elliptical orbit (for a single engine thrust) is by
firing along the direction of travel at perigee.
261
CENTRAL-FORCE MOTION
8-27.
By conservation of angular momentum
mra va = mrp vp
va =
or
rp vp
ra
Substituting gives
va = 1608 m/s
8-28. Use the conservation of energy for a spacecraft leaving the surface of the moon with
just enough velocity vesc to reach r = ∞:
Ti + U i = Tf + U f
1
GM m m
2
−
=0+0
mvesc
2
rm
vesc =
2GMm
rm
where
Mm = mass of the moon = 7.36 × 10 22 kg
rm = radius of the moon = 1.74 × 10 6 m
Substituting gives
vesc = 2380 m/s
8-29.
vmax = v + v0 , vmin = v − v0
From conservation of angular momentum we know
mva ra = mvb rb
or
vmax rmin = vmin rmax ;
rmax vmax
=
rmin vmin
(1)
Also we know
rmin = a (1 − e )
(2)
rmax = a (1 + e )
(3)
Dividing (3) by (2) and setting the result equal to (1) gives
262
CHAPTER 8
rmax 1 + e vmax
=
=
rmin 1 − e vmin
vmin (1 + e ) = vmax (1 − e )
e ( vmin + vmax ) = vmax − vmin
e ( 2v ) = 2v0
e=
v0
v
8-30. To just escape from Earth, a velocity kick must be applied such that the total energy E is
zero. Thus
GMe m
1
=0
mv22 −
2
r
where
v2 = velocity after kick
Me = 5.98 × 10 24 kg
G = 6.67 × 10 −11 Nm 2/kg 2
r = 200 km + re
= 200 km + 6.37 × 10 6 m
= 6.57 × 106 m
Substituting into (1) gives v2 = 11.02 km/sec .
For a circular orbit, the initial velocity v1 is given by Eq. (8.51)
v1 =
GMe
= 7.79 km/sec
r
Thus, to escape from the earth, a velocity
kick of 3.23 km/sec must be applied.
(1)
263
CENTRAL-FORCE MOTION
Since E = 0, the trajectory is a parabola.
parabolic
escape
orbit
Earth
circular orbit
8-31.
From the given force, we find
dF ( r )
2k 4 k ′
= F′ (r) = 3 + 5
dr
r
r
(1)
Therefore, the condition of stability becomes [see Eq. (8.93)]
(
2
)
k ρ 2 + 2k ′
F ′ ( ρ) 3 ρ 5
3
+ =
+ >0
1
F ( ρ) ρ −
kρ 2 + k ′ ρ
4
(2)
kρ2 − k ′
>0
ρ kρ2 + k ′
(3)
ρ
(
)
or,
(
)
Therefore, if ρ 2 k > k ′ , the orbit is stable.
8-32.
For this force, we have
dF ( r )
k
2k
= F′ ( r ) = 3 e−r a + 2 e−r a
dr
r
ar
k
= 3 e−r a
r
r

 2 + a 
(1)
Therefore, the condition of stability [see Eq. (8.93)] becomes
F′ (r) 3
+ =
F (r) r
This condition is satisfied if r < a.
r

− 2 +  + 3
a

>0
r
(2)
264
CHAPTER 8
8-33. The Lagrangian of the particle subject to a gravitational force is written in terms of the
cylindrical coordinates as
L = T −U =
(
)
1
m r 2 + r 2θ 2 + z 2 − mgz
2
(1)
From the constraint r 2 = 4 az , we have
rr
2a
(2)
 mg 2
r2 
1 
m  1 + 2  r 2 + r 2θ 2  −
r
2  4a 
 4a
(3)
d
∂L d ∂L
−
=−
mr 2θ = 0
∂θ dt ∂θ
dt
(4)
z=
Therefore, (1) becomes
L=
Lagrange’s equation for θ is
(
)
This equation shows that the angular momentum of the system is constant (as expected):
mr 2θ = = const.
(5)
mg
m
d 
r2  
∂ L d ∂L
r −  m 1 + 2  r  = 0
−
= 2 rr 2 + mrθ 2 −
dt   4 a  
2a
∂r dt ∂r 4 a
(6)
mg

m 2
r2 
m
2
rr
mr
r
m
r − 2 rr 2 = 0
θ
1
+
−
−
+

2
2
4a
2a
2a
 4a 
(7)
Lagrange’s equation for r is
from which
After rearranging, this equation becomes
2

mg
r2 
m
1
m 1 + 2  r + 2 rr 2 +
r−
=0
m r3
4a
2a
 4a 
(8)
For a circular orbit, we must have r = r = 0 or, r = ρ = constant. Then,
2
mg ρ
=
2a
mρ 3
(9)
m2 g 4
ρ
2a
(10)
or,
2
Equating this with
2
=
= m2 ρ 4θ 2 , we have
m2 ρ 4θ 2 =
m2 g 4
ρ
2a
(11)
265
CENTRAL-FORCE MOTION
or,
θ2 =
g
2a
(12)
Applying a perturbation to the circular orbit, we can write
r → ρ + x where
x
ρ
1
(13)
This causes the following changes:
r 2 → ρ 2 + 2ρ x 

x 
1− 3

1
ρ 
→

ρ3
r3


r→x


r→x

(14)
from which, we have
rr 2 → ( ρ + x ) x 2 ≅ 0, in lowest order



2
2
2
r r → ρ + 2ρ x x ≅ ρ x , in lowest order 
(
)
(15)
Thus, (8) becomes
2
mg
1


+
−
ρ
m 1 + 2 ρ 2  x +
x
(
)
mρ 3
2a
 4a


x
 1 − 3 ρ  = 0
(16)
But
2
mg ρ
=
2a
mρ 3
(17)
mg

ρ2 
3 2
m 1 + 2  x +
x+
x=0
2a
mρ 4
 4a 
(18)
so that (16) becomes
Substituting (17) into (18), we find

2mg
ρ2 
m 1 + 2  x +
x=0
a
 4a 
(19)
or,
x+
2g
a+
ρ2
4a
x=0
(20)
266
CHAPTER 8
Therefore, the frequency of small oscillations is
2g
a + z0
ω=
(21)
where
z0 =
8-34.
ρ2
4a
The total energy of the system is
E=
(
)
1
m r 2 + r 2θ 2 + r 2 cot 2 α + mgr cot α
2
(1)
or,
E=
Substituting
(
)
1
1
m 1 + cot 2 α r 2 + mr 2θ 2 + mgr cot α
2
2
(2)
= mr 2θ , we have
E=
2
1
+ mgr cot α
m 1 + cot 2 α r 2 +
2
2mr 2
(
)
(3)
Therefore, the effective potential is
V (r) =
2
2mr 2
+ mgr cot α
(4)
At the turning point we have r = 0 , and (3) becomes a cubic equation in r:
mgr 3 cot α − Er 2 +
2
2m
=0
V(r
Energy
mg
r1
(5)
)
E
ta
r co
r2
r
This cubic equation has three roots. If we attempt to find these roots graphically from the
intersections of E = const. and V ( r ) = 2 2mr 2 + mgr cot α , we discover that only two of the roots
are real. (The third root is imaginary.) These two roots specify the planes between which the
motion takes place.
267
CENTRAL-FORCE MOTION
8-35.
If we write the radial distance r as
r = ρ + x,
ρ = const.
(1)
then x obeys the oscillatory equation [see Eqs. (8.88) and (8.89)]
x + ω 02 x = 0
(2)
where
ω0 =
3g ( ρ )
ρ
+ g ′ ( ρ)
(3)
The time required for the radius vector to go from any maximum value to the succeeding
minimum value is
∆t =
where τ 0 =
2π
ω0
τ0
(4)
2
, the period of x. Thus,
∆t =
π
ω0
(5)
The angle through which the particle moves during this time interval is
φ = ω∆t =
πω
ω0
(6)
where ω is the angular velocity of the orbital motion which we approximate by a circular
motion. Now, under the force F ( r ) = − µ g ( r ) , ω satisfies the equation
µρω 2 = − F ( r ) = µ g ( ρ )
(7)
Substituting (3) and (7) into (6), we find for the apsidal angle
φ=
πω
=
ω0
π
ρ
3 g ( ρ)
ρ
Using g ( r ) =
g ( ρ)
=
+ g ′ ( ρ)
π
g ′ ( ρ)
3+ρ
g ( ρ)
(8)
k 1
, we have
µ rn
g ′ ( ρ)
n
=−
g ( ρ)
ρ
(9)
Therefore, (8) becomes
φ =π
3−n
(10)
268
CHAPTER 8
In order to have the closed orbits, the apsidal angle must be a rational fraction of 2π. Thus, n
must be
n = 2, − 1, − 6, …
n = 2 corresponds to the inverse-square-force and n = –1 corresponds to the harmonic oscillator
force.
8-36.
The radius of a circular orbit in a force field described by
F (r) = −
k −r a
e
r2
(1)
is determined by equating F(r) to the centrifugal force:
2
k −ρ a
=
e
r2
mρ 3
(2)
Hence, the radius ρ of the circular orbit must satisfy the relation
ρe − ρ a =
2
mk
(3)
Since the orbit in which we are interested is almost circular, we write
r (θ ) = ρ [1 + δ (θ ) ]
(4)
where δ (θ ) 1 for all values of θ. (With this description, the apsides correspond to the
maximum and minimum values of δ.)
We can express the following quantities in terms of δ by using (4):
u=
1 1
= (1 − δ )
r ρ
d2  1 
1 d 2δ
=
−
dθ 2  r 
ρ dθ 2
(5a)
(5b)
F ( u) = − ku2 e −1 au
≅−
k e−ρ a
ρ 2 (1 + δ )
2
(1 − ρδ a)
(5c)
Then, substitution into Eq. (8.20) yields
−
1 d 2δ 1
mke − ρ a
1
δ
+
−
=
(
)
(1 − pδ a)
2
ρ dθ 2 ρ
(6)
Multiplying by ρ, using (3) and simplifying, (6) reduces to
d 2δ
+ (1 − ρ a ) δ = 0
dθ 2
(7)
269
CENTRAL-FORCE MOTION
This equation obviously has two types of solution depending on whether ρ a is larger than or
smaller than 1; we consider only ρ < a. (In fact, there is no stable circular orbit for ρ > a.)
For the initial condition, we choose δ = δ 0 to be a maximum (i.e., an apside) at θ = 0. Then, we
have
δ = δ 0 cos (1 − ρ a) θ , for ρ < a
12
(8)
This solution describes an orbit with well-defined apsides. The advance of the apsides can be
found from (8) by computing for what value of θ is δ again a maximum. Thus,
θ=
2π
1− ρ a
(9)
The advance of the apside is given by
∆ = θ − 2π = 2π 1 − (1 − ρ a )

In the particular case in which ρ
−1 2


(10)
a we obtain, by extending (10),
ρ

∆ ≅ 2π − 2π 1 + 
2a


(11)
so that
∆≅
8-37.
πρ
(12)
a
From the equations in Section 8.8 regarding Hohmann transfers:
∆v = ∆v1 + ∆v2
∆v = vt1 − v1 + v2 − vt2
∆v =
2k  r2 
k
k
2k  r1 
+
−

−


mr1  r1 + r2 
mr1
mr2
mr2  r1 + r2 
Substituting
(
)(
k
= GMe = 6.67 × 10 −11 Nm 2/kg 2 5.98 × 10 24 kg
m
r1 = initial height above center of Earth = 2re
r2 = final height above center of Earth = 3re
re = radius of the Earth = 6.37 × 10 6 m
gives
∆v 1020 m/s
)
(1)
270
8-38.
CHAPTER 8
Substitute the following into Eq. (1) of problem 8-37:
(
)(
k
= GMs = 6.67 × 10 −11 Nm 2/kg 2 1.99 × 10 30 kg
m
)
r1 = mean Earth-sun distance 1.50 × 1011 m
r2 = mean Venus-sun distance 1.08 × 1011 m
The result is ∆v = −5275 m/s . The answer is negative because r2 < r1 ; so the rocket must be fired
in the direction opposite to the motion (the satellite must be slowed down).
∆v = 5275 m/s; opposite to direction of motion.
From Eq. (8.58), the time is given by
T =π
m 32
at = π
k
m
k
 r1 + r2 
 2 
32
(1)
Substituting gives
τ
148 days
8-39. We must calculate the quantity ∆v1 for transfers to Venus and Mars. From Eqs. (8.54),
(8.53), and (8.51):
∆v1 = vt1 − v1
=
2k  r2 
k

−
mr1  r1 + r2 
mr1
where
(
)(
k
= GMs = 6.67 × 10 −11 m 3/s 2 -kg 1.99 × 10 30 kg
m
)
r1 = mean Earth-sun distance = 150 × 10 9 m
 Venus 
108 
− sun distance = 
× 109 m
r2 = mean 


Mars
228




Substituting gives
∆vVenus = −2.53 km/sec
∆vMars = 2.92 km/sec
where the negative sign for Venus means the velocity kick is opposite to the Earth’s orbital
motion.
Thus, a Mars flyby requires a larger ∆v than a Venus flyby.
271
CENTRAL-FORCE MOTION
8-40. To crash into the sun, we calculate ∆v1 from Eq. (8.54) with r1 = mean distance from
sun to Earth, and r2 = radius of the sun. Using Eqs. (8.54), (8.53), and (8.51) we have
( ∆v1 )sun =
2GMs  r2 
GMs

−
r1  r1 + r2 
r1
Substituting
G = 6.67 × 10 −11 Nm 2/kg 2
Ms = 1.99 × 10 30 kg
r1 = rse = 1.5 × 1011 m
r2 = rsun = 6.96 × 10 8 m
gives
( ∆v1 )sun = −26.9 km/sec
To escape from the solar system, we must overcome the gravitational pull of both the sun and
Earth. From conservation of energy ( Efinal = 0 ) we have:
−
GMs m GMe m 1
−
+ mv 2 = 0
rse
re
2
Substituting values gives
v = 43500 m/s
Now
∆v = v − v i
=v−
GMs
rse
= ( 43500 − 29700 ) m/s
( ∆v) escape = 13.8 km/s
To send the waste out of the solar system requires
less energy than crashing it into the sun.
8-41.
From the equations in Section 8.8 regarding Hohmann transfers
∆v = ∆v1 + ∆v2
= vt1 − v1 + v2 − vt2
where
272
CHAPTER 8
vt1 =
2k  r2 

 ; v1 =
mr1  r1 + r2 
k
mr1
vt2 =
2k
mr2
 r1 

 ; v1 =
 r1 + r2 
k
mr2
Substituting
(
)(
k
= GMe = 6.67 × 10 −11 Nm 2/kg 2 5.98 × 10 24 kg
m
r1 = 200 km + re = 6.37 × 10 6 m + 2 × 10 5 m
r2 = mean Earth-moon distance = 3.84 × 10 8 m
gives
∆v = 3966 m/s
From Eq. (8.58), the time of transfer is given by
T =τ
m 32
at = π
k
m  r1 + r2 
k  2 
32
Substituting gives
τ = 429, 000 sec. 5 days
8-42.
G = 6.67 × 10 −11 Nm 2/kg 2
Me = 5.98 × 10 24 kg
r1 = 2 × 10 5 m + 6.37 × 10 6 m
r2 = ?
r3 = mean Earth-moon distance = 3.84 × 10 8 m
We can get r2 from Kepler’s Third Law (with τ = 1 day)
 GMe τ 2 
r2 = 

2
 4π 
We know E = − GMm 2r
13
= 4.225 × 107 m
)
273
CENTRAL-FORCE MOTION
So
E ( r1 ) = −
GMe m
= −3.04 × 1011 J
2r1
E ( r2 ) = −4.72 × 1010 J
E ( r3 ) = −5.19 × 10 9 J
To place the satellite in a synchronous orbit would require a minimum energy of E ( r2 ) − E ( r1 ) =
2.57 × 1011 J
8-43.
In a circular orbit, the velocity v0 of satellite is given by
mv02 GMm
GM
=
⇒ v0 =
2
R
R
R
where M is the Earth’s mass.
Conservation of energy implies
mv12 GMm mv22 GMm
−
=
−
2
2
2R
R
Conservation of angular momentum gives
mRv1 = m2Rv2
From these equations, we find
v1 =
so the velocity need to be increased by a factor
8-44.
The bound motion means that E =
where V = −
4GM
3R
4 3 to change the orbit.
mv 2
+V <0
2
k −r a
e .
r
The orbit of particle moving in this central force potential is given by
274
CHAPTER 8
(
r
θ (r) =
∫
rmin
/ r 2 ) dr
2


2µ  E − V
2
2µ r 

r
1
2µ rmin r 2
∫
=
dr
E+
2
ke − ra
−
2µ r 2
r
In first order of (r / a) , this is
θ( r ) ≈
2µ ∫
dr
r2 E +
2
k
k
−
−
2
r 2µ r
a
=
dr
2µ ∫
2
k k

r2  E −  + −

a  r 2µ r 2
k
k

Now effectively, this is the orbit of particle of total energy  E −  moving in potential − . It is


r
a
well known that this orbit is given by (see Chapter 8)
α
r
where α =
2
µk
and
ε = 1+
= 1 + ε cos θ
k

 E − 
a
µk
2
2
2
If 0 < ε < 1 , the orbit is ellipsoid; if ε = 0 , the orbit is circular.
8-45.
a) In equilibrium, for a circular orbit of radius r0,
F0 = mω φ2 r0 ⇒ ω φ =
F0
mr0
b) The angular momentum (which is conserved) of a particle in circular orbit is
L = mr02ω φ = mFr03
The force acting on a particle, which is placed a distance r (r is very close to equilibrium position
ro ) from the center of force is
F = mω φ2 r − F0 =
≈
L3
− F0
mr 3
L3
3L2
3L2
r
r
F
−
−
−
=
−
(
)
( r − r0 ) = − k ( r − r0 )
0
0
mr03 mr02
mr04
275
CENTRAL-FORCE MOTION
where k ≡ 3L2 / mr04 . So the frequency of oscillation is
ωr =
8-46.
3L2
k
=
=
m
m2 r04
3F0
mr0
In equilibrium circular orbit,
Mv 2 GM 2
=
R
4 R2
⇒ R=
GM
4v 2
where M is the Sun’s mass.
The period is
T=
2π R 4π R R
=
=
v
GM
2π D3 2
≈ 9 × 107 yr
GM
where D = 2R is the separation distance of 2 stars.
8-47.
In equilibrium circular orbit of 1st star
M1v12 GM1 M2
=
L1
L2
mass.
where
L1 =
LM2
is the distance from 1st star to the common center of
M1 + M2
The corresponding velocity is
v1 =
GM2 L1
=
L2
GM22
L( M1 + M2 )
Finally, the period is
T=
2π L1
2π L3 2
=
= 1.2 × 10 8 yr.
v1
G( M1 + M2 )
276
CHAPTER 8
CHAPTER
9
Dynamics of a
System of Particles
9-1. Put the shell in the z > 0 region, with the base in the x-y plane. By symmetry, x = y = 0 .
2π
π 2
∫φ ∫θ ∫
z= π π
∫φ ∫θ ∫
=0
2
=0
=0
r2
r = r1
2 r2
ρ zr 2 dr sin θ dθ dφ
ρ r 2 dr sin θ dθ dφ
= 0 r = r1
Using z = r cos θ and doing the integrals gives
z=
(
8(r
3 r24 − r14
3
2
−r
3
1
)
)
9-2.
z
z=−
h
ρ+h
a
h
a
y
x
By symmetry, x = y = 0 .
Use cylindrical coordinates ρ, φ, z.
ρ0 = mass density
277
278
CHAPTER 9
h
− ρ+ h
a
0
φ =0 ρ=0 z=0
h
a
− ρ+h
2π
4
0
φ =0 ρ =0 z =0
2π
z=
a
∫ ∫ ∫
ρ z ρ dρ dφ dz
∫ ∫ ∫
ρ ρ dρ dφ dz
=
h
4
The center of mass is on the axis
3
of the cone h from the vertex.
4
9-3.
z
h
a
y
x
By symmetry, x = y = 0 .
From problem 9-2, the center of mass of the cone is at z =
1
h.
4
From problem 9-1, the center of mass of the hemisphere is at
z=−
3
a ( r2 = a, r1 = 0 )
8
So the problem reduces to
i z1 =
1
1
h ; m1 = ρ1 π a 2 h
4
3
i z2 = −
z=
3
2
a ; m2 = ρ2 π a 3
8
3
m1 z1 + m2 z2 ρ1 h 2 − 3ρ2 a 2
=
m1 + m2
4 ( ρ1 h + 2ρ2 a )
for ρ1 = ρ2
z=
h 2 − 3a3
4 ( 2a + h)
279
DYNAMICS OF A SYSTEM OF PARTICLES
9-4.
y
a
θ/2
θ′
x
θ/2
a
By symmetry, y = 0 .
If σ = mass length then M = σ aθ
So
x=
x=
1
M
1
M
θ 2
∫θ
′ =−
θ 2
∫θ
′ =−
θ
θ
xdm
2
xσ adθ ′
2
Using M = σ aθ and x = a cos θ ′ ,
x=
=
1
θ 2
θ∫ θ
a cos θ ′dθ ′ =
a
θ
− 2
θ
2 sin
θ
a 
 θ
sin − sin  −  

 2 
θ 
2
2
x=
2a
θ
sin
θ
2
y=0
9-5.
mi
ri
ri – r0
r0
280
CHAPTER 9
ri = position of the i th particle
m1 = mass of the i th particle
M = ∑ mi = total mass
g = constant gravitational field
Calculate the torque about r0
τ = ∑τi
= ∑ ( ri − r0 ) × Fi
= ∑ ( ri − r0 ) × mi g
= ∑ r1 × mi g − ∑ r0 × mi g
=
(∑ m r ) × g − (∑ m ) r
=
( ∑ m r ) × g − Mr
i i
i
0
i i
0
×g
×g
Now if the total torque is zero, we must have
∑m r
= Mr0
1
M
∑m r
i i
or
r0 =
i i
which is the definition of the center of mass. So
τ = 0 about r0 = rCM
or center of gravity = center of mass.
9-6.
Since particle 1 has F = 0, r0 = v 0 = 0 , then r1 = 0 . For particle 2
F2 = F0 xˆ
then r =
F0
xˆ
m
Integrating twice with r0 = v 0 = 0 gives
r2 =
rCM =
F0 2
t xˆ
2m
F
m1r1 + m2 r2
= 0 t 2 xˆ
m1 + m2
4m
281
DYNAMICS OF A SYSTEM OF PARTICLES
rCM =
F0 2
t xˆ
4m
v CM =
F0
t xˆ
2m
a CM =
F0
xˆ
2m
9-7.
y
H
a
52˚
O
x
52˚
a
H
By symmetry y = 0
m0 = 16 mH
Let mH = m, m0 = 16 m
Then
x=
x=
1
M
3
∑m
i
xi
1
a cos 52°
1
2ma cos 52°) =
(
18 m
9
x = 0.068 a
9-8. By symmetry, x = 0 . Also, by symmetry, we may integrate over the x > 0 half of the
triangle to get y . σ = mass/area
y=
∫
a
2
x=0
a
2
x=0
∫
∫
a
−x
2
y =0
a
−x
2
y =0
σ y dy dx
∫
y=
σ dy dx
a
3 2
=
a
3 2
282
CHAPTER 9
9-9.
z
vz
m 2 vy
POW!
m1
v1
45˚
y
Let the axes be as shown with the projectile in the y-z plane. At the top just before the explosion,
v
the velocity is in the y direction and has magnitude v0 y = 0 .
2
v0 y =
v0
=
2
2E0
m1 + m2
=
2
E0
m1 + m2
where m1 and m2 are the masses of the fragments. The initial momentum is


E0
, 0
pi = ( m1 + m2 ) 0,
m1 + m2 

The final momentum is
p f = p1 + p2
p1 = m1 ( 0, 0, v1 )
(
p2 = m2 vx , vy , vz
)
The conservation of momentum equations are
px :
py :
pz :
0 = m2 vx
or
vx = 0
E0 ( m1 + m2 ) = m2 vy
or
0 = m1v1 + m2 vz or v1 = −
vy =
1
E0 ( m1 + m2 )
m2
m2
vz
m1
The energy equation is
(
E0
1
1
1
+ E0 = m1v12 + m2 vy2 + vz2
( m1 + m2 )
2
m1 + m2
2
2
or
(
3E0 = m1v12 + m2 vy2 + vz2
Substituting for vy and v1 gives
)
)
283
DYNAMICS OF A SYSTEM OF PARTICLES
E0 m1 ( 2m2 − m1 )
m22 ( m1 + m2 )
vz =
v1 = −
m2
vz gives
m1
E0 ( 2m2 − m1 )
m1 ( m1 + m2 )
v1 = −
So
m1 travels straight down with speed = v1
m2 travels in the y-z plane
(
v2 = vy2 + vz2
θ = tan −1
)
12
=
E0 ( 4 m1 + m2 )
m2 ( m1 + m2 )
vz
= tan −1
vy
m1 ( 2m2 − m1 )
( m1 + m2 )
The mass m1 is the largest it can be when v1 = 0 , meaning 2m2 = m1 and the mass ratio is
m1 1
=
m2 2
9-10.
y
B
x
θ
A
x1 x2
First, we find the time required to go from A to B by examining the motion. The equation for the
y-component of velocity is
vy = v0 sin θ − gt
(1)
At B, vy = 0 ; thus tB = v0 sin θ g . The shell explodes giving m1 and m2 horizontal velocities v1
and v2 (in the original direction). We solve for v1 and v2 using conservation of momentum and
energy.
px :
( m1 + m2 ) v0 cos θ = m1v1 + m2v2
(2)
E:
1
1
1
( m1 + m2 ) v02 cos2 θ + E = m1v12 + m2v22
2
2
2
(3)
Solving for v2 in (2) and substituting into (3) gives an equation quadratic in v1 . The solution is
284
CHAPTER 9
v1 = v0 cos θ ±
2 m2 E
m1 ( m1 + m2 )
(4)
v2 = v0 cos θ ∓
2 m1E
m2 ( m1 + m2 )
(5)
and therefore we also must have
Now we need the positions where m1 and m2 land. The time to fall to the ocean is the same as
the time it took to go from A to B. Calling the location where the shell explodes x = 0 gives for
the positions of m1 and m2 upon landing:
x1 = v1tB ;
x2 = v2tB
(6)
x1 − x2 =
v0 sin θ
v1 − v2
g
(7)
Thus
Using (4) and (5) and simplifying gives
x1 − x2 =
9-11.
m2 
2E  m1
+


m1 + m2  m2
m1 
v0 sin θ
g
(8)
The term in question is
∑∑ fαβ
α β
α ≠b
For n = 3, this becomes
(
) (
) (
f12 + f13 + f 21 + f 23 + f 31 + f 32 = f12 + f 21 + f13 + f 31 + f 23 + f 32
But by Eq. (9.1), each quantity in parentheses is zero. Thus
3
3
∑∑ fαβ = 0
α =1 β =1
α ≠β
9-12.
a)
v = v0 + u ln
m0
m
Assuming v0 = 0 , we have
m  100

v = 100  ln
s
98

v = 2.02 m/s; yes, he runs out of gas.
)
285
DYNAMICS OF A SYSTEM OF PARTICLES
b) Relative to Stumblebum’s original frame of reference we have:
Before throwing tank
98 kg
→ 2.02 m/s
After throwing the tank we want Stumblebum’s velocity to be slightly greater than 3 m/s (so
that he will catch up to the orbiter).
8 kg
90 kg
V←
→ 3m/s
Conservation of momentum gives
( 98 kg ) ( 2.02 m/s) = ( 90 kg ) ( 3 m/s) − ( 8 kg ) v
v = 9 m/s
(This velocity is relative to Stumblebum’s original reference frame; i.e., before he fires his
pressurized tank.) Since Stumblebum is traveling towards the orbiter at 2.02 m/s, he must
throw the tank at v = 9 m/s + 2.02 m/s
v = 11 m/s
9-13.
From Eq. (9.9), the total force is given by
∑ Fα
(e)
α
+ ∑∑ f αβ
α β
α ≠β
As shown in Section 9.3, the second term is zero. So the total force is
∑ Fα
(e)
α
It is given that this quantity is zero.
Now consider two coordinate systems with origins at 0 and 0′
mα
rα′
rα
r0
O′
O
where
r 0 is a vector from 0 to 0′
rα is the position vector of mα in 0
286
CHAPTER 9
rα′ is the position vector of mα in 0′
We see that ra = r0 + rα′
The torque in 0 is given by
τ = ∑ rα × Fα( e )
α
The torque in 0′ is
τ ′ = ∑ rα′ × Fα( e )
α
(
)
= ∑ rα − r 0 × Fα( e )
α
= ∑ rα × Fα( e ) − ∑ r 0 × Fα( e )
α
α
= τ − r 0 × ∑ Fα( e )
α
But it is given that
∑ Fα
(e)
=0
α
Thus
τ = τ′
9-14. Neither Eq. (9.11) or Eq. (9.31) is valid for a system of particles interacting by magnetic
forces. The derivations leading to both of these equations assumes the weak statement of
Newton’s Third Law [Eq. (9.31) assumes the strong statement of the Third Law also], which is
fαβ = − fβα
That this is not valid for a system with magnetic interactions can be seen by considering two
particles of charge q1 and q2 moving with velocities v1 and v2 :
v2
f12
v1
f21
q2
q1
Now
f ij = qi v i × Bij
where Bij is the magnetic field at qi due to the motion of q j .
287
DYNAMICS OF A SYSTEM OF PARTICLES
Since f ij is perpendicular to both v i and Bij (which is either in or out of the paper), f ij can only
be parallel to f ji if v i and v j are parallel, which is not true in general.
Thus, equations (9.11) and (9.31) are not valid for
a system of particles with magnetic interactions.
σ = mass/length
9-15.
F=
dp
becomes
dt
mg = mv + mv
where m is the mass of length x of the rope. So
m = σ x; m = σ x
σ x g =σ x
xg=x
dv
+σ x v
dt
dv dx
+ v2
dx dt
x g = xv
dv
+ v2
dx
Try a power law solution:
v = ax n ;
dv
= nax n −1
dx
Substituting,
( )(
)
x g = x ax n nax n −1 + a 2 x 2 n
or
x g = a 2 ( n + 1) x 2 n
Since this must be true for all x, the exponent and coefficient of x must be the same on both sides
of the equation.
Thus we have: 1 = 2n or n =
1
2
g = a 2 ( n + 1) or a =
So
v=
2 gx
3
2g
3
288
CHAPTER 9
dv dv dx
dv  2 gx 
=
=v
=
a=
dt dx dt
dx  3 
Ui = 0
Tf =
−1 2
( y = 0 on table)
1
1  2 gL  mgL
mv 2 = m 
=
2
2  3 
3
U f = mgh = − mg
So Ei = 0; E f = −
g  2 gx 
3  3 
g
3
a=
Ti = 0
12
L
2
mgL
6
Energy lost =
mgL
6
9-16.
T
x
T1
T2
The equation of motion for the falling side of the chain is, from the figure,
ρ (b − x)
2
x=
ρ (b − x)
2
g + T2
(1)
From Example 9.2, we have for the energy conservation case
x=g−
(
g 2bx − x 2
2 (b − x)
2
) =g+
x2
2 (b − x)
(2)
Substitution gives us
T2 =
ρx 2
4
(3)
To find the tension on the other side of the bend, change to a moving coordinate system in
which the bend is instantaneously at rest. This frame moves downward at a speed u = x 2 with
respect to the fixed frame. The change in momentum at the bend is
289
DYNAMICS OF A SYSTEM OF PARTICLES
∆p = ( ρ∆x ) ⋅ ( 2u) = 2ρu2 ∆t =
ρx 2
2
∆t
(4)
Equating this with the net force gives
T1 + T2 =
ρx 2
(5)
2
Using equation (3), we obtain
T1 =
ρx 2
(6)
4
as required. Note that equation (5) holds true for both the free fall and energy conservation
cases.
9-17.
As the problem states, we need to perform the following integral
12
τ=
∫
ε
1 − 2α
dα
2α (1 − α )
(1)
Our choice of ε is 10 −4 for this calculation, and the results are shown in the figure. We plot the
natural velocity dα d τ = x 2 gb vs. the natural time τ.
2
1.5
dα
dτ
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
τ
9-18. Once we have solved Problem 9-17, it becomes an easy matter to write the expression
for the tension (Equation 9.18):
T
1 + 2α − 6α 2
=
mg
2 (1 − 2α )
This is plotted vs. the natural time using the solution of Problem 9-17.
(1)
290
CHAPTER 9
20
15
T
mg
10
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
τ
9-19.
ceiling
x
released at
t=0
b
at time t
table
The force that the tabletop exerts on the chain counteracts the force due to gravity, so that we
may write the change in momentum of the center of the mass of the chain as
dp
= ρbg − F
dt
(1)
We can write out what the momentum is, though:
p = ρ (b − x) x
(2)
dp
= ρ  − x 2 + ( b − x ) x  = ρ ( bg − 3 gx )
dt
(3)
which has a time derivative
where we have used x = g and x = 2 gx . Setting this last expression equal to (1) gives us
F = 3ρ gx
(4)
Although M. G. Calkin (personal communication) has found that experimentally the time of fall
for this problem is consistently less than the value one would obtain in the above treatment by
about 1.5%, he also finds evidence that suggests the free fall treatment is more valid if the table
is energy absorbing.
291
DYNAMICS OF A SYSTEM OF PARTICLES
9-20.
a–x
a+x
Let ρ = mass/length
The force on the rope is due to gravity
F = ( a + x) ρ g − ( a − x) ρ g
= 2 xρ g
dp
dv
dv
=m
= 2 aρ
dt
dt
dt
So F =
dp
becomes
dt
xg = a
dv
dt
Now
dv dv dx
dv
=
⋅
=v
dt dx dt
dx
So
xg = av
dv
dx
or
vdv =
g
xdx
a
Integrating yields
1 2 g 2
v =
x +c
2
2a
Since v = 0 when x = 0, c = 0.
Thus
v2 =
g 2
x
a
When the rope clears the nail, x = a. Thus
v = ga
292
CHAPTER 9
9-21. Let us call x the length of rope hanging over the edge of the table, and L the total
length of the rope. The equation of motion is
mgx
gx
dx 2
=m 2 ⇒ x=
L
dt
L
Let us look for solution of the form
x = Aeωt + Be −ωt
Putting this into equation of motion, we find
ω=
g
L
Initial conditions are x(t = 0) = x0 = 0.3 m ; v(t = 0) = 0 m/s .
From these we find A = B = x0 2 .
Finally x = x0 cosh (ω t) . When x = L , the corresponding time is
t=
9-22.
 L
cosh −1   = 0.59 s.
ω
 x0 
1
Let us denote (see figure)
m and 2m
mass of neutron and deuteron respectively
v0
velocity of deuteron before collision
v1 and v2
velocity of neutron and deuteron, respectively after collision
ν2
ν0
2m
2m
m
ψ
α
ν1
a) Conservation of energy:
2mv02 mv12 2mv22
v2 v2
=
+
⇒ v02 = 1 + 2
2
2
2
2
2
Conservation of momentum is
2mv0 = 2mv2 + v1
⇒ 4v22 + 4v02 = v12 + 8v0 v2 cos ψ
Solving these equations, we obtain 2 sets of solutions
v1 =
2v0
6 − 4 cos 2 ψ + 2 cos ψ 4 cos 2 ψ − 3
3
v2 =
2v0 cos ψ + 4v02 cos 2 ψ − 3v02
3
293
DYNAMICS OF A SYSTEM OF PARTICLES
or numerically
v1 = 5.18 km/s
v2 = 14.44 km/s and
v1 = 19.79 km/s
v2 = 5.12 km/s
b) Let us call α the lab scattering angle of the neutron, then from the sine theorem we have
mv1
2mv2
=
sin ψ sin α
⇒ sin α = 2
v2
sin ψ
v1
⇒ α = 74.84° and α = 5.16°
c) From a) we see that cos ψ =
2v02 + 6v22
3
≥
8 v0 v 2
2
=
9-23.
4v02 + 4v22 − v12
8v0 v2
⇒
ψ ≤ 300 ⇒ ψ max = 30°
Conservation of momentum requires v f to be in the same direction as u1 (component
of v f ⊥ to u1 must be zero).
pi = m1u1
p f = ( m1 + m2 ) v f
pi = p f → v f =
m1
u1
m1 + m2
The fraction of original kinetic energy lost is
m12 u12
1
1
m1u12 − ( m1 + m2 )
2
2
Ki − K f 2
m1 + m2 )
(
=
1
Ki
m1u12
2
=
=
m1 −
m12
m1 + m2
m1
m2
m1 + m2
294
CHAPTER 9
ω
x
v
9-24.
O
aθ
θ
v0
ω0
b
m
The energy of the system is, of course, conserved, and so we have the following relation
involving the instantaneous velocity of the particle:
1
1
mv 2 = mv02
2
2
(1)
The angular momentum about the center of the cylinder is not conserved since the tension in
the string causes a torque. Note that although the velocity of the particle has both radial and
angular components, there is only one independent variable, which we chose to be θ. Here
ω = θ is the angular velocity of the particle about the point of contact, which also happens to be
the rate at which the point of contact is rotating about the center of the cylinder. Hence we may
write
v = ω 0 ( b − aθ )
v0 = ω 0 b ;
(2)
From (1) and (2), we can solve for the angular velocity after turning through an angle θ
ω=
ω0
a
1− θ
b
(3)
The tension will then be (look at the point of contact)
T = mω 2 ( b − aθ ) = mω 0ω b
(4)
9-25. The best elements are those that will slow down the neutrons as much as possible. In a
collision between m1 (the neutron) and m2 (moderator atom), we would thus want to minimize
T1 (kinetic energy of the neutron after the collision); or alternatively, maximize T2 (kinetic
energy of the moderator atom after the collision). From Eq. (9.88)
T2
4 m1 m2
cos 2 ζ
=
T0 ( m1 + m2 ) 2
Since one cannot control the angle ζ, we want to maximize the function
f=
m1m2
( m1 + m2 ) 2
295
DYNAMICS OF A SYSTEM OF PARTICLES
with respect to m2 . ( m1 = constant)
(
)
m1 m12 − m22
df
=
= 0 when m1 = m2
dm2
( m1 + m2 ) 4
By evaluating
d2 f 
one can show that the equilibrium point is a maximum. Thus, T2 is a

dm22  m = m
1
2
maximum for m1 = m2 . Back to reactors, one would want elements whose mass is as close as
possible to the neutron mass (thus, as light as possible). Naturally, there are many other factors
to consider besides mass, but in general, the lower the mass of the moderator, the more energy
is lost per collision by the neutrons.
9-26.
The internal torque for the system is
N = r1 × f 12 + r 2 × f 21
(1)
where f12 is the force acting on the first particle due to the second particle. Now
f 21= −f12
(2)
Then,
(
)
N = r1 − r 2 × f12


r
r 2 − r1 
= k r1 − r 2 ×  r 2 − r1 −
v0


(
=
) (
)
) (
)
(
kr
r1 − r 2 × r1 − r 2
v0
(
)
(
(
)
(3)
)
This is not zero in general because r1 − r 2 and r1 − r 2 are not necessarily parallel. The
internal torque vanishes only if the internal force is directed along the line joining two particles.
The system is not conservative.
9-27.
The equation for conservation of py in the lab system is (see fig. 9-10c):
0 = m1v1 sin ψ − m2 v2 sin ζ
Thus
sin ζ =
m1v1
sin ψ
m2 v2
or
sin ζ =
m1T1
sin ψ
m2T2
296
9-28.
CHAPTER 9
Using the notation from the chapter:
m1 : Ti = T0 , T f = T1
m2 : Ti = 0; Tf = T2
Thus
T0 = T1 + T2
or 1 =
T1 T2
+
T0 T0
If we want the kinetic energy loss for m1 to be a maximum, we must minimize
equivalently, maximize
(1)
T1
or,
T0
T2
.
T0
From Eq. (9.88):
T2
4 m1 m2
=
cos 2 ζ
T0 ( m1 + m2 ) 2
To maximize this, ζ = 0 (it can’t = 180°).
T2
4 m1 m2
=
T0 ( m1 + m2 ) 2
The kinetic energy loss for m1 is T0 − T1 . The fraction of kinetic energy loss is thus
T0 − T1
T T
= 1 − 1 = 2 (from (1))
T0
T0 T0
T0 − T1
T0
=
max
4 m1 m2
( m1 + m2 )2
ζ = 0 implies ψ = 0, 180° (conservation of pv ). So the reaction is as follows
v1
Before:
m1
m2
v1
After:
m1
v2
m2
px : m1v = m1v1 + m2 v2
E:
Solving for v1 gives v1 =
1
1
1
m1v 2 = m1v12 + m2 v22
2
2
2
m1 − m2
v
m1 + m2
So
m2 travels in + x direction
297
DYNAMICS OF A SYSTEM OF PARTICLES
 + x direction if m1 > m2
m1 travels in 
 − x direction if m1 < m2
9-29.
From Eq. (9.69)
tan ψ =
sin θ
cos θ + ( m1 m2 )
From Eq. (9.74)
θ = π − 2ζ
Substituting gives
tan ψ =
sin ( π − 2ζ )
( m1 m2 ) + cos (π − 2ζ )
or
tan ψ =
sin ( 2ζ )
( m1 m2 ) − cos ( 2ζ )
9-30.
v1
Before:
m1
v1
After:
m1
a)
m2
v2
m2
∆py = ( 0.06 kg ) 16 m/s cos 15° − ( −8 m/s ) cos 45° 
= 1.27 N ⋅ sec
∆px = ( 0.06 kg ) 16 m/s cos 15° ( −8 m/s ) sin 45°
= −0.09 N ⋅ sec
The impulse P is the change in momentum.
So
P = ( −0.09x + 1.27 y ) N ⋅ sec
b)
P = ∫ F dt = F ∆t
So
F = − ( 9x + 127 y ) N
298
9-31.
CHAPTER 9
From Eq. (9.69)
tan ψ =
sin θ
m1
− cos θ
m2
From Eq. (9.74)
θ =π −φ
Substituting gives
tan ψ =
sin φ
m1
− cos φ
m2
pi = mu1
9-32.
p f = mv1 + 2mv2
Conservation of momentum gives
u1 = v1 + 2v2
∆T =
=
or v1 = u1 − 2v2
1
1
mu12 − mv12 − mv22
2
2
(
)
1
1
mu12 − m u12 − 4u1v2 + 4v22 − mv22
2
2
= 2mu1v2 − 3mv22
d ( ∆T )
= 0 implies 2u1 = 6v2
dv2
or v2 =
 d 2 ( ∆T )

< 0, so this is a maximum 

2
 dv2

v1 = u1 − 2v2 =
v1 = v2 =
u1
3
u1
3
u1
3
299
DYNAMICS OF A SYSTEM OF PARTICLES
9-33.
From Eq. (9.87b) in the text, we have
2
1
T1
m
=
T0 ( m1 + m2 ) 2
2


 
 cos ψ +  m2  − sin 2 ψ 


 m1 


2
2
2


 m2 
 m2 
2
2
2


cos
ψ
sin
ψ
2
cos
ψ
sin
ψ
=
+
−
+
−
 
 
2


m
m
 m2 
 1
 1

1 + m  
1

1
Substituting m2 m1 ≡ α and cos ψ ≡ y we have
T1
−2
= (1 + α )  2 y 2 + α 2 − 1 + 2 y α 2 + y 2 − 1 


T0
1
(1)
α = 12
T1
T0
α=4
0
α=1
α=2
π/2
ψ
π
9-34.
Before
After
v1
45˚
u1
m
m
m
m
x
θ
v2
Cons. of pz :
mu1 = mv1 cos 45° + mv2 cos θ
(1)
Cons. of py :
0 = mv1 sin 45° − mv2 sin θ
(2)
Cons. of energy (elastic collision)
1
1
1
mu12 = mv12 − mv22
2
2
2
Solve (1) for cos θ :
cos θ =
u1 − v1
v2
Solve (2) for sin θ :
sin θ =
v1
2 v2
2
(3)
300
CHAPTER 9
Substitute into cos 2 θ + sin 2 θ = 1 , simplify, and the result is
u12 = v22 − v12 + 2 u1v1
Combining this with (3) gives
2v12 = 2 u1v1
We are told v1 ≠ 0 , hence
v1 = u1
2
v2 = u1
2
Substitute into (3) and the result is
Since v1 = v2 , (2) implies
0 = 45°
9-35.
From the following two expressions for T1 T0 ,
T1 v12
=
T0 u12
2
1
T1
m
=
T ( m1 + m2 ) 2
Eq. (9.82)
2

 
cos ψ ±  m2  − sin 2 ψ

 m1 





2
Eq. (9.87b)
we can find the expression for the final velocity v1 of m1 in the lab system in terms of the
scattering angle ψ :
v1 =
2

m 
m1u1 
cos ψ ±  2  − sin 2 ψ
m1 + m2 
 m1 





(1)
If time is to be constant on a certain surface that is a distance r from the point of collision, we
have
r = v1t0
(2)
Thus,
2

 m2 
m1u1t0 
cos ψ ±   − sin 2 ψ
r=
m1 + m2 
 m1 





This is the equation of the required surface. Let us consider the following cases:
(3)
301
DYNAMICS OF A SYSTEM OF PARTICLES
i)
m2 = m1 :
r=
u1t0 
cos ψ ± 1 − sin 2 ψ  = u1t0 cos ψ


2
(4)
(The possibility r = 0 is uninteresting.)
iI)
m2 = 2m1 :
r=
iII)
u1t0 
cos ψ ± 4 − sin 2 ψ 


3
(5)
m2 = ∞ : Rewriting (3) as
mut
r= 1 10
m
1+ 1
m2
2
 cos ψ
 1  sin 2 ψ

±   −
 m2
m22
 m1 





(6)
and taking the limit m2 → ∞ , we find
r = u1t0
(7)
All three cases yield spherical surfaces, but with the centers displaced:
m2 = ∞
m2 = 2m1
m1 = m2
m1
–v1t0
−
v1t0
3
O
m2
v1t0
This result is useful in the design of a certain type of nuclear detector. If a hydrogenous material
is placed at 0 then for neutrons incident on the material, we have the case m1 = m2 . Therefore,
neutrons scattered from the hydrogenous target will arrive on the surface A with the same time
delay between scattering and arrival, independent of the scattering angle. Therefore, a
coincidence experiment in which the time delay is measured can determine the energies of the
incident neutrons. Since the entire surface A can be used, a very efficient detector can be
constructed.
9-36.
Since the initial kinetic energies of the two particles are equal, we have
1
1
1
m1u12 = m2 u22 = α 2 m2 u12
2
2
2
(1)
m1
= α2
m2
(2)
or,
302
CHAPTER 9
Now, the kinetic energy of the system is conserved because the collision is elastic. Therefore,
1
1
1
m1u12 = m2 u22 = m1u12 = m2 v22
2
2
2
(3)
since v1 = 0 . Momentum is also conserved, so we can write
m1u1 + m2 u2 = ( m1 + α m2 ) u1 = m2 v2
(4)
Substituting the second equality in (4) into (3), we find
2
 m + α m2  2
1
m u = m2  1
 u1
2
 m2

2
1 1
(5)
or,
1
m1 = m2
2
 m1

+ α

 m2

2
(6)
Using m1 m2 = α 2 , (6) becomes
(
2α 2 = α 2 + α
)
2
(7)
solving for α, we obtain
α = −1 ∓ 2;
α2 = 3 ∓ 2 2
(8)
u2
 + :α < 0
= − 1 ± 2 with 
u1
 − :α > 0
(9)
This gives us
m1
= 3 ± 2 2;
m2
9-37.
(
)
Impulse = ∫ F dt
=∫
6 × 10 −3
t=0
( 360 − 10 t ) dt
7
2
3

107
=  360 ⋅ 6 × 10 −3 −
⋅ 6 × 10 −3  N ⋅ s
3


(
Impulse = 1.44
kg m
s
Since the initial velocity is zero, v f = ∆v
Impulse = ∆P = m∆v
So
)
303
DYNAMICS OF A SYSTEM OF PARTICLES
kg m
s
vf =
0.003 kg
1.44
vmuzzle = 480
m
s
9-38.
v1
v1′
θ
ψ
θ−ψ
V
1

m1u12 
2


1
T1 = m1v12 
2

(1)
T1 v12
=
T0 u12
(2)
v1 = V cos ψ + v1′ cos (θ − ψ )
(3)
T0 =
Thus,
Now, from the diagram above, we have
Using Eq. (9.68) in the text, this becomes


m
v1 = V cos ψ + 2 cos (θ − ψ ) 
m1


(4)
Thus,

v12 V 2 
m
= 2 cos ψ + 2 (θ − ψ ) 
2
u1 u1 
m1

2
(5)
Using Eq. (9.84) in the text,
V
m1
=
u1 m1 + m2
(6)
Therefore, we find
T1
m12
=
T0 ( m1 + m2 ) 2
If we define

cos (θ − ψ ) 
 cos ψ +
( m1 m2 ) 

2
(7)
304
CHAPTER 9
S ≡ cos ψ +
cos (θ − ψ )
( m1 m2 )
(8)
we have
T1
m12
=
× S2
T0 ( m1 + m2 ) 2
(9)
as desired.
9-39.
y
θ
x
θ′
u
v
As explained in Section 9.8, the component of velocity parallel to the wall is unchanged. So
vx = u sin θ
vy is given by
ε=
vy
uy
=
vy
u cos θ
or
vy = ε u cos θ
Thus
v =  u2 sin 2 θ + ε 2 u2 cos 2 θ 
v = u sin 2 θ + ε 2 cos 2 θ 
tan θ ′ =
u sin θ
ε u cos θ
or
θ ′ = tan −1  tan θ 
ε

1
12
12
305
DYNAMICS OF A SYSTEM OF PARTICLES
9-40. Because of the string, m2 is constrained to move in a circle of radius a. Thus, initially,
m2 will move straight up (taken to be the y direction). Newton’s rule applies to the velocity
component along u1 . The perpendicular component of velocity (which is zero) is unchanged.
Thus m1 will move in the original direction after the collision.
From conservation of py we have
m1u1 sin α = m1v1 sin α + m2 v2
(1)
From Newton’s rule we have
ε=
v2 cos ( 90° − α ) − v1
u1
or
v1 = v2 sin α − ε u1
(2)
Substituting (2) into (1) and solving for v2 gives
v2 =
( ε + 1) m1u1 sin α
straight up
m1 sin 2 α + m2
(2) then gives
v1 =
9-41.
Using y = v0 t −
(
u1 m1 sin 2 α − ε m2
)
m1 sin α + m2
2
along u1
1 2
gt and v = v0 − gt , we can get the velocities before and after the
2
collision:
Before:
u1 = − gt1
where h1 =
u1 = − g
So
After:
0 = v0 − gt2
h2 = v0 t2 −
=
So
1 2
gt1
2
2 h1
= − 2 gh1
g
or t2 = v0 g
1 2
gt2
2
v02 1 v02
−
g 2 g
or v0 = 2 gh2
v1 = 2 gh2
306
CHAPTER 9
Thus
v2 − v1
=
u2 − u1
ε=
2 gh2
2 gh1
h2
h1
ε=
Tlost = Ti − Tf
Fraction lost =
=
Ti − T f
Ti
u12 − v12 h1 − h2
h
=
= 1− 2
2
u1
h1
h1
Ti − Tf
= 1− ε2
Ti
9-42.
y
θ
30˚
x
5 m/s
As explained in Section 9.8, the velocity component in the y-direction is unchanged.
 m
vy = uy =  5  sin 30° = 2.5 m/s
 s
For the x component we have
0.8 =
vx
ux
=
vx
vx
=
 m
3 m
5
 5 s  cos 30°
2
s
vx = 2 3
vf =
1
73
2
θ = tan −1
m
s
4.3 m/s
2.5
2 3
36°
307
DYNAMICS OF A SYSTEM OF PARTICLES
9-43.
v2
4m
α
T0
m
45˚
4m
m
v1
Conservation of px gives
2mT0 = 4 mv2 cos α +
1
mv1
2
or
cos α =
2mT0 −
1
mv1
2
4 mv2
Conservation of py gives
1
mv1 − 4 mv2 sin α
2
0=
or
sin α =
v1
4 2 v2
Substituting into sin 2 α + cos 2 α = 1 gives
v2
1= 1 2 +
32v2
2mT0 +
1 2 2
m v1 − 2 mT0 mv1
2
16 m2 v22
Simplifying gives
v22 =
v12 T0 v1 T0 m
+
−
16 8 m
8m
(1)
The equation for conservation of energy is
T0 −
T0 1
1
= mv12 + ( 4 m) v22
6 2
2
or
5T0 = 3mv12 + 12mv22
Substituting (1) into (2) gives a quadratic in v1 :
15mv12 − 6 T0 m v1 − 14T0 = 0
(2)
308
CHAPTER 9
Using the quadratic formula (taking the positive sign since v1 > 0 ) gives:
v1 = 1.19
T0
m
Substituting this into the previous expressions for cos α and sin α and dividing gives
tan α =
sin α
= 1.47
cos α
Thus α , the recoil angle of the helium, is 55.8°.
9-44.
v0
x
Fgrav = mg = µ xg where µ = mass/length
Fimpulse = mv + mv + mv0 , since v = v0 , v = 0.
We have
m=
d
dx
( µ x ) = µ = µv0
dt
dt
So the total force is
F ( x ) = µ xg + µv02
We want F(x = a)
F ( a ) = µ ag + µv02
or
 v2 
F = µ ag 1 + 0 
 ag 
9-45. Since the total number of particles scattered into a unit solid angle must be the same in
the lab system as in the CM system [cf. Eq. (9.124) in the text],
σ (θ ) 2π sin θ dθ = σ (ψ ) ⋅ 2π sin ψ dψ
(1)
Thus,
σ (θ ) = σ (ψ )
sin ψ dψ
sin θ dθ
(2)
309
DYNAMICS OF A SYSTEM OF PARTICLES
The relation between θ and ψ is given by Eq. (9.69), which is
sin θ
cos θ + x
tan ψ =
(3)
where x = m1 m2 . Using this relation, we can eliminate ψ from (2):
sin ψ =
1
=
1
1+
tan 2 ψ
1
( cos θ + x ) 2
1+
=
sin θ
1 + 2 x cos θ + x 2
sin 2 θ
d ( tan ψ )
cos θ ( cos θ + x ) + sin 2 θ
dψ
dψ
2
=
= cos ψ
dθ d ( tan ψ )
dθ
( cos θ + x )2
Since cos 2 ψ =
(4)
(5)
1
, (5) becomes
1 + tan 2 ψ
dψ
=
dθ
1 + x cos θ
1 + x cos θ
1
=
2
2
sin θ
( cos θ + x ) 1 + 2x cos θ + x 2
1+
( cos θ + x ) 2
(6)
Substituting (4) and (6) into (2), we find
σ (θ ) = σ (ψ )
1 + x cos θ
(1 + 2x cos θ + x )
2 32
(7)
9-46. The change in angle for a particle of mass µ moving in a central-force field is [cf. Eq.
(9.121)]. Let ψ = capital θ here.
∆ψ = ∫
(
rmax
rmin
(
)
r 2 dr
2µ E − U −
θ
2
2µ r 2
)
(1)
rmin
b
a
ψ
In the scattering from an impenetrable sphere, rmin is the radius of that sphere. Also, we can see
from the figure that θ = π − 2ψ .
For r > rmin , U = 0. Thus (1) becomes
∆ψ = ∫
∞
a
(
)
r 2 dr
2µE −
2
r2
(2)
Substituting
b 2µT0′ = ;
E = T0′
(3)
310
CHAPTER 9
(2) becomes
∆ψ = ∫
dr
∞
α
(4)
r2
r 2 −1
b
This integral can be solved by using Eq. (E. 10b), Appendix E:
∞
−2 
ψ = sin 
4 
r

b2  a

−1
(5)
Thus,
sin ψ =
b
a
(6)
Therefore, we can find the relation between θ and b by substituting (θ = π − 2ψ ) into (6). We
have
b = a cos
θ
(7)
2
Now, the differential cross section is given by Eq. (9.120):
b db
sin θ dθ
(8)
1
θ a
θ a2
a cos × sin =
sin θ
2 2
2 4
(9)
σ (θ ) =
From (7), we have
σ (θ ) =
Total cross section is given by
σ t = ∫ σ (θ ) dΩ =
a2
⋅ 4π
4
(10)
so that
σ t = π a2
(11)
9-47. The number of recoil particles scattered into unit solid angle in each of the two systems,
lab and CM, are the same. Therefore,
σ ( φ ) sin φ dφ = σ (ζ ) sin ζ dζ
(1)
where φ and ζ are the CM and lab angles, respectively, of the recoil particle. From (1) we can
write [cf. Eq. (9.125) in the text]
σ ( φ ) sin ζ dζ
=
σ (ζ ) sin φ dφ
(2)
311
DYNAMICS OF A SYSTEM OF PARTICLES
Now, in general, φ = 2ζ [see Eq. (9.74)]. Hence,
sin ζ
sin ζ
1
=
=
sin φ sin 2ζ 2 cos ζ
(3)
dζ 1
=
dφ 2
(4)
σ (φ )
1
=
σ (ζ ) 4 cos ζ
(5)
and
Using (3) and (4) in (2), we have
For m1 = m2 , the Rutherford scattering cross section is [Eq. (9.141)]
σ (θ ) =
k2
1
×
2
4
4T0 sin (θ 2)
(6)
Also for this case, we have [Eqs. (9.71) and (9.75)]
θ




π
ψ = −ζ 
2

ψ=
2
(7)
Hence,
sin
θ
π

= sin ψ = sin  − ζ  = cos ζ
2

2
(8)
and since the CM recoil cross section σ (φ) is the same as the CM scattering cross section σ (θ ) ,
(6) becomes
k2
1
×
2
4T0 cos 4 ζ
(9)
σ (ζ ) = σ ( φ ) × 4 cos ζ
(10)
σ (φ ) =
Using (5) to express σ (ζ ), we obtain
or,
σ (ζ ) =
k2
1
×
2
T0 cos 3 ζ
(11)
312
CHAPTER 9
9-48. In the case m1 m2 , the scattering angle ψ for the incident particle measured in the lab
system is very small for all energies. We can then anticipate that σ (ψ) will rapidly approach
zero as ψ increases.
Eq. (9.140) gives the Rutherford cross section in terms of the scattering angle in the CM system:
σ CM (θ ) =
From Eq. (9.79) we see that for m1
k2
( 4T0′)
1
2
sin (θ 2)
4
(1)
m2 ,
T0′ =
m2
m
T0 ≅ 2 T0
m1 + m2
m1
(2)
Furthermore, from Eq. (9.69),
tan ψ =
sin θ
m1
+ cos θ
m2
≅
m2
sin θ
m1
(3)
and therefore, since ψ is expected to be small for all cases of interest,
sin θ ≅
m1
m
tan ψ ≅ 1 ψ
m2
m2
(4)
Then,
m 
cos θ = 1 −  1 ψ 
 m2 
2
(5)
and
2

 m1  
1
sin (θ 2) = 1 − 1 − 
ψ 
2
m
 2  

2
(Notice that ψ
to unity.)
1 , but since m1
(6)
m2 , the quantity m1ψ m2 is not necessarily small compared
With the help of (2) and (6), we can rewrite the CM cross section in terms of ψ as
2
 mk 
1
σ CM (ψ ) =  1 
2
2 
 2m2T0  
1 − 1 −  m1 ψ  

 m2  

According to Eq. (9.129),
(7)
313
DYNAMICS OF A SYSTEM OF PARTICLES
2


 
 m1 cos ψ + 1 −  m1 sin ψ  
 m2
 m2
 

σ LAB (ψ ) = σ CM (θ )
2
 m1

1− 
sin ψ 
 m2

2
(8)
We can compute σ LAB (ψ ) with the help of (7) and the simplifications introduced in the righthand side of (8) by the fact that ψ 1 :
2
2


 
 m1 + 1 −  m1 ψ  
2
 m2
 m2  
 m1k 

σ LAB (ψ ) ≅ 

2
2
2
 2m2T0  
 m1  
 m1 
1 − 1 − 
ψ  1−  ψ 
m2  


 m2 


(9)
and so,
σ LAB (ψ ) ≅
(m k
2
1
2m22T0
2


 
1 − 1 −  m1 ψ  

 m2  

2
)
2
m 
1−  1 ψ 
 m2 
(10)
2
This expression shows that the cross section has a second-order divergence at ψ = 0. For values
of ψ > m2 m1 , (9) gives complex values for σ lab . This result is due to the approximations
involved in its derivation, making our result invalid for angles larger than m2 m1 .
9-49. The differential cross section for Rutherford scattering in the CM system is [cf. Eq.
(9.140) in the text]
σ (θ ) =
1
k2
2
16T0′ sin 4 θ
2
(1)
where [cf. Eq. (9.79)]
T0′ =
m2
T0
m1 + m2
(2)
314
CHAPTER 9
Thus,
1  m1 + m2 
k2
σ (θ ) =


2
16T0 sin 4 θ  m2 
2
1 
k2
m1 
=
1 +

2
θ
16T0 sin 4  m2 
2
Since m1 m2
2
2
(3)
1 , we expand
2

m1 
m1
+…
1 +
 ≅ 1+ 2
m2
 m2 
(4)
Thus, to the first order in m1 m2 , we have
σ (θ ) =
m1 
k2
1 
1 + 2

2
θ
16T0 sin 4 
m2 
2
(5)
This result is the same as Eq. (9.140) except for the correction term proportional to m1 m2 .
9-50.
The potential for the given force law is
U (r) =
k
2r 2
(1)
First, we make a change of variable, z = 1 r . Then, from Eq. (9.123), we can write
θ =∫
=
b dz
zmax

k  2
z
1 − b 2 +
mu02 

0
b
k
2 b +
mu02
sin −1
2
=
z
zmax

k 
zmax = b 2 +

mu02 

−1 2
(2)
0
πb
2 b2 +
k
mu02
Solving (2) for b = b(θ ),
bθ =
k
mu02
2θ
π − 4θ 2
2
(3)
315
DYNAMICS OF A SYSTEM OF PARTICLES
According to Fig. 9-22 and Eq. (9.122),
θ=
1
(π − θ )
2
(4)
so that b(θ ) can be rewritten as (θ):
b (θ ) =
k
mu02
π −θ
θ ( 2π − θ )
(5)
The differential cross section can now be computed from Eq. (9.120):
σ (θ ) =
b db
sin θ dθ
(6)
with the result
σ (θ ) =
kπ 2 (π − θ )
(7)
mu02θ 2 ( 2π − θ ) sin θ
2
9-51.
n
θ
φ=π−θ
p
In the CM system, whenever the neutron is scattered through the angle θ, the proton recoils at
the angle φ = π – θ. Thus, the neutron scattering cross section is equal to the recoil cross section
at the corresponding angles:
dN p
dN n
=
dΩ (θ ) dΩ ( φ )
(1)
dN p dTp
dN n
=
dΩ (θ ) dTp dΩ ( φ )
(2)
Thus,
where dN p dTp is the energy distribution of the recoil protons. According to experiment,
dN p dTp = const. Since mp ≅ mn , Tp is expressed in terms of the angle ψ by using Eq. (9.89b):
Tp = T0 sin 2 ψ
We also have ψ =
θ
2
(3)
for the case mp ≅ mn . Thus,
dTp
dΩ ( φ )
=
(
d
1
T0 sin 2 ψ
2π sin φ dφ
)
(4)
316
CHAPTER 9
or,
dTp
dΩ ( φ )
=
T0
d
π − φ 
sin 2 
2π sin φ dφ
 2 
=
T0
φ
φ T
sin cos = 0
2π sin φ
2
2 4π
(5)
Therefore, we find for the angular distribution of the scattered neutron,
dN p T0
dN n
=
⋅
dΩ (θ ) dTp 4π
(6)
Since dN p dTp = const., dN n dΩ is also constant. That is, the scattering of neutrons by protons
is isotropic in the CM system.
9-52. Defining the differential cross section σ (θ ) in the CM system as in Eq. (9.116), the
number of particles scattered into the interval from θ to θ + dθ is proportional to
dN ∝ σ (θ ) sin θ dθ = −σ (θ ) d ( cos θ )
(1)
From Eq. (9.87a) and the assumption of elastic collisions (i.e., T0 = T1 + T2 ), we obtain
T2
2m1 m2
=
(1 − cos θ )
T0 ( m1 + m2 ) 2
(2)
or, solving for cos θ,
cos θ =
where Tm =
4 m1 m2
( m1 + m2 )2
Tm − 2T2
Tm
(3)
T0 is the maximum energy attainable by the recoil particle in the lab
system. Then, (1) can be rewritten as
dN ∝ 2σ (θ )
dT2
Tm
(4)
and consequently, we obtain the desired result for the energy distribution:
dN
∝ σ (θ )
dT2
(5)
317
DYNAMICS OF A SYSTEM OF PARTICLES
9-53.
m1 = mass of particle α 1 ,
m2 = mass of
238
U
u1 , u′1 : velocity of particle α in LAB and CM before collision
v1 , v′1 : velocity of particle ∝ in LAB and CM after collision
u2 , u′2 : velocity of
238
v2 , v′2 ” velocity of
238
u2 = 0 ,
T1 =
U in LAB and CM before collision
U in LAB and CM after collision
m1u12
= 7.7MeV
2
ψ = 90° is angle through which particle α is deflected in LAB
θ
is angle through which particle α and
ζ
is recoil angle of
238
238
U are deflected in CM
U in LAB
v1
m1
u1
m1
ζ
m2
θ
v2′
v2
–vCM
a) Conservation of momentum in LAB:
m1 u1 = m2 v2 cos ζ 
 ⇒ v1 = u1 tan ζ
m1 v1 = m2 v2 sin ζ 
Conservation of energy in LAB:
m1u12 m1v12 m2 v22
=
+
2
2
2
From these equations we obtain the recoil scattering angle of
tan ζ =
m2 − m1
m2 + m1
238
U
⇒ ζ = 44.52°
b) The velocity of CM of system is
vCM =
The velocity of
238
m1u1
m1 + m2
U in CM after collision is v2' = v2 − vCM . From the above figure we can obtain
the scattering angle of particle
238
U in CM to be
318
CHAPTER 9
tan θ =
v2 sin ζ
=
v2 cos ζ − vcm
m22 − m12
m12
⇒ θ = 89.04°
In CM, clearly after collision, particle α moves in opposite direction of that of
c) The kinetic energy of particle
238
238
U.
U after collision in LAB is
2
m2 v22 m2  m1v0 
m12 u12
2m1
T1 = 0.25 MeV
=
=
=


2
2  m2 cos ζ 
m1 + m2 m1 + m2
Evidently, conservation of energy is satisfied.
d) The impact parameter in CM is given in Section 9.10.
b=
where k =
(
k
θ
cot  
 2
2T0′
)
q1q2
1
and T0′ = m1u1′ 2 + m2 u2′ 2 is the total energy of system in CM,
2
9πε 0
b=
so
q1 q2 m1 + m2
θ
cot   = 1.8 × 10 −14 m
2
 2
4πε 0 m1 m2 u1
We note that b is the impact parameter of particle α with respect to CM, so the impact
( m1 + m2 ) b = 1.83 × 10−14 m.
parameter of particle α with respect to 238 U is
m2
e) In CM system, the orbit equation of particle α is
α
r′
= (1 + ε ) cos θ
⇒ rmin
′ =
α
1+ ε
where θ = 0
corresponds to r = rmin
is closest distance from particle α to the center of mass, and
4πε 0 ( m1u1′b )
α=
=
m1k q1q2
m1
2
2
and
ε = 1+ 2
4πε 0 ( m1u1′b )
E 2
= 1 + 2E
m1k
q1 q2
m1
= 1 + m1u1′ 2
4πε 0 ( m1u1′b )
q1q2
m1
2
But the actual minimum distance between particles is
rmin =
m1 + m2
rmin
′ = 0.93 × 10 −14 m.
m2
2
319
DYNAMICS OF A SYSTEM OF PARTICLES
f)
Using formula
σ LAB (ψ ) = σ CM (θ )
where x =
m1
, ψ = 90° ,
m2
σ CM (θ ) =
k2
( 4T0′)
2
(
x cos ψ + 1 − x 2 sin 2 ψ
)
2
1 − x sin ψ
2
2
1
θ
sin 4  
 2
We find this differential cross section in LAB at ψ = 90°:
σ LAB (ψ = 90°) = 3.16 × 10 −28 m 2
g) Since
dN
= σ (θ ) sin ψ dψ dφ we see that the ratio of probability is
N
σ (ψ ) sin ψ
= 11.1
σ (ψ ') sin ψ ′
9-54.
Equation 9.152 gives the velocity of the rocket as a function of mass:
v = v0 + u ln
m0
m
= µ ln 0
m
m
p = mv = mu ln
To maximize p, set
( v0 = 0 )
m0
m
dp
=0
dm
0=
ln
m0
=1
m
dp
 m

= u ln 0 − 1
dm
 m

m0
=e
m
or
m
= e −1
m0
To check that we have a maximum, examine
d2 p 

dm2  m = m e−1
0
d2 p
u
=−
2
dm
m
d2 p 
u
e < 0 , so we have a maximum.
=−
2 
dm  m = m e−1
m0
0
m
= e −1
m0
320
9-55.
CHAPTER 9
The velocity equation (9.165) gives us:
 m 
v ( t ) = − gt + u ln  0 
 m (t) 
(1)
where m ( t ) = m0 − α t , the burn rate α = 9m0 10τ , the burn time τ = 300 s , and the exhaust
velocity u = 4500 m ⋅ s -1 . These equations are good only from t = 0 to t = τ. First, let us check that
the rocket does indeed lift off at t = 0: the thrust α u = 9um0 10τ = 13.5 m ⋅ s −2 ⋅ m0 > m0 g , as
required. To find the maximum velocity of the rocket, we need to check it at the times t = 0 and
t = τ, and also check for the presence of any extrema in the region 0 < t < τ. We have v(0) = 0,
v(τ ) = − gτ + u ln 10 = 7400 m ⋅ s −1 , and calculate
 αu

dv
αu
= −g +
=g
− 1 > 0
dt
m (t)
 m (t) g

(2)
The inequality follows since α u > m0 g > m ( t ) g . Therefore the maximum velocity occurs at t = τ,
where v = − gτ + u ln 10 = 7400 m ⋅ s −1 . A similar single-stage rocket cannot reach the moon since
(
)
v ( t ) < u ln m0 m final = u ln 10 10.4 m ⋅ s −1 , which is less than escape velocity and independent of
fuel burn rate.
9-56.
a) Since the rate of change of mass of the droplet is proportional to its cross-sectional area, we
have
dm
= kπ r 2
dt
(1)
If the density of the droplet is ρ,
m=
4π
ρ r3
3
(2)
so that
dm dm dr
dr
=
= 4πρr 2
= π kr 2
dt
dr dt
dt
(3)
dr
k
=
dt 4 ρ
(4)
Therefore,
or,
r = r0 +
as required.
k
t
4ρ
(5)
321
DYNAMICS OF A SYSTEM OF PARTICLES
b) The mass changes with time, so the equation of motion is
F=
d
dv
dm
= mg
( mv) = m + v
dt
dt
dt
(6)
Using (1) and (2) this becomes
4π 3 dv
4π 3
+ π kr 2 v =
ρr
ρr g
3
3
dt
(7)
dv 3k
v=g
+
dt 4 ρr
(8)
or,
Using (5) this becomes
dv 3k
v
+
=g
dt 4 ρ r + k t
0
4ρ
If we set A =
(9)
3k
k
and B =
, this equation becomes
4ρ
4ρ
dv
A
+
v=g
dt r0 + Bt
(10)
and we recognize a standard form for a first-order differential equation:
dv
+ P (t) v = Q (t)
dt
(11)
in which we identify
P (t) =
A
; Q(t) = g
r0 + Bt
(12)
The solution of (11) is
− P ( t ) dt 
∫ P (t) dt Q dt + constant 
v (t) = e ∫
 ∫ e

(13)
Now,
∫ P (t ) dt = ∫ r
0
A
A
dt = ln ( r0 + Bt )
B
+ Bt
= ln ( r0 + Bt )
since
3
(14)
A
= 3 . Therefore,
B
e∫
Pdt
= ( r0 + Bt )
3
(15)
322
CHAPTER 9
Thus,
−3
3
v ( t ) = ( r0 + Bt )  ∫ ( r0 + Bt ) g dt + constant 


−3  g
4

= ( r0 + Bt )  ( r0 + Bt ) + C 
 4B

(16)
The constant C can be evaluated by setting v ( t = 0 ) = v0 :
v0 =
1 g 4

r0 + C 
r03  4B

(
)
(17)
so that
C = v0 r03 −
g 4
r0
4B
(18)
We then have
v (t) =
1
( r0 + Bt )
3
g 4
4
 g
3
 4B ( r0 + Bt ) + v0 r0 − 4B r0 
(19)
or,
v (t) =
1
( Bt ) 3
 g
4
3 
 4B ( Bt ) + 0 r0 
( )
(20)
( )
where 0 r03 means “terms of order r03 and higher.” If r0 is sufficiently small so that we can
neglect these terms, we have
v (t) ∝ t
(21)
as required.
9-57.
Start from our definition of work:
W = ∫ F dx = ∫
dp
dx = ∫ v dp
dt
(1)
We know that for constant acceleration we must have v = at (zero initial velocity). From
Equation (9.152) this means
m = m0 e − at u
(2)
 at 
dp = d ( mv ) = d ( mat ) = ma dt + at dm = m0 ae − at u 1 −  dt
u

(3)
We can then compute dp:
This makes our expression for the work done on the rocket
Wr =
m0 a t
( at )( u − at ) e − at u dt
u ∫0
(4)
323
DYNAMICS OF A SYSTEM OF PARTICLES
The work done on the exhaust, on the other hand, is given with v → (v – u) and
dp → dmexhaust (v − u) , so that
We =
m0 a t
( at − u) 2 e − at u dt
u ∫0
(5)
The upper limit on the integrals is the burnout time, which we can take to be the final velocity
divided by the acceleration. The total work done by the rocket engines is the sum of these two
quantities, so that
W=
(
)
m0 a v a 2
u − uat e − at u dt = m0 u 2
∫
0
u
−
∫ 0 (1 − x ) e
v u
x
dx
(6)
where the obvious substitution was made in the last expression. Upon evaluating the integral
we find
W = m0 uve − v u = muv
where m is the mass of the rocket after its engines have turned off and v is its final velocity.
9-58.
From Eq. (9.165) the velocity is
v=
dy
m
= − gt + u ln 0
dt
m
m0 

 − gt + u ln m  dt
∫ dy = ∫
Since
m
= −α , dt = − dm α
dt
y+C =−
1 2 u
gt −
2
α

a
∫ ln
m0
dm
m
a
∫ ln x dx = x  1 + ln x  , so we have
y+C =−
1 2 u
gt −
2
α
m0 

 m + m ln m 
Evaluate C using y = 0 when t = 0, m = m0
C=−
y=
u ( m0 − m)
α
−
α
1 2 mu m0
ln
gt −
; m0 − m = α t
m
2
α
y = ut −
At burnout, y = yB , t = tB
um0
1 2 mu m0
ln
gt −
2
α
m
(7)
324
CHAPTER 9
yB = utB −
1 2 mu m0
gtB −
ln
m
2
α
After burnout, the equations are
y = y 0 + v0 t −
1 2
gt and v = v0 − gt
2
Calling the top of the path the final point
v f = 0 = vB − gt f
y − y0 =
or
t f = vB g
vB2 vB2 vB2
−
=
;
g 2g 2g
y=
y0 = 0
vB2
2g
9-59. In order to immediately lift off, the thrust must be equal in magnitude to the weight of
the rocket. From Eq. (9.157):
Thrust = v0α
v0 = velocity of fuel
So
v0α = mg
or
v0 = mg α
9-60.
The rocket will lift off when the thrust just exceeds the weight of the rocket.
Thrust = − u
dm
= uα
dt
Weight = mg = ( m0 − α t ) g
Set thrust = weight and solve for t:
uα = ( m0 − α t ) g ; t =
m0
α
−
u
g
With m0 = 70000 kg , α = 250 kg/s , u = 2500 m/s , g = 9.8 m/s 2
t
25 sec
The design problem is that there is too much fuel on board. The rocket sits on the ground
burning off fuel until the thrust equals the weight. This is not what happens in an actual launch.
A real rocket will lift off as soon as the engines reach full thrust. The time the rocket sits on the
ground with the engines on is spent building up to full thrust, not burning off excess fuel.
325
DYNAMICS OF A SYSTEM OF PARTICLES
9-61.
From Eq. (9.153), the velocity after the first state is:
v1 = v0 + u ln k
After the second stage:
v2 = v1 + u ln k = v0 + 2u ln k
After the third stage:
v3 = v2 + u ln k = v0 + 3u ln k
After the n stages:
vn = vn −1 + u ln k = v0 + nu ln k
vn = v0 + nu ln k
9-62. To hover above the surface requires the thrust to counteract the gravitational force of
the moon. Thus:
−u
−
dm 1
= mg
dt 6
6u dm
= dt
g m
Integrate from m = m0 to 0.8 m0 and t = 0 to T:
T=−
6 ( 2000 m/s )
6u
ln 0.8 = −
ln 0.8
9.8 m/s 2
g
T = 273 sec
9-63.
a) With no air resistance and constant gravity, the problem is simple:
1
mv02 = mgh
2
giving the maximum height of the object as h = v02 2 g
v0 g 610 s .
(1)
1800 km . The time it takes to do this is
b) When we add the expression for air resistance, the differential equation that describes the
projectile’s ascent is

F=m


dv
1
v
= − mg − cW ρ Av 2 = − mg 1 +  
 v 
dt
2

t

2





(2)
326
CHAPTER 9
where vt = 2mg cW ρ A
2.5 km ⋅ s −1 would be the terminal velocity if the object were falling
from a sufficient height (using 1.3 kg ⋅ m −3 as the density of air). Solution of this differential
equation gives



v0  gt 

 −
vt 
 vt 

v ( t ) = vt tan  tan −1 

This gives v = 0 at time τ = ( vt g ) tan −1 ( v0 vt )
(3)
300 s . The velocity can in turn be integrated to
give the y-coordinate of the projectile on the ascent. The height it reaches is the y-coordinate at
time τ:



v2
v 
h = t ln 1 +  0 
2 g   vt 

which is
2





(4)
600 km.
c) Changing the acceleration due to gravity from –g to − GMe
( Re + y )
2
= − g  Re ( Re + y ) 
2
changes our differential equation for y to
 y  2  R  2 
e
y = − g    + 
 


  vt   Re + y  
(5)
Using the usual numerical techniques, we find that the projectile reaches a height of
a flight time of
630 km in
330 s.
d) Now we must replace the ρ in the air resistance equation with ρ ( y ) . Given the dependence
of vt on ρ, we may write the differential equation
 ρ ( y)  y  2  R  2


e
y = −g 

 +
 


ρ
v
R
+
 0  t   e y  
(6)
where we use log 10 ρ ( y ) = 0.11 − (5 × 10 −5 )y and ρ0 = 1.3 , with the ρ ’s in kg ⋅ m −3 and y in meters.
The projectile then reaches a height of
2500 km in a flight time of
940 s. This is close to the
height to which the projectile rises when there is no air resistance, which is
2600 km.
327
DYNAMICS OF A SYSTEM OF PARTICLES
2.5
height (1000 km)
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t (1000 s)
(a)
(b)
(c)
(d)
9-64. We start with the equation of motion for a rocket influenced by an external force, Eq.
(9.160), with Fext including gravity, and later, air resistance.
a) There is only constant acceleration due to gravity to worry about, so the problem can be
solved analytically. From Eq. (9.166), we can obtain the rocket’s height at burnout
yb = utb −
1 2 mu  m0 
gtb −
ln  
2
α
 mb 
(1)
where mb is the mass of the rocket at burnout and α = ( m0 − mb ) tb . Substitution of the given
values gives yb
250 km . After burnout, the rocket travels an additional vb2 2 g , where vb is the
rocket velocity at burnout. The final height the rocket ends up being
is taken into account.
3700 km, after everything
b) The situation, and hence the differential equation, becomes more complicated when air
resistance is added. Substituting Fext = − mg − cW ρ Av 2 2 (with ρ = 1.3 kg ⋅ m −3 ) into Equation
(9.160), we obtain
c ρ Av 2
dv uα
=
−g− W
dt m
2m
(2)
We must remember that the mass m is also a function of time, and we must therefore include it
also in the system of equations. To be specific, the system of equations we must use to do this by
computer are
v


y 
2 
cW ρ Av 
   uα
 v  =  m − g − 2m 
 m  

−α


These must be integrated from the beginning until the burnout time, and therefore must be
integrated with the substitution α = 0. Firstly, we get the velocity and height at burnout to be
vb 7000 m ⋅ s −1 and yb 230 km . We can numerically integrate to get the second part of the
(3)
328
CHAPTER 9
journey, or use the results of Problem 9-63(b) to help us get the additional distance travelled
with air resistance, analytically. The total height to which the rocket rises is 890 km in a total
flight time of
410 s.
c) The variation in the acceleration of gravity is taken into account by substituting
( Re + y )
GMe
vb
2
= g  Re ( Re + y )  for g in the differential equation in part (b). This gives
6900 m ⋅ s −1 , yb
2
230 km , with total height
950 km and time-of-flight
460 s.
d) Now one simply substitutes the given expression for the air density, ρ (y) for ρ, into the
differential equation from part (c). This gives vb 8200 m ⋅ s −1 , yb 250 km , and total height
8900 km with time-of-flight
2900 s.
10
height (km)
8
6
4
2
0
0
0.5
1
1.5
2
2.5
3
t (1000 s)
(a)
(b)
(c)
9-65.
Total impulse
P = 8.5 N ⋅ s
Total mass
m0 = 0.054 kg
Burn time
t f = 1.5 s
Rocket cross section area
S=
Drag coefficient
cw = 0.75
Drag force
D=
π d2
4
= 4.5 × 10 −4 m 2
1
ρcw Sv 2 = Kv 2 = 2 × 10 −4 v 2 N
2
where ρ = 1.2 kg/m 3 is density of air
and v is rocket’s speed
Rocket exhaust speed u = 800 m/s
a) The total mass of propellant is
∆m =
P
= 0.0106 kg
u
(d)
329
DYNAMICS OF A SYSTEM OF PARTICLES
Since m ~ 20% m0 , we will assume that the rocket’s mass is approximately constant in this
problem. The equation of motion of rocket is
m0
(where α =
dv
= − mo g + uα − Kv 2
dt
∆m
= 7.1 × 10 −3 kg/s is fuel burn rate)
tf
⇒
m0 dv
= dt
( uα − m0 g ) − Kv2
Using the initial condition at t = 0, v = 0, we find
( uα − m0 g ) tanh 
v(t) =
K ( uα − m0 g ) 

m0 2


K
At burn-out, t = t f = 1.5 s , we find
v f = v(t f ) = 114.3 m/s
The height accordingly is given by
h(t) = ∫ v(t) dt =
0
 k ( uα − m g )  
m0 
0
ln  cosh  t

2

 
k
m

0



At burn out, t = t f , we find the burn-out height
h f = h(t f ) = 95.53 m
b)
After the burn-out, the equation of motion is
m0
dv
= − m0 g − Kv 2
dt
⇒
m0 dv
= − dt
m0 g + Kv 2
with solution
v(t) = −
 Kg

m0 g
tan  t
− C
K
 m0

Using the initial condition at t = t f , v(t) = v f , we find the constant C = –1.43 rad, so
v(t) = −
 Kg

m0 g
tan  t
− 1.43
K
 m0

and the corresponding height
t
h(t) = h f + ∫ v(t) dt = h f +
tf
m0
K


 Kg
  
− 1.43  
0.88 + ln cos  t
 m0
  


330
CHAPTER 9
When the rocket reaches its maximum height (at t = tmax ) the time tmax can be found by setting
v(tmax ) = 0. We then find tmax = 7.52 s. And the maximum height the rocket can reach is
hmax = h(tmax ) = 334 m
c)
Acceleration in burn-out process is (see v(t) in a))
dv uα − m0 g
=
dt
m0
a(t ) =
1
 k ( uα − m g ) 
0

cosh 2 t
2
m0


Evidently, the acceleration is maximum when t = 0 and
amax = a(t = 0) =
uα − m0 g
= 95.4 m/s 2
m0
d) In the fall-down process, the equation of motion is
m0 dv
= m0 g − Kv 2 ,
dt
With the initial condition t = tmax , v = 0, we find ( t ≥ tmax )
v(t) = −

m0 g
Kg 
tanh ( t − tmax )

K
m0 

(v(t) is negative for t ≥ tmax , because then the rocket falls downward)
The height of the rocket is
t
h = hmax +
∫
tmax
v(t) dt = hmax −

Kg  
m0 
ln cosh ( t − tmax )

K
m0  


To find the total flight-time, we set h = 0 and solve for t. We find ttotal = 17.56 s
e) Putting t = ttotal into the expression of V(t) in part d), we find the speed at ground impact to
be
vg = −

m0 g
Kg 
tanh ( ttotal − tmax )
 = −49.2 m/s
K
m0 

9-66. If we take into account the change of the rocket’s mass with time m = m0 − α t , where α
is the fuel burn rate,
α = 7.1 × 10 −3 kg/s
as calculated in problem 9-65.
The equation of motion for the rocket during boost phase is
331
DYNAMICS OF A SYSTEM OF PARTICLES
dv
( m0 − α t ) dt = uα − Kv2
dv
dt
=
Kv − uα α t − m0
⇒
2
Integrating both sides we obtain finally
 1 + C αt − m 2
(
0)
v(t) = 
 1 − C (α t − m ) 2
0

uα

Ku
K
α 

Ku
α
where C is a constant. Using the initial condition v(t ) = 0 at t = 0 , we can find C and the velocity
is

2
 1 −  1 − αt 

m0 
uα 
v(t) =

K 
2


 1 + 1 − αt


m0 




Ku
∝ 


Ku
∝
a) The rocket speed at burn-out is (note t f = 1.5 s ).
v f = v(t f ) = 131.3 m/s
b) The distance the rocket has traveled to the burn-out is
tf
h f = ∫ v(t) dt = 108.5 m
θ
9-67.
From Equation (9.167) we have
H bo = −
(
g m0 − m2f
2α
2
) + u m

α
f

 mf 
ln 
+ m0 − m f 

 m0 

Using numerical values from Example 9.12
α = −1.42 × 10 4 kg/s
m0 = 2.8 × 106 kg
m f = 0.7 × 106 kg
u = 2.600 m/s
we find H bo = 97.47 km .
From Equation (9.168) we find
vbo = −
g(m0 − m f )
α
m 
+ u ln  o 
 mf 
vbo = 2125 m/s
332
CHAPTER 9
CHAPTER
10
Motion in a
Noninertial Reference Frame
10-1.
The accelerations which we feel at the surface of the Earth are the following:
:
980 cm/sec 2
(1)
Gravitational
(2)
Due to the Earth’s rotation on its own axis:
 2π rad/day 
rω = 6.4 × 10 cm × 

 86400 sec/day 
(
2
)
8
(
) (
= 6.4 × 10 8 × 7.3 × 10 −5
)
2
2
= 3.4 cm/sec 2
(3) Due to the rotation about the sun:
(
rω = 1.5 × 10
2
13


2π rad/year
cm × 

 86400 × 365 sec/day 
)
2
2
 7.3 × 10 −5 
= 1.5 × 1013 × 
= 0.6 cm/sec 2

 365 
(
10-2.
)
The fixed frame is the ground.
y
a
x
θ
The rotating frame has the origin at the center of the tire and is the frame in which the tire is at
rest.
From Eqs. (10.24), (10.25):
+ a + ω × r + ω × ( ω × r ) + 2ω × v
af = R
f
r
r
333
334
CHAPTER 10
Now we have
= − a cos θ i + a sin θ j
R
f
r = r0 i
ω=
vr = ar = 0
V
k
r0
ω =
a
k
r0
Substituting gives
a f = − a cos θ i + a sin θ j + a j −
v2
i
r0
 v2

a f = −i  + a cos θ  + j ( sin θ + 1) a
 r0

2
We want to maximize a f , or alternatively, we maximize a f :
af
2
=
v4
2 av 2
2
2
a
cos
cos θ + a 2 + 2 a 2 sin θ + a 2 sin 2 θ
+
θ
+
2
r0
r0
=
v4
2av 2
2
a
+
+
2
cos θ + a 2 sin 2 θ
r02
r0
d af
dθ
2
=−
2av 2
cos θ + 2a 2 cos θ
r0
= 0 when tan θ =
ar0
v2
(Taking a second derivative shows this point to be a maximum.)
tan θ =
ar0
implies cos θ =
v2
v2
a 2 r02 + v 4
and
sin θ =
ar0
a r + v4
2 2
0
Substituting into (1)
 v2
av 2
a f = −i  +
 r0
a 2 r02 + v 4
 

ar0
 + j
+ 1 a
  a 2 r02 + v 4

This may be written as
a f = a + a 2 + v 4 r02
(1)
335
MOTION IN A NONINERTIAL REFERENCE FRAME
A
θ
This is the maximum acceleration. The point which experiences this acceleration is at A:
where tan θ =
10-3.
ar0
v2
We desire Feff = 0 . From Eq. (10.25) we have
− mω × r − mω × ( ω × r ) − 2mω × v
Feff = F − mR
f
r
r
0
ω
The only forces acting are centrifugal and friction, thus µs mg = mω 2 r , or
r=
µs g
ω2
10-4. Given an initial position of (–0.5R,0) the initial velocity (0,0.5ωR) will make the puck
motionless in the fixed system. In the rotating system, the puck will appear to travel clockwise
in a circle of radius 0.5R. Although a numerical calculation of the trajectory in the rotating
system is a great aid in understanding the problem, we will forgo such a solution here.
10-5.
The effective acceleration in the merry-go-round is given by Equation 10.27:
x = ω 2 x + 2ω y
(1)
y = ω 2 y − 2ω x
(2)
These coupled differential equations must be solved with the initial conditions
x0 ≡ x ( 0 ) = −0.5 m , y0 ≡ y ( 0 ) = 0 m , and x ( 0 ) = y ( 0 ) = v0 2 m ⋅ s −1 , since we are given in the
problem that the initial velocity is at an angle of 45° to the x-axis. We will vary v0 over some
range that we know satisfies the condition that the path cross over ( x0 , y0 ) . We can start by
looking at Figures 10-4e and 10-4f, which indicate that we want v0 > 0.47 m ⋅ s −1 . Trial and error
can find a trajectory that does loop but doesn’t cross its path at all, such as v0 = 0.53 m ⋅ s −1 .
From here, one may continue to solve for different values of v0 until the wanted crossing is
eyeball-suitable. This may be an entirely satisfactory answer, depending on the inclinations of
the instructor. An interpolation over several trajectories would show that an accurate answer to
the problem is v0 = 0.512 m ⋅ s −1 , which exits the merry-go-round at 3.746 s. The figure shows
this solution, which was numerically integrated with 200 steps over the time interval.
336
CHAPTER 10
1
y (m)
0.5
0
0.5
–1
–0.5
–1
0
0.5
1
x (m)
10-6.
z
z = f(r)
m
r
Consider a small mass m on the surface of the water. From Eq. (10.25)
− mω × r − mω × ( ω × r ) − 2mω × v
Feff = F − mR
f
r
In the rotating frame, the mass is at rest; thus, Feff = 0 . The force F will consist of gravity and the
force due to the pressure gradient, which is normal to the surface in equilibrium. Since
= ω = v = 0 , we now have
R
f
r
0 = mg + Fp − mω × ( ω × r )
where Fp is due to the pressure gradient.
Fp
θ
mω2r
θ′
mg
Since Feff = 0 , the sum of the gravitational and centrifugal forces must also be normal to the
surface.
Thus θ ′ = θ.
tan θ ′ = tan θ =
ω 2r
g
337
MOTION IN A NONINERTIAL REFERENCE FRAME
but
tan θ =
dz
dr
Thus
z=
ω2
2g
r 2 + constant
The shape is a circular paraboloid.
10-7. For a spherical Earth, the difference in the gravitational field strength between the poles
and the equator is only the centrifugal term:
g poles − gequator = ω 2 R
For ω = 7.3 × 10 −5 rad ⋅ s −1 and R = 6370 km, this difference is only 34 mm ⋅ s −2 . The disagreement
with the true result can be explained by the fact that the Earth is really an oblate spheroid,
another consequence of rotation. To qualitatively describe this effect, approximate the real Earth
as a somewhat smaller sphere with a massive belt about the equator. It can be shown with more
detailed analysis that the belt pulls inward at the poles more than it does at the equator. The
next level of analysis for the undaunted is the “quadrupole” correction to the gravitational
potential of the Earth, which is beyond the scope of the text.
10-8.
ω
z
y
λ
x
Choose the coordinates x, y, z as in the diagram. Then, the velocity of the particle and the
rotation frequency of the Earth are expressed as
v = ( 0, 0, z )


ω = ( −ω cos λ , 0, ω sin λ ) 
(1)
so that the acceleration due to the Coriolis force is
a = −2ω × r = 2ω ( 0, − z cos λ , 0 )
(2)
338
CHAPTER 10
This acceleration is directed along the y axis. Hence, as the particle moves along the z axis, it
will be accelerated along the y axis:
y = −2ω z cos λ
(3)
Now, the equation of motion for the particle along the z axis is
z = v0 − gt
z = v0 t −
where v0 is the initial velocity and is equal to
h:
(4)
1 2
gt
2
(5)
2gh if the highest point the particle can reach is
v0 = 2 gh
(6)
y = −2ω z cos λ + c
(7)
From (3), we have
but the initial condition y ( z = 0 ) = 0 implies c = 0. Substituting (5) into (7) we find
1


y = −2ω cos λ  v0 t − gt 2 


2
(
= ω cos λ gt 2 − 2v0 t 2
)
(8)
Integrating (8) and using the initial condition y(t = 0) = 0, we find
1

y = ω cos λ  gt 2 − v0 t 2 
3


(9)
From (5), the time the particle strikes the ground (z = 0) is
1 

0 =  v0 − gt t

2 
so that
t=
2v0
g
(10)
Substituting this value into (9), we have
 1 8v 3
4v 2 
y = ω cos λ  g 30 − v0 20 
g 
3 g
v3
4
= − ω cos λ 02
3
g
If we use (6), (11) becomes
(11)
339
MOTION IN A NONINERTIAL REFERENCE FRAME
4
8h3
y = − ω cos λ
3
g
(12)
The negative sign of the displacement shows that the particle is displaced to the west.
10-9. Choosing the same coordinate system as in Example 10.3 (see Fig. 10-9), we see that the
lateral deflection of the projectile is in the x direction and that the acceleration is
ax = x = 2 ω z vy = 2 (ω sin λ )(V0 cos α )
(1)
Integrating this expression twice and using the initial conditions, x ( 0 ) = 0 and x ( 0 ) = 0 , we
obtain
x ( t ) = ω V0 t 2 cos α sin λ
(2)
Now, we treat the z motion of the projectile as if it were undisturbed by the Coriolis force. In
this approximation, we have
z ( t ) = V0 t sin α −
1 2
gt
2
(3)
from which the time T of impact is obtained by setting z = 0:
T=
2V0 sin α
g
(4)
Substituting this value for T into (2), we find the lateral deflection at impact to be
x (T ) =
4ωV03
sin λ cos α sin 2 α
2
g
(5)
10-10. In the previous problem we assumed the z motion to be unaffected by the Coriolis
force. Actually, of course, there is an upward acceleration given by −2ω x vy so that
z = 2ω V0 cos α cos λ − g
(1)
from which the time of flight is obtained by integrating twice, using the initial conditions, and
then setting z = 0:
T′ =
2V0 sin α
g − 2ωV0 cos α cos λ
(2)
Now, the acceleration in the y direction is
ay = y = 2ω x vz
= 2 ( −ω cos λ )(V0 sin α − gt )
Integrating twice and using the initial conditions, y ( 0 ) = V0 cos α and y ( 0 ) = 0 , we have
(3)
340
CHAPTER 10
y (t) =
1
ω gt 3 cos λ − ωV0 t 2 cos λ sin α + V0 t cos α
3
(4)
Substituting (2) into (4), the range R′ is
R′ =
4ωV03 sin 3 α cos λ
2V02 cos α cos λ
8 ωV03 g sin 3 α cos λ
−
+
3 ( g − 2ωV0 cos α cos λ ) 3 ( g − 2ωV0 cos α cos λ ) 2 g − 2ωV0 cos α cos λ
(5)
We now expand each of these three terms, retaining quantities up to order ω but neglecting all
quantities proportional to ω 2 and higher powers of ω. In the first two terms, this amounts to
neglecting 2ωV0 cos α cos λ compared to g in the denominator. But in the third term we must
use
2V02 cos α sin α
 2ωV0

cos α cos λ 
g 1 −
g


≅
 2ωV0

2V02
cos α sin α 1 +
cos α cos λ 
g
g


= R0′ +
4ωV03
sin α cos 2 α cos λ
2
g
(6)
where R′0 is the range when Coriolis effects are neglected [see Example 2.7]:
R0′ =
2V02
cos α sin α
g
(7)
The range difference, ∆R ′ = R ′ − R0′ , now becomes
∆R ′ =
4ωV03
1


cos λ  sin α cos 2 α − sin 3 α 
2


g
3
(8)
Substituting for V0 in terms of R′0 from (7), we have, finally,
∆R′ =
2R0′
1


ω cos λ  cot1 2 α − tan 3 2 α 


g
3
(9)
341
MOTION IN A NONINERTIAL REFERENCE FRAME
10-11.
d=
Rθ
R sin θ
θ
This problem is most easily done in the fixed frame, not the rotating frame. Here we take the
Earth to be fixed in space but rotating about its axis. The missile is fired from the North Pole at
some point on the Earth’s surface, a direction that will always be due south. As the missile
travels towards its intended destination, the Earth will rotate underneath it, thus causing it to
miss. This distance is:
∆ = (transverse velocity of Earth at current latitude) × (missile’s time of flight)
= ω R sin θ × T
=
(1)
dω R
 d
sin  
 R
v
(2)
Note that the actual distance d traveled by the missile (that distance measured in the fixed
frame) is less than the flight distance one would measure from the Earth. The error this causes
in ∆ will be small as long as the miss distance is small. Using R = 6370 km, ω = 7.27 × 10 −5
rad ⋅ s −1 , we obtain for the 4800 km, T = 600 s flight a miss distance of 190 km. For a 19300 km
flight the missile misses by only 125 km because there isn’t enough Earth to get around, or
rather there is less of the Earth to miss. For a fixed velocity, the miss distance actually peaks
somewhere around d = 12900 km.
Doing this problem in the rotating frame is tricky because the missile is constrained to be in a
path that lies close to the Earth. Although a perturbative treatment would yield an order of
magnitude estimate on the first part, it is entirely wrong on the second part. Correct treatment
in the rotating frame would at minimum require numerical methods.
10-12.
ε
r0
λ
z
Fs
x
342
CHAPTER 10
Using the formula
Feff = ma f − mω × ( ω × r ) − 2mω × v r
(1)
we try to find the direction of Feff when ma f (which is the true force) is in the direction of the z
axis. Choosing the coordinate system as in the diagram, we can express each of the quantities in
(1) as
vr = 0


ω = (−ω cos λ , 0, ω sin λ ) 


r = (0, 0, R)


ma f = (0, 0, − mg0 )

(2)
ω × r = Rω cos λ e y
(3)
Hence, we have
and (1) becomes
ex
ey
Feff = − mg0 ez − m −ω cos λ
0
ez
0
ω sin λ
Rω cos λ
0
(4)
from which, we have
Feff = − mg0 ez + mRω 2 sin λ cos λ ex + mRω 2 cos 2 λ e z
(5)
Therefore,
( Ff )x = mRω 2 sin λ cos λ



2
2
( Ff )z = − mg 0 + mRω cos λ 
(6)
The angular deviation is given by
tan ε =
( Ff )x
( Ff )z
=
Rω 2 sin λ cos λ
g 0 − Rω 2 cos 2 λ
(7)
Since ε is very small, we can put ε ≅ ε . Then, we have
ε=
Rω 2 sin λ cos λ
g 0 − Rω 2 cos 2 λ
(8)
It is easily shown that ε is a maximum for λ 45° .
Using R = 6.4 × 10 8 cm , ω = 7.3 × 10 −5 sec −1 , g = 980 cm/sec 2 , the maximum deviation is
ε≅
1.7
≅ 0.002 rad
980
(9)
343
MOTION IN A NONINERTIAL REFERENCE FRAME
10-13.
ω
ε
z
z′
Earth
x
x′
λ
The small parameters which govern the approximations that need to be made to find the
southerly deflection of a falling particle are:
height of fall
h
=
R radius of Earth
(1)
centrifugal force
Rω 2
=
g0
purely gravitational force
(2)
δ≡
and
α≡
The purely gravitational component is defined the same as in Problem 10-12. Note that
although both δ and α are small, the product δα = hω 2 g 0 is still of order ω 2 and therefore
expected to contribute to the final answer.
Since the plumb line, which defines our vertical direction, is not in the same direction as the
outward radial from the Earth, we will use two coordinate systems to facilitate our analysis. The
unprimed coordinates for the Northern Hemisphere-centric will have its x-axis towards the
south, its y-axis towards the east, and its z-axis in the direction of the plumb line. The primed
coordinates will share both its origin and its y′-axis with its unprimed counterpart, with the z′and x′-axes rotated to make the z′-axis an outward radial (see figure). The rotation can be
described mathematically by the transformation
x = x ′ cos ε + z ′ sin ε
(3)
y = y′
(4)
z = − x ′ sin ε + z ′ cos ε
(5)
Rω 2
sin λ cos λ
g
(6)
where
ε≡
as found from Problem 10-12.
a) The acceleration due to the Coriolis force is given by
a X ≡ −2ω × v ′
Since the angle between ω and the z′-axis is π – λ, (7) is most appropriately calculated in the
primed coordinates:
(7)
344
CHAPTER 10
x ′ = 2ω y ′ sin λ
(8)
y ′ = −2ω ( z ′ cos λ + x ′ sin λ )
(9)
z ′ = 2ω y ′ cos λ
(10)
In the unprimed coordinates, the interesting component is
x = 2ω y ( sin λ cos ε + cos λ sin ε )
(11)
At our level approximation this becomes
x 2ω y sin λ
(12)
Using the results for y and z , which is correct to order ω (also found from Example 10.3),
x 2ω 2 gt 2 sin λ cos λ
(13)
Integrating twice and using the zeroth order result for the time-of-fall, t = 2 h g , we obtain for
the deflection
dX =
2 h2 2
ω sin λ cos λ
3 g
(14)
b) The centrifugal force gives us an acceleration of
a c ≡ −ω × ( ω × r ′ )
(15)
The component equations are then
x ′ = ω 2 sin λ x ′ sin λ + ( R + z ′ ) cos λ 
(16)
y′ = ω 2 y′
(17)
z ′ = ω 2 cos λ x ′ sin λ + ( R + z ′ ) cos λ  − g0
(18)
where we have included the pure gravitational component of force as well. Now transform to
the unprimed coordinates and approximate
x ω 2 ( R + z ) sin λ cos λ − g0 sin ε
(19)
We can use Problem 10-12 to obtain sin ε to our level of approximation
sin ε ε Rω 2
sin λ cos λ
g0
(20)
The prompts a cancellation in equation (19), which becomes simply
x ω 2 z sin λ cos λ
(21)
Using the zeroth order result for the height, z = h − gt 2 2 , and for the time-of-fall estimates the
deflection due to the centrifugal force
dc 5 h2 2
ω sin λ cos λ
6 g
(22)
345
MOTION IN A NONINERTIAL REFERENCE FRAME
c) Variation in gravity causes the acceleration
ag ≡ −
GM
r + g0 k
r3
(23)
where r = x ′i + y ′j + ( R + z ′ ) k is the vector pointing to the particle from the center of the
spherical Earth. Near the surface
r 2 = x ′ 2 + y ′ 2 + ( z ′ + R) R2 + 2Rz ′
2
(24)
so that (23) becomes, with the help of the binomial theorem,
ag −
g0
( x ′i + y ′j − 2 z ′k )
R
(25)
Transform and get the x component
g0
( − x ′ cos ε + 2z ′ sin ε )
R
(26)
=
g0
 − ( x cos ε − z sin ε ) cos ε + 2 ( x sin ε + z cos ε ) sin ε 
R
(27)
g0
( − x + 3z sin ε )
R
(28)
x Using (20),
x 3ω 2 z sin λ cos λ
(29)
where we have neglected the x R term. This is just thrice the part (b) result,
dg 5 h2 2
ω sin λ cos λ
2 g
(30)
Thus the total deflection, correct to order ω 2 , is
d4
h2 2
ω sin λ cos λ
g
(31)
(The solution to this and the next problem follow a personal communication of Paul Stevenson,
Rice University.)
10-14. The solution to part (c) of the Problem 10-13 is modified when the particle is dropped
down a mineshaft. The force due to the variation of gravity is now
ag ≡ −
g0
r + g0 k
R
(1)
As before, we approximate r for near the surface and (1) becomes
ag −
In the unprimed coordinates,
g0
( x ′i + y ′j + z ′k )
R
(2)
346
CHAPTER 10
x − g0
x
R
(3)
To estimate the order of this term, as we probably should have done in part (c) of Problem
10-13, we can take x ~ h 2ω 2 g , so that
x ~ ω 2 h ×
h
R
(4)
which is reduced by a factor h R from the accelerations obtained previously. We therefore have
no southerly deflection in this order due to the variation of gravity. The Coriolis and centrifugal
forces still deflect the particle, however, so that the total deflection in this approximation is
d
10-15.
3 h2 2
ω sin λ cos λ
2 g
(5)
The Lagrangian in the fixed frame is
( )
1
mv 2f − U rf
2
L=
(1)
where v f and rf are the velocity and the position, respectively, in the fixed frame. Assuming
we have common origins, we have the following relation
v f = v r + ω × rr
(2)
where vr and rr are measured in the rotating frame. The Lagrangian becomes
L=
m 2
2
vr + 2v r ⋅ ( ω × rr ) + ( ω × rr )  − U ( rr )


2
(3)
∂L
= mv r + m ( ω × rr )
∂v r
(4)
The canonical momentum is
pr ≡
The Hamiltonian is then
H ≡ v r ⋅ pr − L =
1
1
2
mvr2 − U ( rr ) − m ( ω × rr )
2
2
(5)
H is a constant of the motion since ∂L ∂t = 0 , but H ≠ E since the coordinate transformation
equations depend on time (see Section 7.9). We can identify
Uc = −
1
2
m ( ω × rr )
2
(6)
as the centrifugal potential energy because we may find, with the use of some vector identities,
−∇U c =
m  2 2
2
∇ ω rr − ( ω ⋅ rr ) 


2
= m ω 2 rr − ( ω ⋅ rr ) ω 
(7)
(8)
347
MOTION IN A NONINERTIAL REFERENCE FRAME
= − mω × ( ω ⋅ rr )
(9)
which is the centrifugal force. Computing the derivatives of (3) required in Lagrange’s
equations
d ∂L
= ma r + mω × v r
dt ∂v r
∂L
= m∇ ( v r × ω ) ⋅ rr  − ∇ (U c + U )
∂rr
= − m ( ω × v r ) − mω × ( ω × rr ) − ∇U
(10)
(11)
(12)
The equation of motion we obtain is then
ma r = −∇U − mω × ( ω × rr ) − 2m ( ω × v r )
(13)
If we identify Feff = ma r and F = −∇U , then we do indeed reproduce the equations of motion
given in Equation 10.25, without the second and third terms.
10-16. The details of the forces involved, save the Coriolis force, and numerical integrations
in the solution of this problem are best explained in the solution to Problem 9-63. The only thing
we do here is add an acceleration caused by the Coriolis force, and re-work every part of the
problem over again. This is conceptually simple but in practice makes the computation three
times more difficult, since we now also must include the transverse coordinates in our
integrations. The acceleration we add is
a c = 2ω  vy sin λi − ( vx sin λ + vz cos λ ) j + vy cos λ k 
(1)
where we have chosen the usual coordinates as shown in Figure 10-9 of the text.
a) Our acceleration is
a = − gk + a C
(2)
As a check, we find that the height reached is 1800 km, in good agreement with the result of
Problem 9-63(a). The deflection at this height is found to be 77 km, to the west.
b) This is mildly tricky. The correct treatment says that the equation of motion with air
resistance is (cf. equation (2) of Problem 9-63 solution)

v
a = − g k + 2
vt


v  + aC

The deflection is calculated to be 8.9 km.
c)
Adding the vaiation due to gravity gives us a deflection of 10 km.
d)
Adding the variation of air density gives us a deflection of 160 km.
(3)
348
CHAPTER 10
Of general note is that the deflection in all cases was essentially westward. The usual small
deflection to the north did not contribute significantly to the total transverse deflection at this
precision. All of the heights obtained agreed well with the answers from Problem 9-63.
Inclusion of the centrifugal force also does not change the deflections to a significant degree at
our precision.
10-17. Due to the centrifugal force, the water surface of the lake is not exactly perpendicular
to the Earth’s radius (see figure).
B
m
g
β
Ta
n
ge
nt
Wa
ter
sur
β
to
fac
rth
A
α
C
e
Ea
su
r
fa
ce
The length BC is (using cosine theorem)
BC =
AC 2 + (mg ) 2 − 2 ACmg cos α
where AC is the centrifugal force AC = mω 2 R cos α with α = 47° and Earth’s radius
R ≅ 6400 km ,
The angle β that the water surface is deviated from the direction tangential to the Earth’s surface
is
BC
AC
=
sin α sin β
⇒ sin β =
AC sin α
= 4.3 × 10−5
BC
So the distance the lake falls at its center is h = r sin β where r = 162 km is the lake’s radius.
So finally we find h = 7 m.
349
MOTION IN A NONINERTIAL REFERENCE FRAME
10-18.
Let us choose the coordinate system Oxyz as shown in the figure.
α
O
νy
β
νx
y
ν
x
The projectile’s velocity is
 vx   v0 cos β 
G   
v = v y = v0 sin β − gt 
  

0
 0 

where β = 37°
The Earth’s angular velocity is
 −ω cos α 
ω =  −ω sin α 


0


G
where α = 50°
So the Coriolis acceleration is
G
G G
ac = 2v × ω = −2v0 ω cos β sin α + 2 ( v0 sin β − gt ) ω cos α e z
(
)
The velocity generated by Coriolis force is
t
vc = ∫ ac dt = 2v0 ω t ( cos β sin α − sin β cos α ) − gt 2ω cos α
0
And the distance of deviation due to the Coriolis force is
gt 3ω cos α
zc = ∫ vc dt = −v0 ω t sin (α − β ) −
3
0
t
2
The flight time of the projectile is t =
2v0 sin β
. If we put this into zc , we find the deviation
2
distance due to Coriolis force to be
zc ~ 260 m
350
CHAPTER 10
10-19.
The Coriolis force acting on the car is
G
G
G G
Fc = 2m v × ω ⇒ Fc = 2mvω sin α
where α = 65°, m = 1300 kg, v = 100 km/hr.
G
So Fc = 4.76 N.
10-20. Given the Earth’s mass, M = 5.976 × 1024 kg , the magnitude of the gravitational field
vector at the poles is
g pole =
GM
= 9.866 m/s 2
2
R pole
The magnitude of the gravitational field vector at the equator is
g eq =
GM
− ω 2 R e q = 9.768 m/s 2
2
Req
where ω is the angular velocity of the Earth about itself.
If one use the book’s formula, we have
g (λ = 90°) = 9.832 m/s 2 at the poles
and
g (λ = 0°) = 9.780 m/s 2 at the equator
10-21.
The Coriolis acceleration acting on flowing water is
G
G G
ac = 2v × ω ⇒
G
ac = 2vω sin α
Due to this force, the water is higher on the west bank. As in problem 10-17, the angle β that the
water surface is deviated from the direction tangential to Earth’s surface is
sin β =
ac
=
g +a
2
2
c
2vω sin α
g + 4v ω sin α ∝
2
2
2
2
= 2.5 × 10−5
The difference in heights of the two banks is
∆h = A sin β = 1.2 × 10−3 m
where A = 47 m is the river’s width.
10-22.
G
G
G
G
The Coriolis acceleration is ac = 2v × ω . This acceleration ac pushes lead bullets
G
eastward with the magnitude ac = 2vω cos α = 2 gt ω cos α , where α = 42°.
The velocity generated by the Coriolis force is
351
MOTION IN A NONINERTIAL REFERENCE FRAME
vc (t ) = ∫ a dt = gt 2ω cos α
and the deviation distance is
∆xc = ∫ vc (t ) dt =
The falling time of the bullet is t =
∆xc =
gt 3
ω cos α
3
2h g . So finally
ω 8h3
3
g
cos α = 2.26 × 10−3 m
352
CHAPTER 10
CHAPTER
11
Dynamics
of Rigid Bodies
11-1.
The calculation will be simplified if we use spherical coordinates:
x = r sin θ cos φ 

y = r sin θ sin φ 


z = r cos θ

(1)
z
y
x
Using the definition of the moment of inertia,

I ij = ∫ ρ ( r ) δ ij

∑x
k
2
k

− xi x j  dv

(2)
we have
(
)
I 33 = ρ ∫ r 2 − z 2 dv
(
)
= ρ ∫ r − r cos θ r dr d ( cos θ ) dφ
2
2
2
(3)
2
or,
R
I 33 = ρ ∫ r 4 dr
0
= 2πρ
2π
+1
∫ (1 − cos θ ) d ( cos θ ) ∫ dφ
2
−1
R5 4
⋅
5 3
0
(4)
353
354
CHAPTER 11
The mass of the sphere is
M=
4π
ρ R3
3
(5)
I 33 =
2
MR 2
5
(6)
Therefore,
Since the sphere is symmetrical around the origin, the diagonal elements of {I} are equal:
I11 = I 22 = I 33 =
2
MR2
5
(7)
A typical off-diagonal element is
I12 = ρ ∫ ( − xy ) dv
(8)
= − ρ ∫ r sin θ sin φ cos φ r dr d ( cos θ ) dφ
2
2
2
This vanishes because the integral with respect to φ is zero. In the same way, we can show that
all terms except the diagonal terms vanish. Therefore, the secular equation is
I11 − I
0
0
0
I 22 − I
0
0
0
I 33 − I
=0
(9)
From (9) and (7), we have
I1 = I 2 = I 3 =
2
MR2
5
(10)
11-2.
a) Moments of inertia with respect to the xi axes:
x3 = x3′
R h
CM
x2′
x1′
x2
x1
It is easily seen that I ij = 0 for i ≠ j. Then the diagonal elements I ii become the principal
moments I i , which we now calculate.
The computation can be simplified by noting that because of the symmetry, I1 = I 2 ≠ I 3 . Then,
I1 = I 2 =
I1 + I 2 ρ
=
2
2
∫ ( 2x
2
3
)
+ x12 + x22 dv
(1)
355
DYNAMICS OF RIGID BODIES
which, in cylindrical coordinates, can be written as
I1 = I 2 =
ρ
2∫
2π
0
h
Rz h
0
0
dφ ∫ dz ∫
(r
2
)
+ 2z 2 rdr
(2)
where
ρ=
M
3M
=
V π R2 h
(3)
Performing the integration and substituting for ρ, we find
I1 = I 2 =
(
3
M R2 + 4 h 2
20
)
(4)
I 3 is given by
(
)
I 3 = ρ ∫ x12 + x22 dv = ρ ∫ r 2 ⋅ rdr dφ dz
(5)
from which
I3 =
3
MR 2
10
(6)
b) Moments of inertia with respect to the xi′ axes:
Because of the symmetry of the body, the center of mass lies on the x′3 axis. The coordinates of
the center of mass are (0 , 0, z0 ) , where
z0 =
∫ x ′ dv = 3 h
∫ dv 4
3
(7)
Then, using Eq. (11.49),
I ij′ = I ij − M  a 2δ ij − ai a j 
(8)
In the present case, a1 = a2 = 0 and a3 = ( 3 4 ) h , so that
11-3.
I1′ = I1 −
9
3
1 

Mh 2 =
M  R2 + h2 
16
20 
4 
I 2′ = I 2 −
9
3
1 

Mh 2 =
M  R2 + h 2 
16
20 
4 
I 3′ = I 3 −
3
MR2
10
The equation of an ellipsoid is
x12 x22 x32
+
+
=1
a2 b 2 c2
(1)
356
CHAPTER 11
which can be written in normalized form if we make the following substitutions:
x1 = aξ , x2 = bη , x3 = cζ
(2)
ξ2 +η2 +ζ 2 = 1
(3)
Then, Eq. (1) reduces to
This is the equation of a sphere in the (ξ,η,ζ) system.
If we denote by dv the volume element in the xi system and by dτ the volume element in the
(ξ,η,ζ ) system, we notice that the volume of the ellipsoid is
V = ∫ dv = ∫ dx1 dx2 dx3 = abc ∫ dξ dη dζ
(4)
4
= abc ∫ dτ = π abc
3
because ∫ dτ is just the volume of a sphere of unit radius.
The rotational inertia with respect to the x3 -axis passing through the center of mass of the
ellipsoid (we assume the ellipsoid to be homogeneous), is given by
I3 =
=
M
V
∫(x
2
1
)
+ x22 dv
(
)
M
abc ∫ a 2ξ 2 + b 2η 2 dτ
V
(5)
In order to evaluate this integral, consider the following equivalent integral in which z = r cos θ :
∫a
2
z 2 dv = ∫ a 2 z 2 ( r dr r sin θ dθ dφ )
= a2
∫
2π
0
= a 2 × 2π ×
=
π
R =1
0
0
dφ ∫ cos 2 θ sin θ dθ ∫
r 4 dr
2 1
×
3 5
4π a 2
15
(6)
Therefore,
∫ (a ξ
2
2
)
+ b 2η 2 dτ =
4π 2
a + b2
15
(
)
(7)
and
I3 =
(
1
M a2 + b 2
5
)
Since the same analysis can be applied for any axis, the other moments of inertia are
(8)
357
DYNAMICS OF RIGID BODIES
I1 =
(
1
M b2 + c2
5
(
1
I 2 = M a2 + c2
5
)
(9)
)
11-4.
The linear density of the rod is
ρA =
m
A
(1)
For the origin at one end of the rod, the moment of inertia is
A
I = ∫ ρA x 2 dx =
0
m A3 m 2
= A
A 3 3
(2)
If all of the mass were concentrated at the point which is at a distance a from the origin, the
moment of inertia would be
I = ma 2
(3)
A
3
(4)
Equating (2) and (3), we find
a=
This is the radius of gyration.
11-5.
z
M
J
z–a
a
Q
a) The solid ball receives an impulse J; that is, a force F(t) is applied during a short interval of
time τ so that
J = ∫ F ( t ′ ) dt ′
(1)
dp
=F
dt
(2)
dL
=r×F
dt
(3)
The equations of motion are
358
CHAPTER 11
which, for this case, yield
∆p = ∫ F ( t ′ ) dt ′ = J
(4)
∆L = ∫ r × F ( t ′ ) dt ′ = r × J
(5)
Since p(t = 0) = 0 and L(t = 0) = 0, after the application of the impulse, we have
p = MVCM = J; L = I 0 ω = r × J = ( z − a ) J
ω
ω
(6)
so that
VCM =
J
M
(7)
and
ω=
J
ω
( z − a)
ω
I0
(8)
where I 0 = ( 2 5) Ma 2 .
The velocity of any point a on the ball is given by Eq. (11.1):
vα = VCM + ω × rα
(9)
For the point of contact Q, this becomes
v Q = VCM − ω a
=
J
J
J  5 ( z − a) 
1−
2a 
M 
(10)
Then, for rolling without slipping, v Q = 0 , and we have
2a = 5 ( z − a)
(11)
so that
z=
7
a
5
(12)
b) Many billiard tricks are performed by striking the ball at different heights and at different
angles in order to impart slipping and spinning motion (“English”). For the table not to
introduce spurious effects, the rail must be at such a height that the ball will be “reflected” upon
collision.
Consider the case in which the ball is incident normally on the rail, as in the diagram. We have
the following relationships:
359
DYNAMICS OF RIGID BODIES
y
VCM
x
Linear Momentum
Angular Momentum
Before Collision
After Collision
px = − MVCM
px′ = + MVCM
py = 0
py′ = 0
Lx = 0
Lx′ = 0
Ly = *
Ly′ = − Ly
Lz = 0
Lz′ = 0
* The relation between Ly and VCM depends on whether or not slipping occurs.
Then, we have
∆p = −2 px = J = 2 MVCM
(13)
∆L = −2Ly = 2I 0 ω = J ( z − a)
(14)
2I 0 ω = 2 MVCM ( z − a )
(15)
so that
from which
z−a=
I0 ω
2 Ma 2ω 2 a2ω
=
=
MVCM 5 MVCM 5 VCM
(16)
If we assume that the ball rolls without slipping before it contacts the rail, then VCM = ω a , and
we obtain the same result as before, namely,
z−a=
2
a
5
(17)
or,
z=
7
a
5
(18)
Thus, the height of the rail must be at a height of ( 2 5) a above the center of the ball.
11-6. Let us compare the moments of inertia for the two spheres for axes through the centers
of each. For the solid sphere, we have
Is =
2
MR 2
5
(see Problem 11-1)
(1)
360
CHAPTER 11
For the hollow sphere,
R sin θ
θ
2π
π
0
0
I h = σ ∫ dφ ∫ ( R sin θ ) R2 sin θ dθ
2
π
= 2πσ R 4 ∫ sin 3 θ dθ
0
8
= πσ R 4
3
or, using 4πσ R 2 = M , we have
Ih =
2
MR 2
3
(2)
Let us now roll each ball down an inclined plane. [Refer to Example 7.9.] The kinetic energy is
T=
1
1
M y 2 + Iθ 2
2
2
(3)
where y is the measure of the distance along the plane. The potential energy is
U = Mg ( A − y ) sin α
(4)
where A is the length of the plane and α is the angle of inclination of the plane. Now, y = Rθ, so
that the Lagrangian can be expressed as
L=
1
1 I 2
M y2 +
y + Mgy sin α
2
2 R2
(5)
where the constant term in U has been suppressed. The equation of motion for y is obtained in
the usual way and we find
y=
gMR 2 sin α
MR2 + I
(6)
Therefore, the sphere with the smaller moment of inertia (the solid sphere) will have the greater
acceleration down the plane.
361
DYNAMICS OF RIGID BODIES
11-7.
x
θ
R
r
d
φ
The force between the force center and the disk is, from the figure
F = − kr
(1)
Only the component along x does any work, so that the effective force is Fx = − kr sin φ = − kx .
This corresponds to a potential U = kx 2 2 . The kinetic energy of the disk is
T=
1
1
3
Mx 2 + Iθ 2 = Mx 2
2
2
4
(2)
where we use the result I = MR2 2 for a disk and dx = R dθ. Lagrange’s equations give us
3
+ kx = 0
Mx
2
(3)
This is simple harmonic motion about x = 0 with an angular frequency of oscillations
ω=
2k
3m
(4)
11-8.
x3′
h
w
x1′
x2′
d
We let x′3 be the vertical axis in the fixed system. This would be the axis (i.e., the hinge line) of
the door if it were properly hung (no self-rotation), as indicated in the diagram. The mass of the
door is M=ρwhd.
The moment of inertia of the door around the x3′ axis is
h
w
m
1
I3 =
dh ′ ∫ w ′ 2 d dw ′ = Mw 2
∫
whd 0
3
0
where the door is considered to be a thin plate, i.e., d w,h.
(1)
362
CHAPTER 11
The initial position of the self-closing door can be expressed as a two-step transformation,
starting with the position in the diagram above. The first rotation is around the x′1 -axis through
an angle θ and the second rotation is around the x′′
1 -axis through an angle ψ:
x3′
x3′
x3′
x3″
θ
w
x3 = x3″
w3 = w
h
x2′
x2′
x1′
x1′ = x1″
θ
x2″
x1″
ψ
x1
x2′
ψ x2
x2″
The x′1 -axes are the fixed-system axes and the xi -axes are the body system (or rotating) axes
which are attached to the door. Here, the Euler angle φ is zero.
The rotation matrix that transforms the fixed axes into the body axes ( xi′ → xi ) is just Eq. (11.99)
with φ = 0 and θ → – θ since this rotation is performed clockwise rather than counterclockwise
as in the derivation of Eq. (11.99):
 cos ψ

λ =  − sin ψ
 0
cos θ sin ψ
cos θ cos ψ
sin θ
− sin θ sin ψ 

− sin θ cos ψ 

cos θ
(2)
The procedure is to find the torque acting on the door expressed in the fixed coordinate system
and then to obtain the x3 component, i.e., the component in the body system. Notice that when
the door is released from rest at some initial angle ψ 0 , the rotation is in the direction to decrease
ψ. According to Eq. (11.119),
I 3ω 3 = N 3 = I 3 ψ
(3)
where ω 1 = ω 2 = 0 since φ = θ = 0 .
In the body ( xi ) system the coordinates of the center of mass of the door are
0
1 
R = w 
2
 h 
(4)
where we have set the thickness equal to zero. In the fixed ( x′i ) system, these coordinates are
obtained by applying the inverse transformation λ −1 to R; but λ −1 = λ t , so that
− w sin ψ


1

R ′ = λ R =  w cos θ cos ψ + h sin θ 
2
 − w sin θ cos ψ + h cos θ 
t
(5)
Now, the gravitational force acting on the door is downward, and in the x′i coordinate system
is
F ′ = − Mg e′3
(6)
363
DYNAMICS OF RIGID BODIES
There the torque on the door, expressed in the fixed system, is
N′ = R ′ × F ′
e1′

1

= − Mg  − w sin ψ
2

0
e2′
w cos θ cos ψ + h sin θ
0
e′3


− w sin θ cos ψ + h cos θ 

1
 w cos θ cos ψ + h sin θ 
1


= − Mg 
w sin ψ

2


0
(7)
so that in the body system we have
 w cos θ cos 2 ψ + h sin θ cos ψ + w cos θ sin 2 ψ 
1


N ′ = λN ′ = − Mg 
− h sin θ sin ψ

2


θ
ψ
sin
sin
w


(8)
Thus,
−N3 =
1
Mgw sin θ sin ψ
2
(9)
and substituting this expression into Eq. (3), we have
−
1
1
Mgw sin θ sin ψ = I 3 ψ = Mw 2 ψ
2
3
(10)
where we have used Eq. (1) for I 3 . Solving for ψ ,
ψ = −
3 g
sin θ sin ψ
2w
(11)
This equation can be integrated by first multiplying by ψ :
1
∫ ψ ψ dt = 2 ψ
=
2
=−
3 g
sin θ ∫ sin ψ ψ dt
2w
3 g
sin θ cos ψ
2w
(12)
where the integration constant is zero since cos ψ = 0 when ψ = 0 . Thus,
ψ = ±
3g
sin θ cos ψ
w
(13)
We must choose the negative sign for the radical since ψ < 0 when cos ψ > 0 . Integrating again,
from ψ = 90° to ψ = 0°,
0
∫π
2
dψ
=−
cos ψ
T
3g
sin θ ∫ dt
w
0
(14)
364
CHAPTER 11
where T = 2 sec. Rewriting Eq. (14),
π
dψ
=T
cos ψ
2
∫
0
3g
sin θ
w
(15)
Using Eq. (E. 27a), Appendix E, we find
1
Γ 
π 4
−1 2
∫ cos ψ dψ = 2  3 
0
Γ 
4
π
2
(16)
From Eqs. (E.20) and (E.23),
1 1
 1
Γ   = Γ 1  = 0.906
4 4
 4
1
Γ   = 3.624
4
(17)
And from Eqs. (E.20) and (E.24),
3 3
 3
Γ   = Γ 1  = 0.919
4 4
 4
3
Γ   = 1.225
4
(18)
dψ
π 3.624
=
= 2.62
2
1.225
cos ψ
(19)
Therefore,
π
2
∫
0
Returning to Eq. (15) and solving for sin θ,
sin θ =
w
2
× ( 2.62)
3 gT 2
(20)
Inserting the values for g, w(= 1m), and T(= 2 sec), we find
θ = sin −1 ( 0.058 )
or,
θ ≅ 3.33°
(21)
365
DYNAMICS OF RIGID BODIES
11-9.
y
C
C′
Q
P
a
θ R
x
O
The diagram shows the slab rotated through an angle θ from its equilibrium position. At
equilibrium the contact point is Q and after rotation the contact point is P. At equilibrium the
position of the center of mass of the slab is C and after rotation the position is C′.
Because we are considering only small departures from θ = 0, we can write
QP ≅ Rθ
(1)
Therefore, the coordinates of C′ are (see enlarged diagram below)
r = OA + AC ′
(2)
a

 sin θ − Rθ cos θ 
2


a

y =  R +  cos θ + Rθ sin θ 


2
(3)
so that

x = R +

C′
θ
Q
Rθ
θ
O
Consequently,
Rθ
C
A
P
366
CHAPTER 11

x =  R +

a

 cos θ − R cos θ + Rθ sin θ  θ
2

a

=  cos θ + Rθ sin θ  θ
2

a
 

y =  −  R +  sin θ + Rθ cos θ + R sin θ  θ


2


 a

=  − sin θ + Rθ cos θ  θ
 2

from which
 a2

x 2 + y 2 =  + R 2θ 2  θ 2
 4

(4)
The kinetic energy is
T=
(
)
1
1
M x 2 + y 2 + Iθ 2
2
2
(5)
where I is the moment of inertia of the slab with respect to an axis passing through the center of
mass and parallel to the z-axis:
I=
(
1
M A 2 + a2
12
)
(6)
Therefore,
T=
1
f1 (θ ) θ 2
2
(7)
where
 a2

f1 (θ ) = M  + R 2θ 2  + I
 4

(8)
U = Mgy = − f 2 (θ )
(9)
a


f 2 (θ ) = − Mg  R +  cos θ + Rθ sin θ 
2


(10)
The potential energy is
where
and where Eq. (3) has been used for y.
The Lagrangian is
L=
The Lagrange equation for θ is
1
f1 (θ ) θ 2 + f 2 (θ )
2
(11)
367
DYNAMICS OF RIGID BODIES
d ∂ L ∂L
−
=0
dt ∂θ ∂θ
(12)
Now,
∂L
= f1 (θ ) θ
∂θ
d ∂L
= f1 (θ ) θ + f1 (θ ) θ
dt ∂θ
  a2


=  M  + R 2θ 2  + I  θ + 2 MR 2 θ θ 2

  4

(13)
∂L 1
= f1′(θ ) θ 2 + f 2′ (θ )
∂θ 2
a


= MR 2 θ θ 2 + Mg  R +  sin θ − Rθ cos θ − R sin θ 


2


(14)
Combining, we find
  a2


2 2
2
2
 M  + R θ  + I  θ + MR θ θ − Mg  R +


  4

a

 sin θ − Rθ cos θ − R sin θ  = 0
2

(15)
For the case of small oscillations, θ 2 θ and θ 2 θ , so that Eq. (15) reduces to
a

Mg  R − 

2
θ +
θ =0
2
Ma
+I
4
(16)
The system is stable for oscillations around θ = 0 only if
a

Mg  R − 

2
= ω2 > 0
2
Ma
+I
4
(17)
This condition is satisfied if R − a 2 > 0 , i.e.,
a
2
(18)
a

Mg  R − 

2
2
Ma
1
+
M A2 + a2
4
12
(19)
R>
Then, the frequency is
ω=
Simplifying, we have
(
)
368
CHAPTER 11
a

12 g  R − 

2
ω=
(A
+ 4a2
2
(20)
)
According to Eqs. (9) and (10), the potential energy is
a


U (θ ) = Mg  R +  cos θ + Rθ sin θ 
2


(21)
This function has the following forms for R > a 2 and R < a 2 :
R>
U(θ)
a
2
−π/2

Mg  R +

a

2
U(θ)
R<
a
2
π/2
θ
θ
To verify that a stable condition exists only for R > a 2 , we need to evaluate ∂ 2U ∂θ 2 at θ = 0:
∂U
 a

= Mg  − sin θ + Rθ cos θ 
∂θ
 2

(22)
∂ 2U
 a

= Mg  − cos θ + R cos θ − Rθ sin θ 
∂θ 2
2


(23)
a
∂ 2U

= Mg  R − 
2

∂θ θ = 0
2
(24)
∂ 2U
a
> 0 if R >
2
∂θ
2
(25)
and
so that
11-10.
z
θ
m
R
When the mass m is at one pole, the z component of the angular momentum of the system is
Lz = Iω =
2
MR 2ω
5
After the mass has moved a distance vt = Rθ along a great circle on the surface of the sphere,
the z component of the angular momentum of the system is
(1)
369
DYNAMICS OF RIGID BODIES
2

Lz =  MR 2 + mR 2 sin 2 θ  φ
5

(2)
where φ is the new angular velocity. Since there is no external force acting on the system,
angular momentum must be conserved. Therefore, equating (1) and (2), we have
φ =
2
MR 2ω
5
(3)
2
MR 2 + mR2 sin 2 θ
5
Substituting θ = vt R and integrating over the time interval during which the mass travels from
one pole to the other, we have
t=
φ=
πR
2
MR2ω
5
V
∫
t=0
2
MR 2 + mR 2 sin 2 ( vt R)
5
dt
(4)
Making the substitutions,
vt R ≡ u, dt = ( R v ) du
(5)
we can rewrite (4) as
π
φ=∫
0
2
MR 2ω
5
R
du
2
v
2
2
2
MR + mR sin u
5
π 2
2Rω
=
v
du
∫ 1 + β sin
0
2
u
(6)
where β ≡ 5m 2 M and where we have used the fact that the integrand is symmetric around
u = π 2 to write φ as twice the value of the integral over half the range. Using the identity
sin 2 u =
1
(1 − cos 2u)
2
(7)
we express (6) as
2Rω
φ=
v
π 2
du
1  1
 1 + β  − β cos 2u
2
2
∫
0
(8)
or, changing the variable to x = 2u,
φ=
Rω
v
∫
π
0
dx
1  1

 1 + β  − β cos x
2
2
Now, we can use Eq. (E.15), Appendix E, to obtain
(9)
370
CHAPTER 11
 (1 + β ) tan ( x 2) 
2Rω
φ=
tan −1 

v 1+ β
1+ β


=
π Rω
v
π
0
2M
2 M + 5m
2M
2 M + 5m
= ωT
(10)
where T = π R v is the time required for the particle to move from one pole to the other.
If m = 0, (10) becomes
φ ( m = 0) = ωT
(11)
α = φ ( m = 0 ) − φ ( m)
(12)
Therefore, the angle of retardation is
or,

α = ω T 1 −


2M

2 M + 5m 
(13)
11-11.
a) No sliding:
2
2
P
P
From energy conservation, we have
mg
A
A 1
1
2
= mg + mvC.M.
+ Iω 2
2 2
2
2
(1)
where vCM is the velocity of the center of mass when one face strikes the plane; vC.M. is related
to ω by
vCM =
A
ω
2
(2)
I is the moment of inertia of the cube with respect to the axis which is perpendicular to one face
and passes the center:
I=
Then, (1) becomes
1
mA 2
6
(3)
371
DYNAMICS OF RIGID BODIES
mgA
2
1  Aω  1  mA 2  2 1
ω = mA 2ω 2
2 −1 = m   + 
2  2  2  6 
3
(
2
)
(4)
from which, we have
ω2 =
(
3 g
2 A
)
2 −1
(5)
b) Sliding without friction:
In this case there is no external force along the horizontal direction; therefore, the cube slides so
that the center of mass falls directly downward along a vertical line.
P
θ
P
While the cube is falling, the distance between the center of mass and the plane is given by
y=
A
cos θ
2
(6)
Therefore, the velocity of center of mass when one face strikes the plane is
=−
y
A
sin θ θ
2
1
1
= − Aθ = − Aω
2
2
0 =π 4
(7)
θ =π 4
From conservation of energy, we have
2
A
A 1  1 
11

mg
= mg + m  − Aω  +  mA 2  ω 2




2 2
2
2 6
2
(8)
from which we have
ω2 =
11-12.
12 g
5 A
(
)
2 −1
(9)
According to the definition of the principal moments of inertia,
(
)
(
)
(
)
I j + I k = ∫ ρ xi2 + xk2 dv + ∫ ρ xi2 + x 2j dv
= ∫ ρ x 2j + xk2 dv + 2 ∫ ρ xi2 dv
= I i + 2 ∫ ρ xi2 dv
since
∫ρ x
we have
2
i
dv > 0
(1)
372
CHAPTER 11
I j + Ik ≥ Ii
11-13.
(2)
We get the elements of the inertia tensor from Eq. 11.13a:
(
)
( )
( )
I11 = ∑ mα xα2 ,2 + xα2 ,3
α
( )
= 3m b 2 + 4 m 2b 2 + 2m b 2 = 13mb 2
Likewise I 22 = 16 mb 2 and I 33 = 15mb 2
I12 = I 21 = − ∑ mα xα ,1 xα ,2
α
( )
( )
= −4 m b 2 − 2m − b 2 = −2mb 2
Likewise I13 = I 31 = mb 2
and I 23 = I 32 = 4mb 2
Thus the inertia tensor is
 13 −2 1 
{I } = mb  −2 16 4 
 1 4 15 
2
The principal moments of inertia are gotten by solving
−2
1 
13 − λ


mb  −2
16 − λ
4 =0
 1
4
15 − λ 
2
Expanding the determinant gives a cubic equation in λ:
λ 3 − 44λ 2 + 622λ − 2820 = 0
Solving numerically gives
λ1 = 10.00
λ 2 = 14.35
λ 3 = 19.65
Thus the principal moments of inertia are I1 = 10 mb 2
I 2 = 14.35 mb 2
I 3 = 19.65 mb 2
To find the principal axes, we substitute into (see example 11.3):
373
DYNAMICS OF RIGID BODIES
(13 − λi ) ω1i − 2ω 2i + ω 3i = 0
−2 ω 1i + (16 − λ i ) ω 2 i + 4ω 3 i = 0
ω 1i + 4ω 2 i + (15 − λi ) ω 3 i = 0
For i = 1, we have ( λ1 = 10 )
3ω 11 − 2ω 21 + ω 31 = 0
−2ω 11 − 6ω 21 + 4ω 31 = 0
ω 11 − 4ω 21 + 5ω 31 = 0
Solving the first for ω 31 and substituting into the second gives
ω 11 = ω 21
Substituting into the third now gives
ω 31 = −ω 21
or
ω 11 : ω 21 : ω 31 = 1 : 1 : − 1
So, the principal axis associated with I1 is
1
( x + y − z)
3
Proceeding in the same way gives the other two principal axes:
i = 2:
−.81x + .29y − .52z
i = 3:
−.14 x + .77 y + .63z
We note that the principal axes are mutually orthogonal, as they must be.
11-14.
z
y
x
Let the surface of the hemisphere lie in the x-y plane as shown. The mass density is given by
ρ=
M
M
3M
=
=
V 2 π b 3 2π b 3
3
First, we calculate the center of mass of the hemisphere. By symmetry
374
CHAPTER 11
xCM = yCM = 0
zCM =
1
M
∫ ρz dv
v
Using spherical coordinates ( z = r cos θ , dv = r 2 sin θ dr dθ dφ ) we have
zCM =
2π
ρ
M
∫
π 2
dφ
φ =0
∫
sin θ cos θ dθ
θ =0
b
∫r
3
dr
r =0
 3 
1 1  3
2π )    b 4  = b
=
3(
 2π b 
2 4  8
We now calculate the inertia tensor with respect to axes passing through the center of mass:
z′ = z
y′
3
b
8
x′
By symmetry, I12 = I 21 = I13 = I 31 = I 23 = I 32 = 0 . Thus the axes shown are the principal axes.
Also, by symmetry I11 = I 22 . We calculate I11 using Eq. 11.49:
3 
I11 = J11 − M  v 
8 
2
(1)
where J11 = the moment of inertia with respect to the original axes
(
)
J11 = ρ ∫ y 2 + z 2 dv
v
(
)
= ∫ r 2 sin 2 θ sin 2 φ + r 2 cos 2 θ r 2 sin θ dr dθ dφ
v
=
3M
2π b 3
∫
r 4 dr
r =0
π 2
3 Mb 2
=
10π
=
 2π

2
2
2
∫  ∫ sin θ sin φ + cos θ dφ  sin θ dθ
θ = 0 φ = 0

π 2
b
∫ (π sin
3
θ =0
(
)
θ + 2π cos 2 θ sin θ ) dθ
2
Mb 2
5
Thus, from (1)
I11 = I 22 =
Also, from Eq. 11.49
2
9
83
Mb 2 −
Mb 2 =
Mb 2
5
64
320
375
DYNAMICS OF RIGID BODIES
I 33 = J 33 − M ( 0 ) = J 33
( I 33 = J 33 should be obvious physically)
So
(
)
I 33 = ρ ∫ x 2 + y 2 dv
v
= ρ ∫ r 4 sin 3 θ dr dθ dφ =
v
2
Mb 2
5
Thus, the principal axes are the primed axes shown
in the figure. The principal moments of inertia are
I11 = I 22 =
I 33 =
83
Mb 2
320
2
Mb 2
5
11-15.
P
θ
g
We suspend the pendulum from a point P which is a distance A from the center of mass. The
rotational inertia with respect to an axis through P is
I = MR02 + MA 2
(1)
where R0 is the radius of gyration about the center of mass. Then, the Lagrangian of the system
is
L = T −U =
Iθ 2
− MgA (1 − cos θ )
2
(2)
Lagrange’s equation for θ gives
Iθ + MgA sin θ = 0
(3)
For small oscillation, sin θ ≅ θ . Then,
MgA
θ =0
I
(4)
gA
θ =0
R + A2
(5)
θ +
or,
θ +
2
0
376
CHAPTER 11
from which the period of oscillation is
τ=
2π
ω
= 2π
R02 + A 2
gA
(6)
If we locate another point P′ which is a distance A′ from the center of mass such that the period
of oscillation is also τ, we can write
R02 + A 2 R02 + A ′ 2
=
gA
gA ′
(7)
from which R02 = AA ′ . Then, the period must be
τ = 2π
AA ′ + A 2
gA
(8)
A + A′
g
(9)
or,
τ = 2π
This is the same as the period of a simple pendulum of the length A + A′. Using this method, one
does not have to measure the rotational inertia of the pendulum used; nor is one faced with the
problem of approximating a simple pendulum physically. On the other hand, it is necessary to
locate the two points for which τ is the same.
11-16.
The rotation matrix is
 cos θ
( λ ) =  − sin θ
 0
sin θ
cos θ
0
0

0
1 
(1)
The moment of inertia tensor transforms according to
(Ι ′) = (λ ) (Ι ) (λ t )
(2)
That is
 cos θ
( I ′ ) =  − sin θ
 0
sin θ
cos θ
0
1
 ( A + B)
0  2
1
0   ( A − B)
2
1  
0


1
( A − B) 0 
2
 cos θ
1
( A + B) 0   sin θ

2

C  0
0


− sin θ
cos θ
0
0
0 
1 
377
DYNAMICS OF RIGID BODIES
 cos θ
=  − sin θ
 0
sin θ
cos θ
0
1
1
 2 ( A + B) cos θ + 2 ( A − B) sin θ
0 
1
1
0   ( A − B) cos θ + ( A + B) sin θ
2
2
1  
0


1
1
( A + B) sin θ + ( A − B) cos θ
2
2
1
1
− ( A − B) sin θ + ( A + B) cos θ
2
2
0
−

0

0

C


1
1
2
2
 2 ( A + B) cos θ + ( A − B) cos θ sin θ + 2 ( A + B) sin θ

1
1
=
− ( A − B) sin 2 θ + ( A − B) cos 2 θ

2
2

0


1
1

0
( A − B) cos 2 θ − ( A − B) sin 2 θ
2
2

1
1
( A + B) sin 2 θ − ( A − B) sin θ cos θ + ( A + B) cos 2 θ 0 

2
2
C
0


or
 1
 2 ( A + B) + ( A − B) cos θ sin θ

1
1
( I′ ) =  − ( A − B) sin 2 θ + ( A − B) cos 2 θ
 2
2

0


If θ = π 4 , sin θ = cos θ = 1
1
1
( A − B) cos 2 θ − ( A − B) sin 2 θ 0 
2
2

1
0
( A + B) − ( A − B) cos θ sin θ

2
C
0


(3)
2 . Then,
A 0 0 
( I′ ) =  0 B 0 
 0 0 C 
11-17.
x3
x2
x1
(4)
378
CHAPTER 11
The plate is assumed to have negligible thickness and the mass per unit area is ρs . Then, the
inertia tensor elements are
I11 = ρs
∫ (r
2
)
− x12 dx1 dx2
(
)
= ρs ∫ x22 + x32 dx1dx2 = ρs ∫ x22 dx1 dx2 ≡ A
(1)
I 22 = ρs
∫ (r
2
− x22 dx1 dx2 = ρs ∫ x12 dx1 dx2 ≡ B
I 33 = ρs
∫ (r
2
− x32 dx1 dx2 = ρs ∫ x12 + x22 dx1 dx2
)
(
)
(2)
)
(3)
Defining A and B as above, I 33 becomes
I 33 = A + B
(4)
Also,
I12 = ρs
∫ ( − x x ) dx
dx2 ≡ −C
(5)
I 21 = ρs
∫ ( − x x ) dx
dx2 = −C
(6)
I13 = ρs
∫ ( − x x ) dx
dx2 = 0 = I 31
(7)
I 23 = ρs
∫ ( − x x ) dx
dx2 = 0 = I 32
(8)
0 

0 
A + B 
(9)
1 2
2 1
1 3
2 3
1
1
1
1
Therefore, the inertia tensor has the form
 A −C
{I} =  −C B
 0
0
11-18. The new inertia tensor {I′} is obtained from {I} by a similarity transformation [see
Eq. (11.63)]. Since we are concerned only with a rotation around the x3 -axis , the transformation
matrix is just λ φ , as defined in Eq. (11.91). Then,
I′ = λ φ I λ φ−1
(1)
λ φ−1 = λ φt
(2)
where
Therefore, the similarity transformation is
 cos θ

I′ =  − sin θ
 0
sin θ
cos θ
0
0   A −C

0   −C B
1   0
0
0   cos θ

0   sin θ
A + B   0
Carrying out the operations and simplifying, we find
− sin θ
cos θ
0
0

0
1 
379
DYNAMICS OF RIGID BODIES

2
2
 A cos θ − C sin 2θ + B sin θ

1
{I′} =  −C cos 2θ + ( B − A) sin 2θ
2

0


−C cos 2θ +
1
( B − A) sin 2θ
2
A sin 2 θ + C sin 2θ + B cos 2 θ
0

0 

0 

A + B


(3)
Making the identifications stipulated in the statement of the problem, we see that
I11
′ = A′ , I 22
′ = B′ 

I12
′ = I 21
′ = −C ′

(4)
I 33
′ = A + B = A′ + B′
(5)
0 
 A ′ −C ′


0 
{I ′ } =  − C ′ B ′
 0
0
A′ + B′ 
(6)
and
Therefore
In order that x1 and x2 be principal axes, we require C′ = 0:
C cos 2θ −
1
( B − A) sin 2θ = 0
2
(7)
or,
tan 2θ =
2C
B− A
(8)
from which
θ=
1
 2C 
tan −1 
2
 B − A 
(9)
Notice that this result is still valid if A = B. Why? (What does A = B mean?)
11-19.
x2
θ = π/2
θ=π
η
θ
θ=0
x1
The boundary of the plate is given by r = keαθ . Any point (η,θ ) has the components
x1 = η cos θ 

x2 = η sin θ 
(1)
380
CHAPTER 11
The moments of inertia are
I1 = A = ρ ∫
π
0
∫
keαθ
0
x22 η dη dθ
π
keαθ
0
0
= ρ ∫ sin 2 θ dθ ∫
η 3 dη
The integral over θ can be performed by using Eq. (E.18a), Appendix E, with the result
ρk 4
P
2α
(2)
e 4πα − 1
16 1 + 4α 2
(3)
I1 = A =
where
P=
(
)
In the same way,
I2 = B = ρ ∫
π
0
∫
keαθ
0
x12 η dη dθ
π
keαθ
0
0
= ρ ∫ cos 2 θ dθ ∫
η 3 dη
(4)
Again, we use Eq. (E.18a) by writing cos 2 θ = 1 − sin 2 θ , and we find
I2 = B =
ρk 4
P ( 1 + 8α 2 )
2α
(5)
Also
I12 = −C = − ρ ∫
π
0
∫
keαθ
0
x1 x2 η dη dθ
π
keαθ
0
0
= − ρ ∫ cos θ sin θ dθ ∫
(6)
η dη
3
In order to evaluate the integral over θ in this case we write cos θ sin θ = (1 2) sin 2θ and use
Eq. (E.18), Appendix E. We find
I12 = −C = ρk 4 P
(7)
Using the results of problem 11-17, the entire inertia tensor is now known.
According to the result of Problem 11-18, the angle through which the coordinates must be
rotated in order to make {I} diagonal is
θ=
1
 2C 
tan −1 
2
 B − A 
(8)
Using Eqs. (2), (5), and (7) for A, B, C, we find
2C
1
=
B − A 2α
(9)
381
DYNAMICS OF RIGID BODIES
so that
tan 2θ =
1
2α
(10)
Therefore, we also have
1 + 4α 2
2θ
2α
sin 2θ =
cos 2θ =
1
1
1 + 4α
2
2α
1 + 4α 2






(11)
Then, according to the relations specified in Problem 11-18,
I1′ = A′ = A cos 2 θ − C sin 2θ + B sin 2 θ
(12)
Using cos 2 θ = (1 2)(1 + cos 2θ ) and sin 2 θ = (1 2)(1 − cos 2θ ) , we have
I1′ = A′ =
1
1
( A + B) + ( A − B) cos 2θ − C sin 2θ
2
2
(13)
Now,
A+B=

ρk 4 P
1 + 4α 2 ) 
(
α

A − B = −4αρk 4 P


(14)
Thus,
I1′ = A′ =
ρk 4 P
2α
1
1 + 4α 2 ) − 2αρk 4 P ×
− ρk 4 P ×
(
2
2α
1 + 4α
1 + 4α 2
(15)
or,
I1′ = A′ = ρk 4 P (Q − R)
(16)
1 + 4α 2 

2α


R = 1 + 4 a 2 
(17)
I 2′ = B′ = ρk 4 P (Q + R)
(18)
where
Q=
Similarly,
and, of course,
382
CHAPTER 11
I 3′ = A′ + B′ = I1′ + I 2′
(19)
We can also easily verify, for example, that I12
′ = −C ′ = 0 .
11-20. We use conservation of energy. When standing upright, the kinetic energy is zero.
Thus, the total energy is the potential energy
E = U1 = mg
(
b
2
b
is the height of the center of mass above the floor.)
2
When the rod hits the floor, the potential energy is zero. Thus
E = T2 =
1 2
Iω
2
where I is the rotational inertia of a uniform rod about an end. For a rod of length b,
mass/length σ:
b
I end = ∫ σ x 2 dx =
0
1 3 1
σ b = mb 2
3
3
Thus
T2 =
1
mb 2 ω 2
6
By conservation of energy
U1 = T2
mg
b 1
= mb 2 ω 2
2 6
ω=
11-21.
3g
b
Using I to denote the matrix whose elements are those of {I} , we can write
L = Iω
(11.54)
L ′ = I′ω ′
(11.54a)
We also have x ′ = λ x and x ′ = λ t x ′ and therefore we can express L and ω as
L = λ t L′
(11.55a)
ω = λ tω ′
(11.55b)
substituting these expressions into Eq. (11.54), we have
383
DYNAMICS OF RIGID BODIES
λ t L ′ = Iλ t ω ′
and multiplying on the left by λ,
λλ t L ′ = λ Iλ t ω ′
or
(
)
L ′ = λ Iλ t ω ′
by virture of Eq. (11.54a), we identify
I′ = λ Iλ t
11-22.
(11.61)
According to Eq. (11.61),
I ij′ = ∑ λ ik I kA λ A−j1
(1)
k ,l
Then,
tr {I′} = ∑ I ii′ = ∑∑ λ ik I kA λ A−i1
i
i
k ,A
= ∑ I kA ∑ λ A−i1 λ ik
k ,A
i
= ∑ I kA δ Ak = ∑ I kk
(2)
tr {I′} = tr {I}
(3)
k ,A
k
so that
This relation can be verified for the examples in the text by straightforward calculations.
Note: A translational transformation is not a similarity transformation and, in general, tr {I} is not
invariant under translation. (For example, tr {I} will be different for inertia tensors expressed in
coordinate system with different origins.)
11-23.
We have
I′ = λIλ −1
Then,
(1)
384
CHAPTER 11
I′ = λ Iλ −1
= λ × I × λ −1
= λλ −1 × I
so that,
I′ = I
(2)
This result is easy to verify for the various examples involving the cube.
11-24.
x2
–a/2
a/2
−
x1
3
a
2
The area of the triangle is A = 3 a 2 4 , so that the density is
ρ=
M
4M
=
A
3 a2
(1)
a) The rotational inertia with respect to an axis through the point of suspension (the origin) is
x2
x1
θ
(
)
I 3 = ρ ∫ x12 + x22 dx1 dx2
a2
= 2ρ
∫
0
=
0
∫
dx1
−
(x
2
1
+ x22 ) dx2
3
( a − 2 x1 )
2
3 4 1
ρ a = Ma 2
24
6
(2)
When the triangle is suspended as shown and when θ = 0, the coordinates of the center of mass
are (0, x2 , 0) , where
385
DYNAMICS OF RIGID BODIES
1
ρ x2 dx1 dx2
M∫
x2 =
2ρ
=
M
=−
a2
∫
0
dx1
0
∫
− 3 ( a − 2 x1 )
x2 dx2
a
(3)
2 3
The kinetic energy is
1 2 1
I 3θ =
Ma 2θ 2
2
12
(4)
Mga
(1 − cos θ )
2 3
(5)
Mga
1
cos θ
Ma 2θ 2 +
12
2 3
(6)
T=
and the potential energy is
U=
Therefore,
L=
where the constant term has been suppressed. The Lagrange equation for θ is
θ + 3
g
sin θ = 0
a
(7)
and for oscillations with small amplitude, the frequency is
ω=
3
g
a
(8)
b) The rotational inertia for an axis through the point of suspension for this case is
x2
x1
I 3′ = 2ρ
0
∫
dx2
3a
−
2
=
−
3
a
2
− x2
3
∫ (x
2
1
)
+ x22 dx1
0
5
Ma 2
12
(9)
The Lagrangian is now
L=
Mga
5
cos θ
Ma 2θ 2 +
24
3
(10)
386
CHAPTER 11
and the equation of motion is
θ +
12 g
sin θ = 0
5 3 a
(11)
so that the frequency of small oscillations is
ω=
12 g
5 3 a
(12)
which is slightly smaller than the previous result.
11-25.
x2
R
ρ
r
θ
x1
2ρ
The center of mass of the disk is (0, x2 ) , where



x2 =
2 ∫ x2 dx1 dx2 + ∫ x2 dx1 dx2 

M
lower
upper
 semicircle

semicircle


ρ 
=
ρ 
R
M  ∫ 0
=−
∫ ( r sin θ ) ⋅ rdrdθ + 2 ∫ ∫ π ( r sin θ ) ⋅ rdrdθ 
π
R
0
0
2π
2 ρR 3
3 M
(1)
Now, the mass of the disk is
1
1
M = ρ ⋅ π R 2 + 2ρ ⋅ π R 2
2
2
=
3
ρπ R2
2
(2)
so that
x2 = −
4
R
9π
(3)
The direct calculation of the rotational inertia with respect to an axis through the center of mass
is tedious, so we first compute I with respect to the x3 -axis and then use Steiner’s theorem.
387
DYNAMICS OF RIGID BODIES
I 3 = ρ ∫
 0
R
=
∫
π
0
r 2 ⋅ rdr dθ + 2 ∫
R
0
2π
∫π
r 2 ⋅ rdr dθ 

3
1
πρR 4 = MR2
4
2
(4)
Then,
I 0 = I 3 − Mx22
=
1
16
MR2 − M ⋅
R2
2
81π 2
=
1
32 

MR2 1 −
π 2 
2
81

(5)
When the disk rolls without slipping, the velocity of the center of mass can be obtained as
follows:
Thus
ρ
2ρ
θ
x2
CM
θ
R
xCM = Rθ − x2 sin θ
yCM = R − x2 cos θ
x CM = Rθ − x2 θ cos θ
y CM = x2 θ sin θ
( x
2
CM
)
2
+ y CM
= V 2 = R 2θ 2 + x22 θ 2 − 2θ 2 R x2 cos θ
V 2 = a 2θ 2
(6)
a = R 2 + x22 − 2R x2 cos θ
(7)
16
8
cos θ
−
2
81π
9π
(8)
where
Using (3), a can be written as
a = R 1+
The kinetic energy is
388
CHAPTER 11
T = Ttrans + Trot
1
1
Mv 2 + I 0θ 2
2
2
(9)
1
3 8

MR2θ 2  −
cos θ 
2
 2 9π

(10)
=
Substituting and simplifying yields
T=
The potential energy is
1

U = Mg  R + x2 cos θ 
2

1
8


MgR 1 −
cos θ 
2
 9π

(11)
1
8

3 8



MR  Rθ 2  −
cos θ  − g 1 −
cos θ  
2
 2 9π

 9π


(12)
=
Thus the Lagrangian is
L=
11-26.
Since ω φ = φ lies along the fixed x′3 -axis , the components of ω φ along the body axes
( xi ) are given by the application of the transformation matrix λ [Eqs. (11.98) and (11.99)]:
( )  φ  0 
 
( )  = φ  = λ 0 
 
 
( )  φ  φ 
 ωφ

ω
 φ
ω
 φ
1
1
2
2
(1)
3
3
Carrying out the matrix multiplication, we find
( )   sin ψ sin θ 
( )  = φ cos ψ sin θ 
( )   cos θ 
 ωφ

ω
 φ
ω
 φ
1
2
(2)
3
which is just Eq. (11.101a).
The direction of ω θ = θ coincides with the line of nodes and lies along the x′′′
axis. The
1
components of ωθ along the body axes are therefore obtained by the application of the
transformation matrix λ ψ which carries the xi′′′ system into the xi system:
 (ω θ ) 1 


(ω θ ) 2  = λ ψ
(ω θ ) 
3

θ 
 cos ψ 
  

 0  = θ  − sin ψ 
0 
 0 
 
(3)
389
DYNAMICS OF RIGID BODIES
which is just Eq. (11.101b).
Finally, since ω ψ lies along the body x3 -axis , no transformation is required:
( )  0 
( )  = ψ 0 
( )  1
ω
 ψ
ω
 ψ
ω
 ψ
1
(4)
2
3
which is just Eq. (11.101c).
Combining these results, we obtain
( ) + (ω ) + (ω ) 
( ) + (ω ) + (ω ) 
( ) + (ω ) + (ω ) 
 ωφ

ω =  ωφ

ω
 φ
1
θ 1
ψ 1
2
θ 2
ψ 2
3
θ 3
ψ 3
φ sin ψ sin θ + θ cos ψ 


= φ  φ cos ψ sin θ − θ sin ψ 


θ cos θ + ψ


(5)
which is just Eq. (11.02).
11-27.
x3′
x3
L
ω
x2
α
θ
Initially:
L1 = 0 = I1ω 1
L2 = L sin θ = I1ω 2 = I1ω sin α
L3 = L xos θ = I 3ω 3 = I 3ω cos α
Thus
tan θ =
L2 I1
= tan α
L3 I 3
From Eq. (11.102)
ω 3 = φ cos θ + ψ
Since ω 3 = ω cos α , we have
(1)
390
CHAPTER 11
φ cos θ = ω cos α − ψ
(2)
From Eq. (11.131)
ψ = −Ω = −
I 3 − I1
ω3
I1
(2) becomes
φ cos θ =
I3
ω cos α
I1
(3)
From (1), we may construct the following triangle
I3 tan α
θ
I3
from which cos θ =
I3
 I + I tan 2 α 
2
3
2
1
12
Substituting into (3) gives
φ =
ω
I1
I12 sin 2 α + I 32 cos 2 α
From Fig. 11-7c we see that ω φ = φ is along the x3′ -axis , ω θ = θ is along the line of
nodes, and ω ψ = ψ is along the x3 -axis . Then,
11-28.
ω φ′ = φ e′3
(1)
whee e′3 is the unit vector in the x3′ direction.
Projecting the lines of nodes into the x1′ - and x2′ -axes , we obtain
ω θ′ = θ ( e1′ cos φ + e′2 sin φ )
(2)
ωψ′ has components along all three of the xi′ axes. First, we write ω ψ′ in terms of a component
along the x3′ -axis and a component normal to this axis:
ω ψ′ = ψ ( e12
′ sin θ + e′3 cos θ )
(3)
e12
′ = e1′ sin φ − e′2 cos φ
(4)
ω ψ′ = ψ ( e1′ sin θ sin φ − e′2 sin θ cos φ + e′3 cos θ )
(5)
where
Then,
391
DYNAMICS OF RIGID BODIES
Collecting the various components, we have
ω 1′ = θ cos φ + ψ sin θ sin φ
ω 2′ = θ sin φ − ψ sin θ cos φ
(6)
ω 3′ = ψ cos θ + φ
11-29.
When the motion is vertical θ = 0. Then, according to Eqs. (11.153) and (11.154),
(
)
Pφ = I 3 φ + ψ = Pψ
(1)
Pψ = Pφ = I 3 ω 3
(2)
and using Eq. (11.159), we see that
Also, when θ = 0 (and θ = 0 ), the energy is [see Eq. (11.158)]
E=
1
I 3 ω 32 + Mgh
2
(3)
Furthermore, referring to Eq. (11.160),
E′ = E −
1
I 3 ω 32 = Mgh
2
(3)
If we wish to examine the behavior of the system near θ = 0 in order to determine the conditions
for stability, we can use the values of Pψ , Pφ , and E′ for θ = 0 in Eq. (11.161). Thus,
I 32 ω 32 (1 − cos θ )
1
+ Mgh cos θ
Mgh = I12θ 2 +
2
2I12 sin 2 θ
2
(5)
Changing the variable to z = cos θ and rearranging, Eq. (5) becomes
z 2 =
(1 − z ) 2 
2
2
2
 2 Mgh I12 (1 + z ) − I 3 w3 
I12
(6)
The questions concerning stability can be answered by examining this expression. First, we note
that for physically real motion we must have z 2 ≥ 0 . Now, suppose that the top is spinning very
rapidly, i.e., that ω 3 is large. Then, the term in the square brackets will be negative. In such a
case, the only way to maintain the condition z 2 ≥ 0 is to have z = 1, i.e., θ = 0. Thus, the motion
at θ = 0 will be stable as long as
4 Mgh I12 − I 32 ω 32 < 0
(7)
4 Mgh I12
<1
I 32 ω 32
(8)
or,
392
CHAPTER 11
Suppose now that the top is set spinning with θ = 0 but with ω 3 sufficiently small that the
condition in Eq. (8) is not met. Any small disturbance away from θ = 0 will then give z a
negative value and θ will continue to increase; i.e., the motion is unstable. In fact, θ will
continue until z reaches a value z0 that again makes the square brackets equal to zero. This is a
turning point for the motion and nutation between z = 1 and z = z0 will result.
From this discussion it is evident that there exists a critical value for the angular velocity, ω c ,
such that for ω 3 > ω c the motion is stable and for ω < ω c there is nutation:
2 Mgh I12
ωc =
(9)
I3
If the top is set spinning with ω 3 > ω c and θ = 0, the motion will be stable. But as friction slows
the top, the critical angular velocity will eventually be reached and nutation will set in. This is
the case of the “sleeping top.”
11-30.
If we set θ = 0 , Eq. (1.162) becomes
(P − P
E′ = V (θ ) =
φ
(
ψ
cos θ
)
2
2 I12 1 − cos 2 θ
)
+ Mgh cos θ
(1)
Re-arranging, this equation can be written as
( 2 Mgh I12 ) cos3 θ − ( 2E′ I12 + Pψ2 ) cos2 θ + 2 ( Pφ Pψ − Mgh I12 ) cos θ + ( 2E′ I12 − Pφ2 ) = 0
(2)
which is cubic in cos θ.
V(θ ) has the form shown in the diagram. Two of the roots occur in the region −1 ≤ cos θ ≤ 1 , and
one root lies outside this range and is therefore imaginary.
V(θ)
+1
–1
cos θ
393
DYNAMICS OF RIGID BODIES
11-31.
The moments of inertia of the plate are



I2 =


I 3 = I1 + I 2


= I 2 (1 + cos 2α ) 


= 2I 2 cos 2 α

I1 = I 2 cos 2α
(1)
We also note that
I1 − I 2 = − I 2 (1 − cos 2α )
(2)
= −2I 2 sin α
2
Since the plate moves in a force-free manner, the Euler equations are [see Eq. (11.114)]
( I1 − I 2 ) ω1ω 2 − I 3ω 3 = 0 

( I 2 − I 3 ) ω 2ω 3 − I1ω 1 = 0 

( I 3 − I1 ) ω 3ω1 − I 2ω 2 = 0 
(3)
Substituting (1) and (2) into (3), we find
( −2I
)
(
)
sin 2 α ω 1ω 2 − 2I 2 cos 2 α ω 3 = 0 


( − I 2 cos 2α ) ω 2ω 3 − ( I 2 cos 2α ) ω 1 = 0 


I 2 ω 3ω 1 − I 2ω 2 = 0

2
(4)
These equations simplify to
ω 3 = −ω 1ω 2 tan 2 α 
ω 1 = −ω 2ω 3
ω 2 = ω 3ω 1





(5)
From which we can write
ω 1ω 2ω 3 = ω 2ω 2 = −ω 1ω 1 = −ω 3ω 3 cot 2 α
(6)
ω 22 − ω 22 ( 0 ) = −ω 12 + ω 12 ( 0 ) = −ω 32 cot 2 α + ω 32 ( 0 ) cot 2 α
(7)
Integrating, we find
Now, the initial conditions are
394
CHAPTER 11
ω 1 ( 0 ) = Ω cos α 


ω 2 ( 0) = 0


ω 3 ( 0 ) = Ω sin α 
(8)
ω 22 = −ω 12 + Ω 2 cos 2 α = −ω 32 cot 2 α + Ω 2 cos 2 α
(9)
ω 22 = ω 32 ω 12
(10)
Therefore, the equations in (7) become
From (5), we can write
and from (9), we have ω 12 = ω 32 cot 2 α . Therefore, (10) becomes
ω 2 = ω 32 cot α
(11)
and using ω 32 = Ω 2 sin 2 α − ω 22 tan 2 α from (9), we can write (11) as
ω 2
= − cot α
ω tan α − Ω 2 sin 2 α
2
2
2
(12)
Since ω 2 = dω 2 dt , we can express this equation in terms of integrals as
∫ω
2
2
dω 2
= − cot α ∫ dt
tan 2 α − Ω 2 sin 2 α
(13)
Using Eq. (E.4c), Appendix E, we find
−
 ω tan α 
1
tanh −1  2
 = −t cot α
( tan α )( Ω sin α )
 Ω sin α 
(14)
Solving for ω 2 ,
ω 2 ( t ) = Ω cos α tanh ( Ω t sin α )
11-32.
a) The exact equation of motion of the physical pendulum is
Iθ + MgL sin θ = 0
where I = Mk 2 , so we have
θ = −
gL
sin θ
k2
or
gL d ( cos θ )
d θ = 2
dt
k
dθ
()
or
(15)
395
DYNAMICS OF RIGID BODIES
θ d (θ ) =
gL
d ( cos θ )
k2
so
θ 2 =
2 gL
cos θ + a
k2
where a is a constant determined by the initial conditions. Suppose that at t = 0 , θ = θ 0 and at
−2 gl
that initial position the angular velocity of the pendulum is zero, we find a = 2 cos θ 0 . So
k
finally
θ =
2 gl
( cos θ − cos θ 0 )
k2
b) One could use the conservation of energy to find the angular velocity of the pendulum at
any angle θ , but it is exactly the result we obtained in a), so at θ = 10 , we have
ω = θ =
2 gL
( cos θ − cos θ 0 ) = 53.7 s−1
k2
11-33. Cats are known to have a very flexible body that they can manage to twist around to a
feet-first descent while falling with conserved zero angular momentum. First they thrust their
back legs straight out behind their body and at the same time they tuck their front legs in.
Extending their back legs helps to resist spinning, since rotation velocity evidently is inversely
proportional to inertia momentum. This allows the cat to twist their body differently to preserve
zero angular momentum: the front part of the body twisting more than the back. Tucking the
front legs encourages spinning to a downward direction preparing for touchdown and as this
happens, cats can easily twist the rear half of their body around to catch up with the front.
However, whether or not cats land on their feet depends on several factors, notably the distance
they fall, because the twist maneuver takes a certain time, apparently around 0.3 sec. Thus the
minimum height required for cats falling is about 0.5m.
11-34. The Euler equation, which describes the rotation of an object about its symmetry axis,
say 0x, is
(
)
I x ω x − I y − I z ω yω z = N x
where N x = − b ω x is the component of torque along Ox. Because the object is symmetric about
Ox, we have I y = I z , and the above equation becomes
Ix
dω x
= −b ω x
dt
⇒ ωx = e
−
b
t
Ix
ω x0
396
CHAPTER 11
CHAPTER
12
Coupled Oscillations
12-1.
m1 = M
m2 = M
k1
k12
k2
x1
x2
The equations of motion are
1 + (κ 1 + κ 12 ) x1 − κ 12 x2 = 0 
Mx

Mx2 + (κ 2 + κ 12 ) x2 − κ 12 x1 = 0 
(1)
We attempt a solution of the form
x1 ( t ) = B1 e iωt 

x2 ( t ) = B2 e iωt 
(2)
Substitution of (2) into (1) yields
(κ
)
+ κ 12 − Mω 2 B1 − κ 12 B2 = 0 


−κ 12 B1 + κ 2 + κ 12 − Mω 2 B2 = 0 
1
(
)
(3)
In order for a non-trivial solution to exist, the determinant of coefficients of B1 and B2 must
vanish. This yields
(
(
)
)
2
κ 1 + κ 12 − Mω 2  κ 2 + κ 12 − Mω 2  = κ 12



(4)
from which we obtain
ω2 =
κ 1 + κ 2 + 2κ 12
2M
±
1
2M
(κ 1 − κ 2 )2 + 4κ 122
(5)
This result reduces to ω 2 = (κ + κ 12 ± κ 12 ) M for the case κ 1 = κ 2 = κ (compare Eq. (12.7)].
397
398
CHAPTER 12
If m2 were held fixed, the frequency of oscillation of m1 would be
2
ω 01
=
1
(κ 1 + κ 12 )
M
(6)
while in the reverse case, m2 would oscillate with the frequency
2
ω 02
=
1
(κ 2 + κ 12 )
M
(7)
Comparing (6) and (7) with the two frequencies, ω + and ω − , given by (5), we find
ω +2 =
>
1 
κ 1 + κ 2 + 2κ 12 +
2M 
(κ 1 − κ 2 )2 + 4κ 122


1
1
κ 1 + κ 2 + 2κ 12 + (κ 1 − κ 2 )  =
(κ 1 + κ 12 ) = ω 021
2M
M
(8)
so that
ω + > ω 01
(9)
Similarly,
ω −2 =
<
1
2M
κ + κ + 2κ −
2
12
 1
(κ 1 − κ 2 )2 + 4κ 122


1
1
κ 1 + κ 2 + 2κ 12 − (κ 1 − κ 2 )  =
(κ 2 + κ 12 ) = ω 022
2M
M
(10)
so that
ω − < ω 02
(11)
If κ 1 > κ 2 , then the ordering of the frequencies is
ω + > ω 01 > ω 02 > ω −
12-2.
(12)
From the preceding problem we find that for κ 12 κ 1 , κ 2
ω1 ≅
κ 1 + κ 12
M
; ω2 ≅
κ 2 + κ 12
M
(1)
If we use
ω 01 =
κ1
M
then the frequencies in (1) can be expressed as
; ω 02 =
κ2
M
(2)
399
COUPLED OSCILLATIONS
ω 1 = ω 01 1 +

κ 12
≅ ω 01 (1 + ε 1 ) 
κ1


ω 2 = ω 02
(3)

κ
1 + 12 ≅ ω 02 (1 + ε 2 ) 
κ2

where
ε1 =
κ 12
κ
; ε 2 = 12
2κ 1
2κ 2
(4)
For the initial conditions [Eq. 12.22)],
x1 ( 0 ) = D, x2 ( 0 ) = 0, x 1 ( 0 ) = 0, x 2 ( 0 ) = 0 ,
(5)
the solution for x1 ( t ) is just Eq. (12.24):
ω + ω2
x1 ( t ) = D cos  1
 2

ω − ω2
t  cos  1

 2

t

(6)
Using (3), we can write
ω 1 + ω 2 = (ω 01 + ω 02 ) + ( ε1ω 01 + ε 2ω 02 )
≡ 2Ω + + 2 ε +
ω 1 − ω 2 = (ω 01 − ω 02 ) + ( ε 1ω 01 − ε 2ω 02 )
≡ 2Ω − + 2ε −
(7)
(8)
Then,
x1 ( t ) = D cos ( Ω + t + ε + t ) cos ( Ω − t + ε − t )
(9)
Similarly,
ω + ω2
x2 ( t ) = D sin  1
 2

ω − ω2
t  sin  1

 2

t

= D sin ( Ω + t + ε + t ) sin ( Ω − t + ε − t )
(10)
Expanding the cosine and sine functions in (9) and (10) and taking account of the fact that ε +
and ε − are small quantities, we find, to first order in the ε’s,
x1 ( t ) ≅ D cos Ω + t cos Ω − t − ε + t sin Ω + t cos Ω − t − ε − t cos Ω + t sin Ω − t 
(11)
x2 ( t ) ≅ D sin Ω + t sin Ω − t + ε + t cos Ω + t sin Ω − t + ε − t sin Ω + t cos Ω − t 
(12)
When either x1 ( t ) or x2 ( t ) reaches a maximum, the other is at a minimum which is greater
than zero. Thus, the energy is never transferred completely to one of the oscillators.
400
12-3.
CHAPTER 12
The equations of motion are
m

x2 + ω 02 x1 = 0 
M


m
x2 +
x2 + ω 02 x2 = 0 
M

(1)
x1 ( t ) = B1 e iωt ; x2 ( t ) = B2 e iωt
(2)
x1 +
We try solutions of the form
We require a non-trivial solution (i.e., the determinant of the coefficients of B1 and B2 equal to
zero), and obtain
2
2
(ω 02 − ω 2 ) − ω 4  m  = 0
M
(3)
so that
ω 02 − ω 2 = ±ω 2
m
M
(4)
and then
ω2 =
ω 02
(5)
m
1±
M
Therefore, the frequencies of the normal modes are
ω 02
ω1 =
1+
ω
ω2 =
1−
m
M
(6)
2
0
m
M
where ω 1 corresponds to the symmetric mode and ω 2 to the antisymmetric mode.
By inspection, one can see that the normal coordinates for this problem are the same as those for
the example of Section 12.2 [i.e., Eq. (12.11)].
12-4.
The total energy of the system is given by
E = T +U
=
Therefore,
(
)
(
)
1
1
1
2
M x 12 + x 22 + κ x12 + x22 + κ 12 ( x2 − x1 )
2
2
2
(1)
401
COUPLED OSCILLATIONS
dE
= M ( x 1 x1 + x 2 x2 ) + κ ( x1 x 1 + x2 x 2 ) + κ 12 ( x 2 − x 1 )( x2 − x1 )
dt
1 + κ x1 − κ 12 ( x2 − x1 )   x 1 +  Mx2 + κ x2 + κ 12 ( x2 − x1 )   x 2
=  Mx
1 + (κ + κ 12 ) x1 − κ 12 x2  x 1 +  Mx2 − κ 12 x1 + (κ + κ 12 ) x2  x 2
=  Mx
(2)
which exactly vanishes because the coefficients of x 1 and x 2 are the left-hand sides of Eqs.
(12.1a) and (12.1b).
An analogous result is obtained when T and U are expressed in terms of the generalized
coordinates η1 and η2 defined by Eq. (12.11):
(
)
T=
1
M η 12 + η 22
4
U=
1
1
κ η12 + η22 + κ 12η12
4
2
(
)
(3)
(4)
Therefore,
2⋅
dE
1 + (κ + 2κ 12 ) η1  η 1 + [ Mη
2 + κη2 ] η 2
=  Mη
dt 
(5)
which exactly vanishes by virtue of Eqs. (12.14).
When expressed explicitly in terms of the generalized coordinates, it is evident that there is only
one term in the energy that has κ 12 as a coefficient (namely, κ 12η12 ), and through Eq. (12.15) we
see that this implies that such a term depends on the C1 ’s and ω 1 , but not on the C2 ’s and ω 2 .
To understand why this is so, it is sufficient to recall that η1 is associated with the
anitsymmetrical mode of oscillation, which obviously must have κ 12 as a parameter. On the
other hand, η2 is associated to the symmetric mode, x1 ( t ) = x2 ( t ) , x 1 ( t ) = x 2 ( t ) , in which both
masses move as if linked together with a rigid, massless rod. For this mode, therefore, if the
spring connecting the masses is changed, the motion is not affected.
12-5.
We set κ 1 = κ 2 = κ 12 ≡ κ . Then, the equations of motion are
m1 x1 + 2κ x1 − κ x2 = 0 

m2 x2 + 2κ x2 − κ x1 = 0 
(1)
x1 ( t ) = B1e iωt 

iω t 
x2 ( t ) = B2 e 
(2)
Assuming solutions of the form
we find that the equations in (1) become
402
CHAPTER 12
( 2κ − m ω ) B
− κ B2 = 0 


−κ B1 + 2κ − m2ω 2 B2 = 0 
2
1
1
(
(3)
)
which lead to the secular equation for ω 2 :
( 2κ − m ω )( 2κ − m ω ) = κ
2
2
1
2
(4)
2
Therefore,
ω2 =
κ
µ


3µ
1 ± 1 −

m1 + m2 

(5)
where µ = m1 m2 ( m1 + m2 ) is the reduced mass of the system. Notice that (5) agrees with Eq.
(12.8) for the case m1 = m2 = M and κ 12 = κ . Notice also that ω 2 is always real and positive since
the maximum value of 3µ ( m1 + m2 ) is 3 4 . (Show this.)
Inserting the values for ω 1 and ω 2 into either of the equations in (3), we find
1

m1
2 −
µ



3µ
1 + 1 −
  a11 = a21
m1 + m2  

(6)


m 
3µ
a12 =  2 − 1 1 − 1 −
  a22
µ 
m1 + m2  

(7)
and
Using the orthonormality condition produces
a11 =
2−
a21 =
1
D1
(8)
m1 + m2 
3m1m2
1 + 1 −
m2 
( m1 + m2 ) 2

D1



(9)
where
D1 ≡ 2 ( m2 − m1 ) + 2
m12
2
m12 − m22
+
m2 m2
(
)
1−
3m1m2
( m1 + m2 ) 2
(10)
The second eigenvector has the components
2−
a12 =
1
m1 + m2 
3m1m2
1 − 1 −
m2 
( m1 + m2 ) 2

D2



Recall that when we use ω = ω 1 , we call the coefficients β1 (ω = ω 1 ) = a11 and β 2 (ω = ω 1 ) = a21 , etc.
(11)
403
COUPLED OSCILLATIONS
1
D2
a22 =
(12)
where
 m2 
D2 ≡ 2m1 1 + 12  + m2
 m2 
 3m12 
1 − 2  + 2m1
m2 

 m12 
3m1 m2
1 − 2  1 −
( m1 + m2 )2
 m2 
(13)
The normal coordinates for the case in which q j ( 0 ) = 0 are
η1 ( t ) = ( m1 a11 x10 + m2 a21 x20 ) cos ω 1t
η2 ( t ) = ( m1 a12 x10 + m2 a22 x20 ) cos ω 2t
(14)
12-6.
k
m
k
m
x1
x2
If the frictional force acting on mass 1 due to mass 2 is
f = − β ( x 1 − x 2 )
(1)
1 + β ( x 1 − x 2 ) + κ x1 = 0 
mx

mx2 + β ( x2 − x1 ) + κ x2 = 0 
(2)
then the equations of motion are
Since the system is not conservative, the eigenfrequencies will not be entirely real as in the
previous cases. Therefore, we attempt a solution of the form
x1 ( t ) = B1eα t ;
x2 ( t ) = B2 eα t
(3)
where α = λ + iω is a complex quantity to be determined. Substituting (3) into (1), we obtain the
following secular equation by setting the determinant of the coefficients of the B’s equal to zero:
( mα
2
+ βα + κ
)
2
= β 2α 2
(4)
from which we find the two solutions
α1 = ± i
κ
κ 
ω1 = ±

m 

1
2

−β ± β − mκ
α2 =

m
(
The general solution is therefore
m
;
)
(5)
404
CHAPTER 12
x1 ( t ) = B11+ e
i κ mt
+ B11− e
i κ mt
(
+ e − βt m B12+ e
β 2 − mκ t m
+ B12− e
β 2 − mκ t m
)
(6)
and similarly for x2 ( t ) .
The first two terms in the expression for x1 ( t ) are purely oscillatory, whereas the last two terms
+
contain the damping factor e − βt . (Notice that the term B12
exp
(
)
β 2 − mκ t increases with time if
β 2 > mκ , but B12+ is not required to vanish in order to produce physically realizable motion
because the damping term, exp(–βt), decreases with time at a more rapid rate; that is
− β + β 2 − mκ < 0 .)
To what modes do α 1 and α 2 apply? In Mode 1 there is purely oscillating motion without
friction. This can happen only if the two masses have no relative motion. Thus, Mode 1 is the
symmetric mode in which the masses move in phase. Mode 2 is the antisymmetric mode in which
the masses move out of phase and produce frictional damping. If β 2 < mκ , the motion is one of
damped oscillations, whereas if β 2 > mκ , the motion proceeds monotonically to zero amplitude.
12-7.
k
m
x1
k
m
x2
We define the coordinates x1 and x2 as in the diagram. Including the constant downward
gravitational force on the masses results only in a displacement of the equilibrium positions and
does not affect the eigenfrequencies or the normal modes. Therefore, we write the equations of
motion without the gravitational terms:
1 + 2κ x1 − κ x2 = 0 
mx

2 + κ x2 − κ x1 = 0 
mx
(1)
Assuming a harmonic time dependence for x1 ( t ) and x2 ( t ) in the usual way, we obtain
( 2κ − mω ) B
− κ B2 = 0 


−κ B1 + κ − mω 2 B2 = 0 
2
1
(
)
Solving the secular equation, we find the eigenfrequencies to be
(2)
405
COUPLED OSCILLATIONS
ω 12 =
ω 22 =
3+ 5 κ
2 m
3− 5 κ
2 m
(3)
Substituting these frequencies into (2), we obtain for the eigenvector components

1− 5
a11 = a21 
2


1+ 5
a12 = a22 

2
(4)
For the initial conditions x 1 ( 0 ) = x 2 ( 0 ) = 0 , the normal coordinates are

η1 ( t ) = ma11  x10 +


1− 5
x20  cos ω 1t
2



1+ 5
x20  cos ω 2t
η2 ( t ) = ma12  x10 +
2


(5)
Therefore, when x10 = −1.6180 x20 , η2 ( t ) = 0 and the system oscillates in Mode 1, the
antisymmetrical mode. When x10 = 0.6180 x20 , η1 ( t ) = 0 and the system oscillates in Mode 2, the
symmetrical mode.
When mass 2 is held fixed, the equation of motion of mass 1 is
mx1 + 2κ x1 = 0
(6)
and the frequency of oscillation is
ω 10 =
2κ
m
(7)
When mass 1 is held fixed, the equation of motion of mass 2 is
2 + κ x2 = 0
mx
(8)
and the frequency of oscillation is
ω 20 =
κ
m
Comparing these frequencies with ω 1 and ω 2 we find
ω1 =
3+ 5
4
ω2 =
3− 5
4

2κ
2κ
= 1.1441
> ω 10 
m
m



κ
κ
= 0.6180
< ω 20 
m
m

(9)
406
CHAPTER 12
Thus, the coupling of the oscillators produces a shift of the frequencies away from the
uncoupled frequencies, in agreement with the discussion at the end of Section 12.2.
12-8. The kinetic and potential energies for the double pendulum are given in Problem 7-7. If
we specialize these results to the case of small oscillations, we have
T=
(
1
mA 2 2φ12 + φ22 + 2φ1 φ2
2
U=
(
1
mgA 2φ12 + φ 22
2
)
)
(1)
(2)
where φ1 refers to the angular displacement of the upper pendulum and φ 2 to the lower
pendulum, as in Problem 7-7. (We have also discarded the constant term in the expression for
the potential energy.)
Now, according to Eqs. (12.34),
T=
1
∑ mjk q j qk
2 j,k
(3)
U=
1
∑ Ajk qj qk
2 j,k
(4)
Therefore, identifying the elements of {m} and {A} , we find
 2 1

 1 1
(5)
2 0

0 1
(6)
{m} = mA 2 
{A} = mgA 
and the secular determinant is
2
g
− 2ω 2
A
−ω
2
−ω 2
=0
g
− ω2
A
(7)
or,
 g
2  g
2
4
 2 − 2ω   − ω  − ω = 0
A
A

(8)
Expanding, we find
2
g
g
ω − 4 ω2 + 2   = 0
A
A
4
which yields
(9)
407
COUPLED OSCILLATIONS
(
ω2 = 2 ± 2
) gA
(10)
and the eigenfrequencies are
g
= 1.848
A
ω1 = 2 + 2
g
A
(11)
g
= 0.765
A
ω2 = 2 − 2
g
A
To get the normal modes, we must solve
∑(A
jk
)
− ω r2 m jk a jr = 0
j
For k = 1, this becomes:
(A
11
)
(
)
− ω r2 m11 a1r + A21 − ω r2 m21 a2 r = 0
For r = 1:
(
)
(
)
g
g

2
2
 2mgA − 2 + 2 A 2mA  a11 − 2 + 2 A mA a21 = 0
Upon simplifying, the result is
a21 = − 2 a11
Similarly, for r = 2, the result is
a22 = 2 a12
The equations
x1 = a11 η1 + a12 η2
x2 = a21 η1 + a22 η2
can thus be written as
x1 = a11 η1 +
1
a22 η2
2
x1 = − 2 a11 η1 + a22 η2
Solving for η1 and η2 :
η1 =
2 x1 − x2
; η2 =
2 2 a11
2 x1 + x2
2 a22
408
CHAPTER 12
η1 occurs when η2 = 0; i.e. when x1 = −
η2 occurs when η1 = 0; i.e. when x1 =
x2
2
x2
2
Mode 2 is therefore the symmetrical mode in which both pendula are always deflected in the
same direction; and Mode 1 is the antisymmetrical mode in which the pendula are always
deflected in opposite directions. Notice that Mode 1 (the antisymmetrical mode), has the higher
frequency, in agreement with the discussion in Section 12.2.
12-9. The general solutions for x1 ( t ) and x2 ( t ) are given by Eqs. (12.10). For the initial
conditions we choose oscillator 1 to be displaced a distance D from its equilibrium position,
while oscillator 2 is held at x2 = 0 , and both are released from rest:
x1 ( 0 ) = D, x2 ( 0 ) = 0, x 1 ( 0 ) = 0, x 2 ( 0 ) = 0
(1)
Substitution of (1) into Eq. (12.10) determines the constants, and we obtain
x1 ( t ) =
D
( cos ω1t + cos ω 2t )
2
(2)
x2 ( t ) =
D
( cos ω 2t − cos ω1t )
2
(3)
where
ω1
κ + 2κ 12
M
> ω2 =
κ
M
(4)
As an example, take ω 1 = 1.2 ω 2 ; x1 ( t ) vs. x2 ( t ) is plotted below for this case.
It is possible to find a rotation in configuration space such that the projection of the system
point onto each of the new axes is simple harmonic.
By inspection, from (2) and (3), the new coordinates must be
x1′ ≡ x1 − x2 = D cos ω 1t
(5)
x2′ ≡ x1 + x2 = D cos ω 2t
(6)
These new normal axes correspond to the description by the normal modes. They are
represented by dashed lines in the graph of the figure.
409
COUPLED OSCILLATIONS
x2′
x2(t)/D
ω 2t =
5π
2
ω1 = 1.2 ω2
4π
0.8
0.6
ω 2t =
π
2
0.4
ω2t = 2π
π
ω 2t =
3
0.2
1.0
0.2
ω 2t = π
0.4
0.6
7π
2
–0.5
ω 2t =
ω 2t = 0
0.8
1.0
x1(t)/D
3π
2
3π
x1′
12-10.
The equations of motion are
1 + bx 1 + (κ + κ 12 ) x1 − κ 12 x2 = F0 cos ω t 
mx


2 + bx 2 + (κ + κ 12 ) x2 − κ 12 x1 = 0
mx

(1)
The normal coordinates are the same as those for the undamped case [see Eqs. (12.11)]:
η1 = x1 − x2 ;
η2 = x1 + x2
(2)
Expressed in terms of these coordinates, the equations of motion become
2 + η
1 ) + b (η 2 + η 1 ) + (κ + κ 12 )(η2 + η1 ) − κ 12 (η2 − η1 ) = 2F0 cos ω t 
m (η


m (η2 − η1 ) + b (η2 − η1 ) + (κ + κ 12 )(η2 − η1 ) − κ 12 (η2 + η1 ) = 0

(3)
By adding and subtracting these equations, we obtain the uncoupled equations:
F
b
κ + 2κ 12

η 1 +
η1 = 0 cos ω t 
m
m
m


F
b
κ

η2 + η 2 + η2 = 0 cos ω t
m
m
m

η1 +
With the following definitions,
(4)
410
CHAPTER 12
2β =
ω 12 =
ω =
2
2
A=
b
m
κ + 2κ 12
m
κ
m
F0
m













(5)
the equations become
η1 + 2βη 1 + ω 12η1 = A cos ω t
η2 + 2βη 2 + ω 22η2 = A cos ω t
Referring to Section 3.6, we see that the solutions for η1 ( t ) and η2 ( t ) are exactly the same as
that given for x(t) in Eq. (3.62). As a result η1 ( t ) exhibits a resonance at ω = ω 1 and η2 ( t )
exhibits a resonance at ω = ω 2 .
12-11.
Taking a time derivative of the equations gives ( q = I )
I
LI1 + 1 + MI2 = 0
C
I
LI2 + 2 + MI1 = 0
C
Assume I1 = B1e iω t , I 2 = B2 e iωt ; and substitute into the previous equations. The result is
−ω 2 LB1e iω t +
1
B1 e iωt − Mω 2 B2 e iωt = 0
C
−ω 2 LB2 e iωt +
1
B2 e iωt − Mω 2 B1e iωt = 0
C
These reduce to
1

B1  − ω 2 L  + B2 − Mω 2 = 0
C

(
)
1

B1 − Mω 2 + B2  − ω 2 L  = 0
C

(
)
This implies that the determinant of coefficients of B1 and B2 must vanish (for a non-trivial
solution). Thus
(6)
411
COUPLED OSCILLATIONS
1
− ω 2L
C
− Mω 2
− Mω 2
1
− ω 2L
C
2
1
2 
2
 C − ω L  − Mω
(
)
=0
2
=0
1
− ω 2 L = ± Mω 2
C
or
ω2 =
1
C (L ± M)
Thus
12-12.
ω1 =
1
C (L + M)
ω2 =
1
C (L − M)
From problem 12-11:
1
LI1 + I1 + MI2 = 0
C
(1)
1
LI2 + I 2 + MI1 = 0
C
(2)
Solving for I1 in (1) and substituting into (2) and similarly for I 2 , we have


M 2  1
M
−
L
I1 + I1 −
I2 = 0 


L 
C
CL




M 2  1
M
 L − L  I 2 + C I 2 − CL I1 = 0 

(3)




M

κ 12 =

LC

M 
1
κ = 1 −  
C
L 
(4)
If we identify
m= L−
M2
L
412
CHAPTER 12
then the equations in (3) become
mI1 + (κ + κ 12 ) I1 − κ 12 I 2 = 0 


mI 2 + (κ + κ 12 ) I 2 − κ 12 I1 = 0 
(5)
which are identical in form to Eqs. (12.1). Then, using Eqs. (12.8) for the characteristic
frequencies, we can write
M
1
L
=
ω1 =
2

M 
C (L − M)
CL −

L 

1+
(6)
M
1−
1
L
ω2 =
=
2

M 
C (L + M)
CL −
L 

which agree with the results of the previous problem.
12-13.
q1
C1
L1
q2
C2
L12
I1
I2
L2
The Kirchhoff circuit equations are
q1

+ L12 I1 − I2 = 0 
C1


q
L2 I2 + 2 + L12 I2 − I1 = 0 

C2
L1 I1 +
(
(
)
)
(1)
Differentiating these equations using q = I , we can write
1

I1 − L12
I2 = 0 
C1


1
( L2 + L12 ) I 2 + I 2 − L12 I1 = 0 
C2

( L1 + L12 ) I1 +
(2)
As usual, we try solutions of the form
I1 ( t ) = B1 e iωt ;
which lead to
I 2 ( t ) = B2 e iωt
(3)
413
COUPLED OSCILLATIONS

 2
1
2
ω ( L1 + L12 ) −  B1 − L12 ω B2 = 0 
C1 




 2
1
2
− L12 ω B1 + ω ( L2 + L12 ) −  B2 = 0 
C1 


(4)
Setting the determinant of the coefficients of the B’s equal to zero, we obtain
 2
1  2
1 
4 2
ω ( L1 + L12 ) −  ω ( L2 + L12 ) −  = ω L12
C
C
1 
2

(5)
with the solution
ω
2
( L1 + L12 ) C1 + ( L2 + L12 ) C2 ± ( L1 + L12 ) C1 − ( L2 + L12 ) C2 
=
2C1C2 ( L1 + L12 )( L2 + L12 ) − L212 
2
+ 4 L212C1C2
(6)
We observe that in the limit of weak coupling ( L12 → 0 ) and L1 = L2 = L , C1 = C2 = C , the
frequency reduces to
ω=
1
LC
(7)
which is just the frequency of uncoupled oscillations [Eq. (3.78)].
12-14.
L1
C1
L2
I1
C12
I2
C2
The Kirchhoff circuit equations are (after differentiating and using q = I )

1
1 
1
L1
I1 +  +
I2 = 0 
 I1 −
C12
 C1 C12 





1
1
1
L2
I2 +  +
I1 = 0 
 I2 −
C12

 C2 C12 
(1)
Using a harmonic time dependence for I1 ( t ) and I 2 ( t ) , the secular equation is found to be

C1 + C12  
C2 + C12 
1
2
2
 L1ω −
  L2ω −
= 2
C1 C12  
C2 C12  C12

(2)
Solving for the frequency,
C1L1 ( C2 + C12 ) + C2 L2 ( C1 + C12 ) ± C1L1 ( C2 + C12 ) − C2 L2 ( C1 + C12 )  + 4C12C22 L1L2
ω =
2L1L2C1C2C12
2
2
(3)
414
CHAPTER 12
Because the characteristic frequencies are given by this complicated expression, we examine the
normal modes for the special case in which L1 = L2 = L and C1 = C2 = C . Then,
ω 12 =
2C + C12
LCC12
(4)
1
ω 22 =
LC
Observe that ω 2 corresponds to the case of uncoupled oscillations. The equations for this
simplified circuit can be set in the same form as Eq. (12.1), and consequently the normal modes
can be found in the same way as in Section 12.2. There will be two possible modes of oscillation:
(1) out of phase, with frequency ω 1 , and (2) in phase, with frequency ω 2 .
Mode 1 corresponds to the currents I1 and I 2 oscillating always out of phase:
I2
I1
;
I2
I1
Mode 2 corresponds to the currents I1 and I 2 oscillating always in phase:
I2
I1
;
I2
I1
(The analogy with two oscillators coupled by a spring can be seen by associating case 1 with
Fig. 12-2 for ω = ω 1 and case 2 with Fig. 12-2 for ω = ω 2 .) If we now let L1 ≠ L2 and C1 ≠ C2 , we
do not have pure symmetrical and antisymmetrical symmetrical modes, but we can associate
ω 2 with the mode of highest degree of symmetry and ω 1 with that of lowest degree of
symmetry.
12-15.
C1
L1
I1
C2
R
I2
L2
Setting up the Kirchhoff circuit equations, differentiating, and using q = I , we find
(
)
1

L1
I1 + R I1 − I2 +
I1 = 0 
C1


1
L2
I 2 + R I2 − I1 +
I2 = 0 

C2
(
)
(1)
Using a harmonic time dependence for I1 ( t ) and I 2 ( t ) , the secular equation is
 2
 2

1
1
2 2
 ω L1 − C − iω R  ω L2 − C − iω R + ω R = 0
1
2
(2)
415
COUPLED OSCILLATIONS
From this expression it is clear that the oscillations will be damped because ω will have an
imaginary part. (The resistor in the circuit dissipates energy.) In order to simplify the analysis,
we choose the special case in which L1 = L2 = L and C1 = C2 = C . Then, (2) reduces to
2
1
 2

2 2
 ω L − − iω R + ω R = 0
C
(3)
which can be solved as in Problem 12-6. We find
ω1 = ±
i
ω2 =
L
1
LC
(4)

L 
2
R ± R − 
C 

The general solution for I1 ( t ) is
+
I1 ( t ) = B11
e
i 1 LC t
−
e
+ B11
− i 1 LC t
+
e
+ e − Rt L B12

R2 − L C t L
−
e
+ B12
− R2 − L C t L


(5)
and similarly for I 2 ( t ) . The implications of these results follow closely the arguments presented
in Problem 12-6.
Mode 1 is purely oscillatory with no damping. Since there is a resistor in the circuit, this means
that I1 and I 2 flow in opposite senses in the two parts of the circuit and cancel in R. Mode 2 is the
mode in which both currents flow in the same direction through R and energy is dissipated. If
R2 < L C , there will be damped oscillations of I1 and I 2 , whereas if R2 > L C , the currents will
decrease monotonically without oscillation.
12-16.
y
O
x
R
θ
P
φ
Mg
R
Q(x,y)
Mg
Let O be the fixed point on the hoop and the origin of the coordinate system. P is the center of
mass of the hoop and Q(x,y) is the position of the mass M. The coordinates of Q are
x = R ( sin θ + sin φ )



y = − R ( cos θ + cos φ ) 
The rotational inertia of the hoop through O is
(1)
416
CHAPTER 12
I O = I CM + MR 2 = 2 MR 2
(2)
The potential energy of the system is
U = U hoop + U mass
= − MgR ( 2 cos θ + cos φ )
(3)
Since θ and φ are small angles, we can use cos x ≅ 1 − x 2 2 . Then, discarding the constant term in
U, we have
U=
(
1
MgR 2θ 2 + φ 2
2
)
(4)
The kinetic energy of the system is
T = Thoop + Tmass
=
(
1 2 1
I Oθ + M x 2 + y 2
2
2
= MR 2θ 2 +
)
(5)
1

MR 2 θ 2 + φ 2 + 2θφ

2
where we have again used the small-angle approximations for θ and φ. Thus,
T=
1

MR 2  3θ 2 + φ 2 + 2θφ

2
(6)
T=
1
∑ mjk q j qk
2 j,k
(7)
U=
1
∑ Ajk qj qk
2 j,k
(8)
Using Eqs. (12.34),
we identify the elements of {m} and {A} :
 3 1

 1 1
(9)
2 0

0 1 
(10)
{m} = MR2 
{A} = MgR 
The secular determinant is
2
g
− 3ω 2
R
−ω
from which
2
−ω 2
g
− ω2
R
=0
(11)
417
COUPLED OSCILLATIONS
 g
2  g
2
4
 2 − 3ω   − ω  − ω = 0
R
R
(12)
Solving for the eigenfrequencies, we find
ω1 = 2
ω2 =
2
2
g
R
(13)
g
R
To get the normal modes, we must solve:
∑(A
jk
)
− ω r2 m jk a jr = 0
j
For k = r = 1, this becomes:
g
g

2
2
 2mgR − 2 3mR  a11 − 2 mR a21 = 0
R
R
or
a21 = −2a11
For k = 1, r = 2, the result is
a12 = a22
Thus the equations
x1 = a11 η1 + a12 η2
x2 = a21 η1 + a22 η2
can be written as
x1 = a11 η1 + a22 η2
x2 = −2 a11 η1 + a22 η2
Solving for η1 , η2
η1 =
x1 − x2
;
3a11
η1 =
2x1 + x2
3a22
1
2
η1 occurs when the initial conditions are such that η2 = 0 ; i.e., x10 = − x20
This is the antisymmetrical mode in which the CM of the hoop and the mass are on opposite
sides of the vertical through the pivot point.
η2 occurs when the initial conditions are such that η1 = 0 ; i.e., x10 = x20
418
CHAPTER 12
This is the symmetrical mode in which the pivot point, the CM of the hoop, and the mass
always lie on a straight line.
12-17.
k
k
m
k
m
x1
k
m
x2
x3
Following the procedure outlined in section 12.6:
1
1
1
mx 12 + mx 22 + mx 32
2
2
2
T=
U=
1 2 1
1
1
2
2
kx1 + k ( x2 − x1 ) + k ( x3 − x2 ) + kx32
2
2
2
2
= k  x12 + x22 + x32 − x1 x2 − x2 x3 
Thus
m 0 0 


m= 0 m 0
 0 0 m 
 2k − k 0 


A =  − k 2k − k 
 0 − k 2k 
Thus we must solve
2k − ω 2 m
−k
−k
2k − ω m
−k
0
−k
2k − ω m
0
2
=0
2
This reduces to
( 2 k − ω m)
2
3
(
)
− 2k 2 2k − ω 2 m = 0
or
( 2k − ω m) ( 2k − ω m)
2
2
2
− 2k 2  = 0

If the first term is zero, then we have
ω1 =
2k
m
If the second term is zero, then
2k − ω 2 m = ± 2 k
419
COUPLED OSCILLATIONS
which leads to
(2 + 2 ) k ;
ω2 =
ω3 =
m
(2 − 2) k
m
To get the normal modes, we must solve
∑(A
jk
)
− ω r2 m jk a jr = 0
j
For k = 1 this gives:
( 2 k − ω m) a
2
r
1r
+ ( − k ) a2 r = 0
Substituting for each value of r gives
r = 1:
( 2k − 2k ) a11 − ka21 = 0 → a21 = 0
r = 2:
(−
r = 3:
(
)
2 k a12 − ka22 = 0 → a22 = − 2 a12
)
2 k a13 − ka23 = 0 → a23 = 2 a13
Doing the same for k = 2 and 3 yields
a11 = − a31
a21 = 0
a12 = a32
a22 = − 2 a32
a13 = a33
a23 = 2 a33
The equations
x1 = a11 η1 + a12 η2 + a13 η3
x2 = a21 η1 + a22 η2 + a23 η3
x3 = a31 η1 + a32 η2 + a33 η3
can thus be written as
x1 = a11 η1 −
1
a22 η2 + a33 η3
2
x2 = a22 η2 + 2 a33 η3
x3 = − a11 η1 −
1
a22 η2 + a33 η3
2
We get the normal modes by solving these three equations for η1 , η2 , η3 :
420
CHAPTER 12
η1 =
x1 − x3
2a11
η2 =
− x1 + 2 x2 − x3
2 2 a22
and
η3 =
x1 + 2 x2 + x3
4 a33
The normal mode motion is as follows
η1 :
• →
•
← •
η2 : ← •
• → ← •
η3 :
• →
• →
x1 = − x3
x2 = − 2 x1 = − 2 x3
• → x2 = 2 x1 = 2 x3
12-18.
x
M
θ
b
m
y1
x1
x1 = x + b sin θ ; x 1 = x + bθ cos θ
y1 = b − b cos θ ;
y 1 = bθ sin θ
Thus
T=
=
(
1
1
Mx 2 + m x 12 + y 12
2
2
)
(
1
1
Mx 2 + m x 2 + b 2 θ 2 + 2b xθ cos θ
2
2
)
U = mgy1 = mgb (1 − cos θ )
For small θ, cos θ 1 −
θ2
2
. Substituting and neglecting the term of order θ 2θ gives
421
COUPLED OSCILLATIONS
(
T=
1
1
( M + m) x 2 + m b 2θ 2 + 2b xθ
2
2
U=
mgb 2
θ
2
)
Thus
 M + m mb 
m=
mb 2 
 mb
0 
0
A=

0 mgb 
We must solve
−ω 2 ( M + m)
−ω 2 mb
−ω 2 mb
mgb − ω 2 mb 2
=0
which gives
ω 2 ( M + m) (ω 2 mb 2 − mgb ) − ω 4 m2 b 2 = 0
ω 2 ω 2 Mb 2 − mgb ( m + M )  = 0
Thus
ω1 = 0
ω2 =
∑(A
jk
g
( M + m)
mb
)
− ω r2 m jk a jr = 0
j
Substituting into this equation gives
( k = 2, r = 1)
a21 = 0
a12 = −
bm
a
( m + M ) 22
( k = 2, r = 2)
Thus the equations
x = a11 η1 + a12 η2
θ = a21 η1 + a22 η2
become
x = a11 η1 −
mb
a η
( m + M ) 22 2
422
CHAPTER 12
θ = a22 η2
Solving for η1 , η2 :
η2 =
θ
a22
x+
n1 =
bm
θ
(m + M)
a11
n1 occurs when n2 = 0; or θ = 0
n2 occurs when n1 = 0; or x = −
12-19.
bm
(m + M)
θ
With the given expression for U, we see that { A} has the form
− ε 12
 1
{A} =  −ε12
 − ε 13
1
− ε 23
− ε 13 

− ε 23 
1 
(1)
The kinetic energy is
T=
(
1 2 2 2
θ1 + θ 2 + θ 3
2
)
(2)
so that {m} is
1 0 0
{m} = 0 1 0 
 0 0 1 
(3)
The secular determinant is
1− ω2
− ε 12
− ε 12
1−ω
− ε 13
− ε 23
− ε 13
2
− ε 23 = 0
1−ω
(4)
2
Thus,
(1 − ω ) − (1 − ω )( ε
2 3
)
(5)
x 3 − 3α 2 x − 2β 2 = 0
(6)
2
2
12
2
2
+ ε 13
+ ε 23
− 2ε 12 ε 13 ε 23 = 0
This equation is of the form (with 1 − ω 2 ≡ x )
which has a double root if and only if
(α )
2 32
= β2
(7)
423
COUPLED OSCILLATIONS
Therefore, (5) will have a double root if and only if
2
2
2
 ε 12

+ ε 13
+ ε 23


3


32
= ε 12 ε 13 ε 23
(8)
This equation is satisfied only if
ε 12 = ε13 = ε 23
(9)
Consequently, there will be no degeneracy unless the three coupling coefficients are identical.
12-20.
If we require a11 = 2a21 , then Eq. (12.122) gives a31 = −3a21 , and from Eq. (12.126) we
obtain a21 = 1
14 . Therefore,
1
3 
 2
,
,−
a1 = 
14 
 14 14
(1)
The components of a2 can be readily found by substituting the components of a1 above into Eq.
(12.125) and using Eqs. (12.123) and (12.127):
−5
1 
 4
,
,
a2 = 
42
42 
 42
(2)
These eigenvectors correspond to the following cases:
a1
12-21.
a2
The tensors {A} and {m} are:

 κ1

1
{A} =  κ 3
2

 0

1
κ3
2
κ2
1
κ3
2

0 

1 
κ3
2 

κ1 

m 0 0 
{m} =  0 m 0 
 0 0 m 
thus, the secular determinant is
(1)
(2)
424
CHAPTER 12
κ 1 − mω 2
1
κ3
2
0
1
κ3
2
κ 2 − mω 2
1
κ3
2
0
1
κ3
2
κ 1 − mω 2
=0
(3)
)
(4)
from which
(κ
1
− mω 2
) (κ
2
2
(
)
1
− mω 2 − κ 32 κ 1 − mω 2 = 0
2
In order to find the roots of this equation, we first set (1 2)κ 32 = κ 1κ 2 and then factor:
(κ
1
− mω 2  κ 1 − mω 2 κ 2 − mω 2 − κ 1κ 2  = 0
)(
(κ
1
− mω 2  m2ω 4 − (κ 1 + κ 2 ) mω 2  = 0
(κ
1
− mω 2 mω 2  mω 2 − (κ 1 + κ 2 )  = 0
)(
)
)
(5)
)
Therefore, the roots are
ω1 =
ω2 =
κ1
m
κ1 +κ 2
m
(6)
ω3 = 0
Consider the case ω 3 = 0 . The equation of motion is
η3 + ω 32η3 = 0
(7)
η3 = 0
(8)
η3 ( t ) = at + b
(9)
so that
with the solution
That is, the zero-frequency mode corresponds to a translation of the system with oscillation.
425
COUPLED OSCILLATIONS
12-22. The equilibrium configuration is shown in diagram (a) below, and the nonequilibrium configurations are shown in diagrams (b) and (c).
(a)
x3
3
4
O
1
x2
2B
2A
x1
2
x3
(b)
2
1
O′
A
O
A



θ
}Aθ
x3
x2
x3
(c)
4
Bφ {
φ
B
1
O′

3

x
B
x2
O
The kinetic energy of the system is
T=
1
1
1
Mx 32 + I1θ 2 + I 2φ 2
2
2
2
(1)
where I1 = (1 3) MA2 and I 2 = (1 3) MB2 .
The potential energy is
1
2
2
2
2
U = κ ( x3 − Aθ − Bφ ) + ( x3 + Aθ − Bφ ) + ( x3 + Aθ + Bφ ) + ( x3 − Aθ + Bφ ) 


2
(
1
= κ 4 x32 + 4 A2θ 2 + 4B2φ 2
2
)
(2)
Therefore, the tensors {m} and {A} are

M

{m} =  0


0

 4κ
{A} =  0
 0
The secular equation is
0
1
MA2
3
0
0
4κ A
0



0 


1
MB2 
3

(3)
0 

0 
4κ B2 
(4)
0
2
426
CHAPTER 12
( 4κ − Mω )  4κ A
2
−
2
1
1


MA2ω 2   4κ B2 − MB2ω 2  = 0


3
3
(5)
Hence, the characteristic frequencies are
ω1 = 2
κ
M
ω2 = 2
3κ
M
ω3 = 2
3κ
= ω2
M
(6)
We see that ω 2 = ω 3 , so the system is degenerate.
The eigenvector components are found from the equation
∑(A
jk
)
− ω r2 m jk a jr = 0
j
(7)
Setting a32 = 0 to remove the indeterminacy, we find


0


1 M 
0






a1 =  0  ; a 2 =  3 MA2  ; a 3 = 

0




 0 
0
 3 MB2 




(8)
The normal coordinates are (for x 3 ( 0 ) = θ ( 0 ) = φ ( 0 ) = 0 )
η1 ( t ) = x30 M cos ω 1t
η2 ( t ) =
θ0 A M
η3 ( t ) =
φ0 B M
3
3
cos ω 2t
(9)
cos ω 3t
Mode 1 corresponds to the simple vertical oscillations of the plate (without tipping). Mode 2
corresponds to rotational oscillations around the x1 axis, and Mode 3 corresponds to rotational
oscillations around the x2 -axis.
The degeneracy of the system can be removed if the symmetry is broken. For example, if we
place a bar of mass m and length 2A along the x2 -axis of the plate, then the moment of inertia
around the x1 -axis is changed:
I1′ =
The new eigenfrequencies are
1
( M + m) A2
3
(10)
427
COUPLED OSCILLATIONS
ω1 = 2
κ
M
ω2 = 2
3κ
M+m
ω3 = 2
3κ
M
(11)
and there is no longer any degeneracy.
12-23.
The total energy of the r-th normal mode is
Er = Tr + U r
=
(1)
1 2 1 2 2
η r + ω r ηr
2
2
where
ηr = β r e iωr t
(2)
η r = iω r β r e iωr t
(3)
Thus,
In order to calculate Tr and U r , we must take the squares of the real parts of η r and ηr :
η r2 = ( Re η r ) =  Re i ω r ( µ r + iν r ) ( cos ω r t + i sin ω r t ) 
2
2
= −ω r ν r cos ω r t − ω r µ r sin ω r t 
2
(4)
so that
Tr =
1 2
2
ω r ν r cos ω r t + µr sin ω r t 
2
(5)
Also
ηr2 = ( Re ηr ) =  Re ( µ r + iν r ) ( cos ω r t + i sin ω r t ) 
2
= µr cos ω r t − ν r sin ω r t 
2
2
(6)
so that
Ur =
1 2
2
ω r µ r cos ω r t − ν r xin ω r t 
2
Expanding the squares in Tr and U r , and then adding, we find
(7)
428
CHAPTER 12
Er = Tr + U r
=
(
1 2 2
ω r µ r + ν r2
2
)
Thus,
Er =
1 2
ω r βr
2
2
(8)
So that the total energy associated with each normal mode is separately conserved.
For the case of Example 12.3, we have for Mode 1
M
( x10 − x20 ) cos ω1t
2
η1 =
(9)
Thus,
η 1 = −ω 1
M
( x10 − x20 ) sin ω1t
2
(10)
1 2 1 2 2
η 1 + ω 1 η1
2
2
(11)
Therefore,
E1 =
But
ω 12 =
κ + 2κ 12
(12)
M
so that
E1 =
=
1 κ + 2κ 12 M
1 κ + 2κ M
( x10 − x20 ) 2 sin 2 ω1t + 2 M 12 2 ( x10 − x20 ) 2 cos2 ω1t
M
2
2
1
(κ + 2κ 12 ) ( x10 − x20 ) 2
4
(13)
which is recognized as the value of the potential energy at t = 0. [At t = 0, x 1 = x 2 = 0 , so that the
total energy is U1 ( t = 0 ) .]
12-24. Refer to Fig. 12-9. If the particles move along the line of the string, the equation of
motion of the j-th particle is
(
) (
j = −κ x j − x j −1 − κ x j − x j+ 1
mx
)
(1)
Rearranging, we find
xj =
κ
m
(x
j −1
− 2x j + x j +1
which is just Eq. (12.131) if we identify τ md with κ m .
)
(2)
429
COUPLED OSCILLATIONS
12-25.
The initial conditions are
q1 ( 0 ) = q2 ( 0 ) = q3 ( 0 ) = a 

q1 ( 0 ) = q 2 ( 0 ) = q 3 ( 0 ) = 0 
(1)
Since the initial velocities are zero, all of the ν r [see Eq. (12.161b)] vanish, and the µr are given
by [see Eq. (12.161a)]
µr =
a
2
πr
πr
3π r 

sin 4 + sin 2 + sin 4 
(2)
so that
2 +1 
a
2


µ2 = 0


2 −1 
µ3 =
a

2
µ1 =
(3)
The quantities sin  jrπ ( n + 1)  are the same as in Example 12.7 and are given in Eq. (12.165).
The displacements of the particles are
q1 ( t ) =
1
2
a ( cos ω 1t + cos ω 3t ) +
a ( cos ω 1t − cos ω 3t )
2
4
q2 ( t ) =
2
1
a ( cos ω 1t − cos ω 3t ) + a ( cos ω 1t + cos ω 3t )
2
2
q3 ( t ) =
1
2
a ( cos ω 1t − cos ω 3t ) +
a ( cos ω 1t + cos ω 3t )
2
4
(4)
where the characteristic frequencies are [see Eq. (12.152)]
ωr = 2
τ
 rπ 
sin   , r = 1, 2, 3
md
8
(5)
Because all three particles were initially displaced, there can exist no normal modes in which
any one of the particles is located at a node. For three particles on a string, there is only one
normal mode in which a particle is located at a node. This is the mode ω = ω 2 (see Figure 12-11)
and so this mode is absent.
12-26.
(
mb 2 2 2 2
Kinetic energy T =
θ1 + θ 2 + θ 3
2
Potential energy
)
 mb 2

⇒ [ m] = 


mb
2



+ mb 2 
430
CHAPTER 12
k
2
2
U = mgb (1 − cos θ 1 ) + (1 − cos θ 2 ) + (1 − cos θ 3 )  + b 2 ( sin θ 2 − sin θ 1 ) + ( sin θ 3 − sin θ 2 ) 


2
≈
mgb 2
kb 2 2
θ1 + θ 22 + θ 32 +
θ 1 + 2θ 22 + θ 32 − 2θ 1θ 2 − 2θ 2θ 3
2
2
(
)
 mgb + kb 2

⇒ [ A] =  − kb 2

0

(
− kb 2
mgb + kb
)


− kb

mgb + kb 2 
0
2
− kb 2
2
The proper frequencies are solutions of the equation
(
 mgb + kb 2 − mb 2ω 2

− kb 2
0 = Det [ A] − ω 2 [ m] = D et 

0

(
)
)
− kb 2
( mgb + 2kb
2
− mb 2ω 2
)
− kb 2


− kb 2


mgb + kb 2 − mb 2ω 2 
0
(
)
We obtain 3 different proper frequencies
mg + kb
mb
⇒ ω1 =
mg + kb
= 4.64 rad/s
mb
mg + 3kb
mb
⇒ ω2 =
mg + 3kb
= 4.81 rad/s
mb
mg
mb
⇒ ω3 =
g
= 4.57 rad/s
f
ω 12 =
ω 22 =
ω 32 =
Actually those values are very close to one another, because k is very small.
12-27.
The coordinates of the system are given in the figure:
θ1
L1
m1
θ2
L2
m2
Kinetic energy:
T=
(
1 2 2 2 1
m1 θ 1 L1 + m2 L21 θ12 + L22 θ22 − 2L1L2θ1θ2 cos (θ 1 − θ 2 )
2
2
(
)
)
1
1
1
≈ θ12 m1L21 + m2 L21 + m2 L22θ22 − m2 L1L2θ1θ2 = ∑ m jk θ j θk
2
2
2 jk
431
COUPLED OSCILLATIONS
( m + m2 ) L21
⇒  m jk  =  1
− m2 L1 L2

− m2 L1L2 

m2 L22 
Potential energy:
U = m1 gL1 (1 − cos θ 1 ) + m2 g  L1 (1 − cos θ 1 ) + L2 (1 − cos θ 2 ) 
≈ ( m1 + m2 ) gL1
θ12
2
+ m2 gL2
θ 22
2
=
1
∑ Ajkθ jθ k
2 jk
( m + m2 ) gL1
⇒  Ajk  =  1
0



m2 gL2 
0
Proper oscillation frequencies are solutions of the equation
(
)
Det [ A] − ω 2 [ m] = 0
⇒ ω 1,2 =
( m1 + m2 ) g ( L1 + L2 ) + ( m1 + m2 ) g 2  m1 ( L1 − L2 )2 + m2 ( L1 + L2 ) 2 
2m1 L1 L2
 a11 
The eigenstate corresponding to ω 1 is   where
 a21 

a21
( m + m2 ) L1  1 −
= 1
m1L2



 ×a
 11
2
2
( m1 + m2 ) g ( L1 + L2 ) + ( m1 + m2 ) g 2  m1 ( L1 − L2 ) + m2 ( L1 + L2 )  
2 gm1 L2
 a12 
The eigenstate corresponding to ω 2 is   where
 a22 

a22
( m + m2 ) L1  1 −
= 1
m1L2



 ×a
 12
2
2
( m1 + m2 ) g ( L1 + L2 ) − ( m1 + m2 ) g 2  m1 ( L1 − L2 ) + m2 ( L1 + L2 )  
2 gm1 L2
These expressions are rather complicated; we just need to note that a11 and a21 have the same
a

a

sign  11 > 0 while a12 and a22 have opposite sign  11 < 0 .
 a21

 a21

The relationship between coordinates (θ 1 , θ 2 ) and normal coordinates η1 , η2 are
θ1 = a11 η1 + a12 η2 

θ 2 = a21 η1 + a22 η2 
⇔
a12

η1 ~ θ 1 − a θ 2

22

a
η2 ~ θ 1 − 11 θ 2

a21
432
CHAPTER 12
To visualize the normal coordinate η1 , let η2 = 0 . Then to visualize the normal coordinate η2 ,
a
a
we let η1 = 0 . Because 11 > 0 and 12 < 0 , we see that these normal coordinates describe two
a21
a22
oscillation modes. In the first one, the two bobs move in opposite directions and in the second,
the two bobs move in the same direction.
12-28.
T=
Kinetic energy:
1
1
m1b 2θ12 + m2 b 2θ22
2
2
⇒
 m2 b 2
[ m] = 
 0
U = m1 gb (1 − cos θ 1 ) + m2 gb (1 − cos θ 2 ) +
Potential energy:
 m1 gb + kb 2
[ A] ≈ 
⇒

(
− kb
2
0 

m2 b 2 
k
( b sin θ1 − b sin θ 2 )
2


m2 gb + kb 
− kb 2
2
)
Solving the equation, Det [ A] − ω 2 [ m] = 0 , gives us the proper frequencies of oscillation,
ω 12 =
g
= 25 (rad/s)2
b
ω 22 =
g k
k
+
+
= 25.11 (rad/s)2
b m1 m2
 a11 
The eigenstate corresponding to ω 1 is   with a21 = 7.44 a11
 a22 
 a12 
The eigenstate corresponding to ω 2 is   with a22 = 8.55 a12
 a22 
From the solution of problem 12-27 above, we see that the normal coordinates are
η1 ~ θ1 −
a12
θ 2 = θ 1 + 0.12 θ 2
a22
η2 ~ θ 1 −
a11
θ 2 = θ 1 + 0.13 θ 2
a21
Evidently η1 then characterizes the in-phase oscillation of two bobs, and η2 characterizes the
out-of-phase oscillation of two bobs.
Now to incorporate the initial conditions, let us write the most general oscillation form:
θ1 = Re (α a11 e iω1t − iδ1 + α a12 e iω2t − iδ 2 )
θ 2 = Re (α a21 e iω1t − iδ1 + α a22 e iω 2t − iδ 2 )
(
= Re 7.44α a11 e iω1t − iδ1 − 8.35α a12 e iω 2t − iδ 2
)
where α is a real normalization constant. The initial conditions helps to determine parameters
α’s, a’s, δ’s.
433
COUPLED OSCILLATIONS
(
 Re θ 1 ( t = 0 ) = −7 ° ⇒ α a11 cos δ 1 + α a12 cos δ 2 = −0.122 rad

 Re (θ 2 ( t = 0 ) = 0° ⇒ 7.44α a11 cos δ 1 + 8.35α a12 cos δ 2 = 0 )
)
⇒ sin δ 1 = sin δ 1 = 0 . Then
θ1 = α a11 cos ω 1t + α a12 cos ω 2t = −0.065 cos ω 1t − 0.057 cos ω 2
θ 2 = 7.44α a11 cos ω 1t + 8.35α a12 cos ω 2t = 0.48 ( cos ω 2t − cos ω 1t )
where ω 1 = 5.03 rad/s , ω 2 = 4.98 rad/s
(found earlier)
Approximately, the maximum angle θ 2 is 0.096 rad and it happens when
 cos ω 2t = 1

cos ω 1t = −1
which gives
ω 2 t = 2n π

 ⇒
ω 1t = ( 2k + 1) π 
because
ω 1 2k + 1
=
2n
ω2
ω 1 101
100π
=
we finally find k = n = 50 and t =
= 63 s .
ω 2 100
ω2
Note: θ 2 max = 0.96 rad and at this value the small-angle approximation breaks down, and
the value θ 2max we found is just a rough estimate.
434
CHAPTER 12
CHAPTER
13
Continuous Systems;
Waves
13-1. The initial velocities are zero and so all of the ν r vanish [see Eq. (13.8b)]. The µr are
given by [see Eq. (13.8a)]
µr =
2A
3π x
rπ x
sin
sin
dx
L ∫0
L
L
L
(1)
= Aδ 3 r
so that
µ3 = A


µr = 0, r ≠ 3 
(2)
The characteristic frequency ω 3 is [see Eq. (13.11)]
ω3 =
3π
L
τ
ρ
(3)

 3π x 
t  sin 
 L 

(4)
and therefore,
 3π
q ( x , t ) = A cos 
 L
τ
ρ
For the particular set of initial conditions used, only one normal mode is excited. Why?
13-2.
L
3
h
L
435
436
CHAPTER 13
The initial conditions are
L
 3h
 L x, 0 ≤ x ≤ 3
q ( x , 0) = 
 3h
L
 ( L − x) ,
≤x≤L
3
 2L
(1)
q ( x , 0) = 0
(2)
Because q ( x , 0 ) = 0 , all of the ν r vanish. The µr are given by
6h
µr = 2
L
=
L3
∫
0
9h
rπ
2
2
rπ x
3h
x sin
dx + 2
L
L
sin
L
∫ ( L − x ) sin
L3
rπ x
dx
L
rπ
3
(3)
We see that µr = 0 for r = 3, 6, 9, etc. The displacement function is
q ( x, t) =
9 3h
2π 2
1
2π x
1
4π x
πx


 sin L cos ω 1t + 4 sin L × cos ω 2t − 16 sin L cos ω 4 t − …
(4)
where
ωr =
rπ
L
τ
ρ
(5)
The frequencies ω 3 , ω 6 , ω 9 , etc. are absent because the initial displacement at L 3 prevents
that point from being a node. Thus, none of the harmonics with a node at L 3 are excited.
13-3.
The displacement function is
q ( x, t)
πx
1
3π x
1
5π x
= sin
cos ω 1t − sin
cos ω 3t +
sin
cos ω 5t + …
8h π 2
L
9
L
25
L
(1)
where
ω1 =
π
L
ω r = rω 1
τ 

ρ 


(2)
For t = 0,
q ( x, 0)
πx 1
3π x 1
5π x
= sin
− sin
+
+…
sin
2
8h π
L 9
L
25
L
The figure below shows the first term, the first two terms, and the first three terms of this
function. It is evident that the triangular shape is well represented by the first three terms.
(3)
437
CONTINUOUS SYSTEMS; WAVES
1 term
1
0
L
2 terms
1
0
L
3 terms
1
0
L
The time development of q(x,t) is shown below at intervals of 1 8 of the fundamental period.
t = 0, T
13-4.
t=
1 7
T, T
8
8
t=
1
3
T, T
4
4
t=
3
5
T, T
8
8
t=
1
T
2
The coefficients ν r are all zero and the µr are given by Eq. (13.8a):
8
rπ x
µr = 2 ∫ x ( L − x ) sin
dx
L 0
L
L
=
16
1 − ( −1) r 

rπ 
3
3
(1)
so that
0,


µ r =  32
,
 3
 rπ 3
Since
r even
(2)
r odd
438
CHAPTER 13
q ( x , t ) = ∑ µ r sin
r
rπ x
cos ω r t
L
(3)
the amplitude of the n-th mode is just µn .
The characteristic frequencies are given by Eq. (13.11):
ωn =
13-5.
nπ
L
τ
ρ
(4)
The initial conditions are
q ( x , 0) = 0

 v0 ,
q ( x, t) = 

 0,
1

≤s 
2


otherwise 
x−
(1)
The µr are all zero and the ν r are given by [see Eq. (13.8b)]
νr = −
=
( L 2) + s
2
rπ x
v sin
dx
ω r L ( L ∫2) − s 0
L
4 v0
rπ
rπ s
sin
sin
2
rπω r
L
(2)
from which
 0,

ν r =  4v
rπ s
0
,
( −1)( r −1) 2 sin
−
πω
r
L
r

r even
r odd
(3)
(Notice that the even modes are all missing, as expected from the symmetrical nature of the
initial conditions.)
Now, from Eq. (13.11),
ω1 =
π
L
τ
ρ
(4)
and ω r = rω 1 . Therefore,
νr =
According to Eq. (13.5),
−4v0
rπ s
, r odd
( −1)( r −1) 2 sin
r 2πω 1
L
(5)
439
CONTINUOUS SYSTEMS; WAVES
q ( x , t ) = ∑ β r e iω r t sin
r
rπ x
L
= − ∑ ν r sin ω r t sin
r
rπ x
L
(6)
Therefore,
q ( x, t) =
4 v0 

πω 1 
πs
sin ω 1t sin
L
sin
πx
L
1
3π s
3π x

− sin ω 3t sin
sin
+ …

9
L
L
(7)
Notice that some of the odd modes—those for which sin ( 3π s L) = 0 —are absent.
13-6.
The initial conditions are
q ( x , 0) = 0
 4 v0
 L

 4v
q ( x , 0) =  0
 L


0

L
4

L
L
≤x≤ 
4
2


L
≤x≤L
2

0≤x≤
x
L

 2 − x 
(1)
The velocity at t = 0 along the string, q ( x , 0 ) , is shown in the diagram.
v
v0
L
4
L
2
3L
4
L
The µr are identically zero and the ν r are given by:
νr = −
=
2
rπ x
q ( x , 0 ) sin
dx
∫
rLω 1 0
L
L
rπ
rπ 

sin
− 2 sin 

r π ω1 
2
4
8 v0
3
2
Observe that for r = 4n, ν r is zero. This happens because at t = 0 the string was struck at L 4 ,
and none of the harmonics with modes at that point can be excited.
Evaluation of the first few ν r gives
(2)
440
CHAPTER 13
ν1 = −0.414 ⋅




2.414 8v0 
⋅
ν5 =
125 π 2ω 1 


2 8 v0

⋅ 2
ν6 =
216 π ω 1 
8 v0
ν4 = 0
π ω1
2
1 8 v0
4 π 2ω 1
ν2 = − ⋅
ν3 = −
2.414 8v0
⋅
27 π 2ω 1
(3)
and so,
q ( x, t) =
8 v0 
2π x
πx 1
+ sin ω 2t sin
 0.414 sin ω 1t sin
2
L 4
L
π ω1
2.414
3π x 2.414
5π x

+
sin ω 3t sin
−
sin ω 5t sin
− …

L
L
27
135
(4)
From these amplitudes we can find how many db down the fundamental are the various
harmonics:
Second harmonic:
2
 0.250 
= −4.4 db
10 log 
 0.414 
(5)
Third harmonic:
2
 2.414 27 
10 log 
= −13.3 db
 0.414 
(6)
These values are much smaller than those found for the case of example (13.1). Why is this so?
(Compare the degree of symmetry of the initial conditions in each problem.)
13-7.
3
L
7
h
L
O
h
3
L
7
Since q ( x , 0 ) = 0 , we know that all of the ν r are zero and the µr are given by Eq. (13.8a):
µr =
The initial condition on q ( x , t ) is
2
rπ x
q ( x , 0 ) sin
dx
∫
L0
L
L
(1)
441
CONTINUOUS SYSTEMS; WAVES
 7h
 − 3L x ,

7h
q ( x , 0 ) =  ( 2 x − L) ,
L

 7h
 3L ( L − x ) ,



3
4 
L ≤ x ≤ L
7
7 

4

L≤ x≤L 
7

0≤x≤
3
L
7
(2)
Evaluating the µr we find
µr =
4 rπ
3rπ 

sin 7 − sin 7 
98 h
3 r 2π 2
(3)
Obviously, µr = 0 when 4 r 7 and 3r 7 simultaneously are integers. This will occur when r is
any multiple of 7 and so we conclude that the modes with frequencies that are multiples of 7ω 1
will be absent.
13-8.
For the loaded string, we have [see Eq. (12.152)]
τ
ωr = 2
md
sin
rπ
2 ( n + 1)
(1)
Using ρ = m d and L = ( n + 1) d , we have
ωr =
=
2
d
τ
rπ
sin
2 ( n + 1)
ρ
2 ( n + 1)
L
rπ
τ
sin
2 ( n + 1)
ρ
(2)
The function
ωr
2
L
τ
ρ
= ( n + 1) sin
rπ
2 ( n + 1)
(3)
is plotted in the figure for n = 3, 5, and 10. For comparison, the characteristic frequency for a
continuous string is also plotted:
ωr
2
L
τ
ρ
=
rπ
2
(4)
442
CHAPTER 13
12
n = 10
nti
nu
Co
τ
ρ
ou
8
ωr
2
L
ss
tri
ng
10
6
n=5
4
n=3
2
0
0
2
4
6
8
10
Of course, the curves have meaning only at the points for which r is an integer.
13-9.
From Eq. (13.49), we have:
β=
D
s 2π 2τ
; ω 02 =
2ρ
ρb
(1)
From section 3.5, we know that underdamped motion requires:
β 2 < ω 02
Using (1) this becomes
D 2 s 2π 2τ
<
4ρ 2
ρb
D2 <
4 ρ s 2 π 2τ
b
underdamped
Likewise D 2 =
4 ρ s 2π 2 τ
b
critically damped
D2 >
4 ρ s 2 π 2τ
b
overdamped
or
The complementary solution to Eq. (13.48) for underdamped motion can be written down using
Eq. (3.40). The result is:
ηs ( t ) = Cs e − βt cos (ω 1t − φs )
443
CONTINUOUS SYSTEMS; WAVES
where ω 12 = ω 02 − β 2 , ω 0 and β are as defined in (1), and Cs and φs are arbitrary constants
depending on the initial conditions. The complete solution to Eq. (13.48) is the sum of the
particular and complementary solutions (analogous to Eq. (13.50)):
ηs ( t ) = C s e − β t
 sπ 
2 F0 sin   cos (ω t − δ s )
2
cos (ω 1t − φ s ) +
2 2
s π τ
 D
− ω2  + ω2
ρb 
 ρb
 ρ
where
Dω

2

s π τ 
ρ
− ω2 

  ρb 

δ s = tan −1 
2
From Eq. (13.40):
q ( x , t ) = ∑ ηr ( t ) sin
r
rπ x
b
Thus


 rπ 


 s 2π 2τ D 2
 2 F0 sin  2  cos (ω t − δ r ) 
 Dt 
rπ x

sin
q ( x , t ) = ∑ Cr exp  −  cos 
− 2 t − φr  +

2
2
ρ
ρ
ρ
2
4
b
b
r


r π τ
 D



ρb 
− ω2  + ω2 


 ρb
 ρ


(underdamped)
13-10.
From Eq. (13.44) the equation for the driving Fourier coefficient is:
b
f s ( t ) = ∫ F ( x , t ) sin
0
sπ x
dx
b
If the point x is a node for normal coordinate s, then
x n
= where n is an integer ≤ s
b s
(This comes from the fact that normal mode s has s-half wavelengths in length b.)
For
x n
= ,
b s
sin
sπ x
= sin nπ = 0; hence f s ( t ) = 0
b
444
CHAPTER 13
Thus, if the string is driven at an arbitrary point,
none of the normal modes with nodes at the driving
point will be excited.
13-11.
From Eq. (13.44)
sπ x
dx
b
b
f s ( t ) = ∫ F ( x , t ) sin
0
(1)
where F ( x , t ) is the driving force, and f s ( t ) is the Fourier coefficient of the Fourier expansion of
F ( x , t ) . Eq. (13.45) shows that f s ( t ) is the component of F ( x , t ) effective in driving normal
coordinate s. Thus, we desire F ( x , t ) such that
f s ( t ) = 0 for s ≠ n
≠ 0 for s = n
From the form of (1), we are led to try a solution of the form
F ( x , t ) = g ( t ) sin
nπ x
b
where g(t) is a function of t only.
Thus
b
f s ( t ) = ∫ g ( t ) sin
0
For n ≠ s, the integral is proportional to sin
nπ x
sπ x
sin
dx
b
b
( n ± s) π x 
b
b
 ; hence f s ( t ) = 0 for s ≠ n.
x=0
For n = s, we have
b
f s ( t ) = g ( t ) ∫ sin 2
0
nπ x
b
dx = g ( t ) ≠ 0
2
b
Only the nth normal coordinate will be driven.
Thus, to drive the nth harmonic only,
nπ x
F ( x , t ) = g ( t ) sin
b
445
CONTINUOUS SYSTEMS; WAVES
13-12.
The equation to be solved is
ηs +
D
ρ
ηs +
s 2π 2τ
η =0
ρb s
(1)
Compare this equation to Eq. (3.35):
x + 2β x + ω 02 x = 0
The solution to Eq. (3.35) is Eq. (3.37):
x ( t ) = e − βt  A1 exp
(
)
(
)
β 2 − ω 02 t + A2 exp − β 2 − ω 02 t 
Thus, by analogy, the solution to (1) is

 D 2 s 2π 2τ 

D 2 s 2 π 2τ  
−
+
−
−
ns ( t ) = e − Dt 2 ρ  A1 exp 
t
A
t
exp


2
2
2
ρb 
ρb  

 4 ρ
 4 ρ

13-13.
Assuming k is real, while ω and v are complex, the wave function becomes
ψ ( x , t ) = Ae i(αt + iβt − kx)
= Ae (α t − kx ) e − βt
(1)
whose real part is
ψ ( x , t ) = Ae − βt cos (α t − kx )
(2)
and the wave is damped in time, with damping coefficient β.
From the relation
k2 =
ω2
v2
(3)
we obtain
(α + iβ ) 2 = k 2 ( u + iw ) 2
(4)
By equating the real and imaginary part of this equation we can solve for α and β in terms of u
and w:
α=
k 2 uw
β
(5)
and
 kw
β=
 iku
Since we have assumed β to be real, we choose the solution
(6)
446
CHAPTER 13
β = kw
(7)
α = ku
(8)
Substituting this into (5), we have
as expected.
Then, the phase velocity is obtained from the oscillatory factor in (2) by its definition:
V=
Re ω α
=
k
k
(9)
That is,
V=u
13-14.
Vn–2
In–2
Vn–1
Vn
In+1
L′
L′
C′
Vn+1
In+2
L′
Qn–1
Qn–2
C′
In
L′
Qn
C′
Vn+2
Qn+1
C′
Qn+2
C′
Consider the above circuit. The circuit in the nth inductor is I n , and the voltage above ground
at the point between the nth elements is Vn . Thus we have
Vn =
Qn
C′
and
L′
dI n
= Vn −1 − Vn
dt
=
Qn −1 Qn
−
C′
C′
(1)
We may also write
dQn
= I n − I n+1
dt
(2)
Differentiating (1) with respect to time and using (2) gives
d2 I n
1
=
[ I n −1 − 2I n + I n + 1 ]
2
dt
C′
(3)
d2 In
1
=
[ I n − 1 − 2I n + I n + 1 ]
2
dt
L′ C ′
(4)
L′
or
447
CONTINUOUS SYSTEMS; WAVES
Let us define a parameter x which increases by ∆x in going from one loop to the next (this will
become the coordinate x in the continuous case), and let us also define
L≡
L′
;
∆x
C≡
C′
∆x
(5)
which will become the inductance and the capacitance, respectively, per unit length in the limit
∆x → 0 .
From the above definitions and
∆I r = I r + 1 − I r
(6)
d2 In
1
+
( ∆I n−1 − ∆I n ) = 0
2
dt
L′C ′
(7)
d 2 I n ∆ ( ∆I n )
−
=0
dt 2
L′C ′
(8)
(4) becomes
or,
Dividing by ( ∆x ) , and multiplying by (–L′C′), we find
2
∆ ( ∆I n )
( ∆x )
2
− LC
d2 In
=0
dt 2
(9)
But by virtue of the above definitions, we can now pass to the continuous limit expressed by
In (t) → I ( x, t)
(10)
Then,
∆ ( ∆I ( x , t ) )
∆x 2
− LC
∂2 I ( x, t)
=0
∂t 2
(11)
and for ∆x → 0 , we obtain
∂2I 1 ∂2I
−
=0
∂x 2 v 2 ∂t 2
(12)
where
v=
13-15.
1
LC
(13)
Consider the wave functions
ψ 1 = A exp  i (ω t − kx ) 



ψ 2 = B exp i ( (ω + ∆ω ) t − ( k + ∆k ) x )  
where ∆ω
ω ; ∆k
k . A and B are complex constants:
(1)
448
CHAPTER 13
A = A exp ( iφ a ) 


B = B exp ( iφb ) 
(2)
The superposition of ψ 1 and ψ 2 is given by
ψ = ψ1 + ψ2
 ∆ω t −∆kx 
 ∆ω t −∆kx  
−i
i
 
∆ω  
∆k   
2
2




= exp  i  ω +
−
+
×
+
exp
exp
t
k
x
A
i
e
B
i
e
φ
φ
( a)
( b)


 

2  
2   
 

(3)
which can be rewritten as

 

 
ψ = exp i  w +

 ∆ω t −∆kx − φ a + φb 
 ∆ω t −∆kx + φb − φ a  
i
−i
φ a + φb    
∆ω  
∆k 
2
2




−
+
+
×
+
t
k
x
A
e
B
e

 

 
2  
2
2    

(4)
Define
t∆ω − x∆k ≡ δ 

φb − φ a ≡ α 
(5)
A e − i(δ +α ) 2 + B e i(δ +α ) 2 = Γ e iθ
(6)
and
Therefore,
(
Γ =2 A + B
2
cos θ =
sin θ =
(
2
A+B
)
2 A + B 


(
2
B− A
2
)
2 A 2 + B 2 


2
)
12
cos
12
sin
(7)
(δ + a )
(8)
2
(δ + a )
(9)
2
That is, θ is a function of ( ∆ω ) t − ( ∆k ) x . Using (6) and (7) – (9), we can rewrite (4) as
 
φ + φb  iθ
∆ω  
∆k 
t −k +  x + a
ψ = Γ exp i  ω +
 e

2  
2
2 
 
(10)
and then,

φ a + φb 
∆ω  
∆k  


Re ψ = Γ cos  ω +
 t −  k +  x  cos  θ +



2
2
2 



∆ω  
∆k  


− sin  ω +
 t −  k +  x  sin  θ +


2
2


φ a + φb  
2
 
(11)
449
CONTINUOUS SYSTEMS; WAVES
From this expression we see that the wave function is modulated and that the phenomenon of
beats occurs, but for A ≠ B, the waves never beat to zero amplitude; the minimum amplitude is,
from Eq. (11), A − B , and the maximum amplitude is A + B . The wave function has the form
shown in the figure.
A+B
A−B
Reψ
wt – kx
13-16. As explained at the end of section 13.6, the wave will be reflected at x = x0 and will
then propagate in the –x direction.
13-17.
We let
j = 2n
 m′ ,
mj = 
 m′′ ,
(1)
j = 2n + 1
where n is an integer.
Following the procedure in Section 12.9, we write
F2 n = m′ q2 n =
τ
d
F2 n +1 = m′′ q2 n +1 =
( q2 n−1 − 2q2n + q2n+1 )
τ
d
( q2n − 2q2n+1 + q2n+ 2 )
(2a)
(2b)
Assume solutions of the form
q2 n = Ae i(ωt − 2 nkd)
(3a)
q2 n + 1 = Be [
i ω t − ( 2 n + 1) kd ]
(3b)
Substituting (3a,b) into (2a,b), we obtain
−ω 2 A =
−ω B =
2
from which we can write
τ
m′ d
τ
m′′ d
( Be
ikd
( Ae
− 2 A + Be − ikd
ikd
− 2B + Ae
)
− ikd
)






(4)
450
CHAPTER 13
2τ

 2τ

− ω2  − B
A
cos kd = 0 
m
d
m
d
′
 ′



2τ
 2τ

cos kd + B 
−A
− ω2  = 0 
m′′ d
 m′′ d


(5)
The solution to this set of coupled equations is obtained by setting the determinant of the
coefficients equal to zero. We then obtain the secular equation
1
 2τ
2   2τ
2
 m′ d − ω   m′′ d − ω  − m′ m′′
 2τ

 d cos kd  = 0
2
(6)
Solving for ω, we find
τ  1
2

1   1
1 
4
±
+
−
ω =  +
sin 2 kd 




d  m′ m′′   m′ m′′ 
m′ m′′


2
12



(7)
from which we find the two solutions
τ  1
2

1   1
1 
4
+
+
−
sin 2 kd 
ω =  +




d  m′ m′′   m′ m′′ 
m′ m′′


2
1
12



12
2
 
τ  1
1   1
1 
4
2
2
+
ω 2 =  +
sin kd  
 − 
 −
d  m′ m′′   m′ m′′ 
m′ m′′
 

(8)
If m′ < m″, and if we define
ωa ≡
2τ
, ωb ≡
m′′ d
2τ
, ω c = ω a2 + ω b2
m′ d
(9)
Then the ω vs. k curve has the form shown below in which two branches appear, the lower
branch being similar to that for m′ = m″ (see Fig. 13-5).
ωc
ω
ωb
ωa
0
k
π/2d
Using (9) we can write (6) as
sin 2 kd =
ω2
ω 2 + ω b2 − ω 2 ) ≡ W (ω )
2 2 ( a
ωaωb
(10)
From this expression and the figure above we see that for ω > ω c and for ω a < ω < ω b , the wave
number k is complex. If we let k = κ + iβ , we then obtain from (10)
sin 2 (κ + iβ ) d = sin 2 κ d cosh 2 β d − cos 2 κ d sinh 2 β d + 2i sin κ d cos κ d sinh β d cosh β d = W (ω )
Equating the real and imaginary parts, we find
(11)
451
CONTINUOUS SYSTEMS; WAVES
sin κ d cos κ d sinh β d cosh β d = 0


2
2
2
2
sin κ d cosh β d − cos κ d sinh β d = W (ω ) 
(12)
We have the following possibilities that will satisfy the first of these equations:
a) sin κd = 0, which gives κ = 0. This condition also means that cos κd = 1; then β is
determined from the second equation in (12):
− sinh 2 β d = W (ω )
(13)
Thus, ω > ω c , and κ is purely imaginary in this region.
b) cos κd = 0, which gives κ = π/2d. Then, sin κd = 1, and cosh 2 β d = W (ω ) . Thus, ω a < ω < ω b ,
and κ is constant at the value π/2d in this region.
c) sinh βd = 0, which gives β = 0. Then, sin 2 κ d = W (ω ) . Thus, ω < ω a or ω b < ω < ω c , and κ is
real in this region.
Altogether we have the situation illustrated in the diagram.
k
π
2d
β
κ
β
κ
κ
κ
ωa
13-18.
ωb
ωc
ω
The phase and group velocities for the propagation of waves along a loaded string are
ω c d sin ( kd 2)
V (k) =
ω
U (k) =
dω ω c d
=
cos ( kd 2)
dk
2
k
=
kd 2
2
(1)
(2)
where
ω = ω c sin ( kd 2)
(3)
The phase and group velocities have the form shown below.
ωc d
2
V(k)
U(k)
V,U
0
π/d
k
When k = π d , U = 0 but V = ω c d π . In this situation, the group (i.e., the wave envelope) is
stationary, but the wavelets (i.e., the wave structure inside the envelope) move forward with the
velocity V.
452
13-19.
CHAPTER 13
The linear mass density of the string is described by
 ρ1
ρ=
 ρ2 > ρ1
if x < 0; x > L
if 0 < x < L
I
II
III
ρ1
ρ2
ρ1
x
0
Consider the string to be divided in three different parts: I for x < 0, II for 0 < x < L, and III for
x > L.
Let φ = A e (
i ω t − k1 x )
be a wave train, oscillating with frequency ω, incident from the left on II. We can write
for the different zones the corresponding wave functions as follows:
ψ I = Ae i(ωt − k1x ) + Be i(ωt + k1x) = e iωt  Ae − ik1x + Be ik1x  
ψ II = Ce i(ωt − k2 x) + De i(ωt + k2 x ) = e iωt Ce − ik2 x
ψ III = Ee i(ωt − k1x )


+ De ik2 x  



(1)
Where
k1 =
ω
V1
, k2 =
ω
V2
τ
τ
, V2 =
ρ1
ρ2
, V1 =
(2)
and where τ is the tension in the string (constant throughout). To solve the problem we need to state first
the boundary conditions; these will be given by the continuity of the wave function and its derivative at
the boundaries x = 0 and x = L. For x = 0, we have
ψ I ( x = 0 ) = ψ II ( x = 0 ) 
∂ψ I
∂x
=
x=0
∂ψ II
∂x
x=0




(3)
and for x = L, the conditions are
ψ II ( x = 0 ) = ψ III ( x = L) 
∂ψ II
∂x
=
x=L
∂ψ III
∂x
x=L




(4)
Substituting ψ as given by (1) into (3) and (4), we have
A+B=C+D



k2
− A + B = ( −C + D) 
k1

and
(5)
453
CONTINUOUS SYSTEMS; WAVES
C e − ik2 L + D e ik2 L = E e − ik1L
−C e
− ik2 L
+De
k
= − 1 E e − ik1L
k2
ik2 L





(6)
From (6) we obtain

k1 
1
i ( k2 − k1 ) L
+
E
e
1

2 
k 2 





1
k
D =  1 − 1  E e − i( k2 + k1 ) L 
2
k2 

C=
(7)
Hence,
k 2 + k1 i 2 k 2 L
e
D
k 2 − k1
(8)
1
k 
1
k 
1 + 2  C + 1 − 2  D

k1 
k1 
2
2
(9)
C=
From (5) we have
A=
Using (7) and rearranging the above equation
2

1  ( k1 + k 2 ) i 2 k2 L
A=
e
− ( k 2 − k1 )  D

2k1  ( k 2 − k1 )

(10)
1
 − ( k 2 + k1 ) e i 2 k2 L + ( k1 + k 2 )  D
2 k1
(11)
In the same way
B=
From (10) and (11) we obtain
(
)
k12 − k 22  e i 2 k2 L − 1
B
=
A ( k1 + k 2 ) 2 e i 2 k 2 L − ( k1 − k 2 ) 2
(12)
On the other hand, from (6) and (8) we have
E=
2k 2 D i( k2 + k1 ) L
e
k 2 − k1
(13)
which, together with (10) gives
4 k1 k 2 e i( k1 + k2 ) L
E
=
A ( k1 + k 2 ) 2 e i 2 k 2 L − ( k1 − k 2 ) 2
(14)
Since the incident intensity I 0 is proportional to A , the reflected intensity is I r = B , and the total
2
transmitted intensity is I t = E , we can write
2
2
454
CHAPTER 13
Ir = I0
B
2
A
2
, It = I 0
E
2
A
2
(15)
Substituting (12) and (14) into (15), we have, for the reflected intensity,
(k
)
− k 22  e i 2 k2 L − 1
Ir = I0
( k1 + k 2 ) 2 e i 2 k 2 L − ( k1 − k 2 ) 2
2
1
2
(16)
From which
(
)
2


k12 − k 22 (1 − cos 2k 2 L)


Ir = I0
 k14 + k 24 + 6 k12 k 22 − k12 − k 22 2 cos 2k2 L 


(
)
(17)
and for the transmitted intensity, we have
It = I0
2
4 k1k 2 e i( k1 + k2 ) L
( k1 + k 2 ) 2 e i 2 k L − ( k1 − k 2 ) 2
(18)
2
so that
It = I0
8 k12 k 22
(
k14 + k 24 + 6k12 k 22 − 2 k12 − k 22
)
2
(19)
cos 2k 2 L
We observe that I r + I t = I 0 , as it must.
For maximum transmission we need minimum reflection; that is, the case of best possible transmission is
that in which
It = I0 

I r = 0 
(20)
In order that I r = 0 , (17) shows that L must satisfy the requirement
1 − cos 2k2 L = 0
(21)
so that we have
L=
mπ mπ
=
k2
ω
τ
, m = 0,1, 2,…
ρ2
(22)
The optical analog to the reflection and transmission of waves on a string is the behavior of light
waves which are incident on a medium that consists of two parts of different optical densities
(i.e., different indices of refraction). If a lens is given a coating of precisely the correct thickness
of a material with the proper index of refraction, there will be almost no reflected wave.
455
CONTINUOUS SYSTEMS; WAVES
13-20.
y
I
II
M
0
We divide the string into two zones:
I: x < 0
II: x > 0
Then,
ψ I = A1e i(ωt − kx) + B1e i(ωt + kx) 
ψ II = A2 e



i (ω t − kx )
(1)
The boundary condition is
ψ I ( x = 0 ) = ψ II ( x = 0 )
(2)
That is, the string is continuous at x = 0. But because the mass M is attached at x = 0, the
derivative of the wave function will not be continuous at this point. The condition on the
derivative is obtained by integrating the wave equation from x = –ε to x = +ε and then taking
the limit ε → 0.
Thus,
M
∂ 2ψ
∂t 2
x=0
∂ψ I 
 ∂ψ
= τ  II −
∂x 
 ∂x
(3)
x=0
Substituting the wave functions from (1), we find
A1 + B1 = A2
(4)
ikτ ( − A2 + A1 − B1 ) = −ω 2 MA2
(5)
which can be rewritten as
( ikτ − ω M)
2
A1 − B1 = A2
ikτ
(6)
From (4) and (6) we obtain
A1 + B1
ikτ
=
A1 − B1 ikτ − ω 2 M
(7)
ω 2 M 2ikτ
B1
ω2M
=
=
A1 2ikτ − ω 2 M 1 − ω 2 M 2ikτ
(8)
from which we write
Define
456
CHAPTER 13
ω2M
= P = tan θ
2kτ
(9)
− iP
B1
=
A1 1 + iP
(10)
Then, we can rewrite (8) as
And if we substitute this result in (4), we obtain a relation between A1 and A2 :
A2
1
=
A1 1 + iP
(11)
2
B
The reflection coefficient, R = 1 , will be, from (10),
A1
2
tan 2 θ
B
P2
=
R= 1 =
1 + P 2 1 + tan 2 θ
A1
(12)
R = sin 2 θ
(13)
or,
2
A
and the transmission coefficient, T = 2 , will be from (10)
A1
T=
A2
A1
2
=
1
1
=
2
1+ P
1 + tan 2 θ
(14)
or,
T = cos 2 θ
(15)
The phase changes for the reflected and transmitted waves can be calculated directly from (10)
and (11) if we substitute
B1 = B1 e iφ B1



iφ
A1 = A1 e A1 

A2 = A2 e iφ A2 
(16)
B i(φ −φ )
B1
P
i tan −1 ( 1 P )
= 1 e B1 A1 =
e
A1 A1
1 + P2
(17)
−1
A2 A2 i(φA2 −φA1 )
1
=
=
e
e i tan ( − P)
A1
A1
1 + P2
(18)
Then,
and
457
CONTINUOUS SYSTEMS; WAVES
Hence, the phase changes are
1
φB1 − φ A1 = tan −1   = tan −1 ( cot θ )
P
φ A2 − φ A1 = − tan ( P ) = − tan
−1
13-21.
−1
(19)
( tan θ ) = −θ
The wave function can be written as {see Eq. (13.111a)]
ψ ( x, t) =
+∞
∫ A(k) e
i (ω t − kx )
dk
(1)
−∞
Since A(k) has a non-vanishing value only in the vicinity of k = k0 , (1) becomes
ψ ( x, t) =
k0 +∆k
∫
e i(ωt − kx ) dk
(2)
k0 −∆k
According to Eq. (13.113),
ω = ω 0 + ω 0′ ( k − k0 )
(3)
Therefore, (2) can now be expressed as
ψ ( x , t ) = e i(ω0 − w0′ k0 )t
k0 +∆k
∫
e(
i ω 0′ t − x ) k
dk
k0 −∆k
=e(
i ω 0 −ω 0′ k0 ) t
=
 e i( k0 +∆k )(ω0′ t − x ) − e i( k0 −∆k )(ω 0′ t − x) 


i (ω 0′ t − x )


2e (
i ω 0 −ω 0′ k0 ) t
ω 0′ t − x
 e i(ω 0′ t − x) ∆k − e i(ω 0′ t − x) ∆k 


2i


(4)
and writing the term in the brackets as a sine, we have
ψ ( x, t) =
2 sin (ω 0′ t − x ) ∆k  i(ω 0t − k0 x)
e
ω 0′ t − x
(5)
The real part of the wave function at t = 0 is
Re ψ ( x , 0 ) =
2 sin ( x∆k )
cos k0 x
x
(6)
If ∆k k0 , the cosine term will undergo many oscillations in one period of the sine term. That
is, the sine term plays the role of a slowly varying amplitude and we have the situation in the
figure below.
458
CHAPTER 13
ReΨ(x,0)
x
13-22.
a)
Using Eq. (13.111a), we can write (for t = 0)
ψ ( x, 0) =
+∞
∫ A(k) e
− ikx
dk
−∞
+∞
=B
∫e
− σ ( k − k0 )
2
e − ikx dk
−∞
= Be
− ik0 x
+∞
∫e
− σ ( k − k0 )
2
−i k−k x
e ( 0 ) dk
−∞
= Be − ik0 x
+∞
∫e
− σ u2
e − iux du
(1)
−∞
This integral can be evaluated by completing the square in the exponent:
+∞
∫e
+∞
− ax 2
e dx =
bx
−∞
∫e
b 

− a  x 2 − x

a 
dx
−∞
=
2
+∞ − a  x 2 − b x + b 
a
4 a2 

∫
e
b2
e 4 a dx
−∞
=e
b 

b 2 +∞
− a x − 
 2a 
4a
2
∫e
dx
(2)
dy
(3)
−∞
and letting y = x − b 2a , we have
+∞
∫e
− ax 2
e dx = e
bx
b 2 +∞
− ay 2
4a
−∞
∫e
−∞
Using Eq. (E.18c) in Appendix E, we have
+∞
∫e
−∞
Therefore,
− ax 2
e dx =
bx
π
a
e
b2
4a
(4)
459
CONTINUOUS SYSTEMS; WAVES
π − ik0 x − x2 4σ
e
e
σ
ψ ( x , 0) = B
(5)
The form of ψ ( x , 0 ) (the wave packet) is Gaussian with a 1 e width of 4 σ , as indicated in the
diagram below.
Ψ ( x, 0 )
B
π
σ
2 σ
−2 σ
1
π
⋅B
e
σ
x
b) The frequency can be expressed as in Eq. (13.113a):
ω ( k ) = ω 0 + ω 0′ ( k − k0 ) + …
(6)
and so,
ψ ( x, t) =
+∞
∫ A(k) e
i (ω t − kx )
dk
−∞
+∞
=
∫ A(k) e
i ω 0 t + ω 0′ ( k − k0 ) t − kx 
dk
−∞
= Be (
i ω 0 t − k0 x )
+∞
∫e
− σ ( k − k0 )
2
i ω 0′ ( k − k0 ) t − ( k − k0 ) x 
e
dk
−∞
= Be (
i ω 0 t − k0 x )
+∞
∫e
− σ u2
e(
i ω 0′ t − x ) u
du
(7)
−∞
Using the same integral as before, we find
ψ ( x, t) = B
π i(ω0t − k0 x) − (ω0′ t − x )2
e
e
σ
4σ
(8)
c) Retaining the second-order term in the Taylor expansion of ω (k), we have
1
2
ω ( k ) = ω 0 + ω 0′ ( k − k0 ) + ω 0′′( k − k0 ) + …
2
(9)
Then,
ψ ( x , t ) = e i (ω 0 t − k 0 x )
+∞
∫
1
2


i ω 0′ ( k − k0 ) t + ω 0′′( k − k0 ) t − ( k − k0 ) x 
2

A(k) e 
dk
−∞
= Be (
i ω 0 t − k0 x )
+∞
∫
e
ω ′′t 

−  σ − i 0  u2

2 
e(
i w0′ t − x ) u
du
−∞
We notice that if we make the change σ − iω 0′′t 2 → σ , then (10) becomes identical to (7).
Therefore,
(10)
460
CHAPTER 13
ψ ( x, t) = B
2π
i ω t−k x
i ( 0 0 ) e −α ( x ,t)
2σ − iw0′′t
(11)
where
(ω 0′t − x ) 2  σ + 2 iω 0′′t
1
α ( x, t) =
(12)
4σ 2 + (ω 0′′) t 2
2
The 1 e width of the wave packet will now be
w1 e ( t ) = 2
4σ 2 + (ω 0′′) t 2
2
(13)
σ
or,
W1 e ( t ) = 4 σ
 ω ′′t 
1+  0 
 2σ 
2
(14)
In first order, W1 e , shown in the figure above, does not depend upon the time, but in second
order, W1 e depends upon t through the expression (14). But, as can be seen from (8) and (11),
the group velocity is ω 0′ , and is the same in both cases. Thus, the wave packet propagates with
velocity ω 0′ but it spreads out as a function of time, as illustrated below.
Ψ (x, t)
t=0
t = t1
O
x
CHAPTER
14
The Special Theory
of Relativity
14-1.
Substitute Eq. (14.12) into Eqs. (14.9) and (14.10):
v 

x1′ = γ  x1 − x1 

c 
v 

x1 = γ  x1′ + x1′ 

c 
(1)
(γ = γ ′ )
(2)
From (1)
x1′
 v
= γ 1 − 
x1
 c
From (2)
x1′
=
x1
1
v
γ 1 + 
c

So
v
1
γ 1 −  =
c


 γ 1+ v

c 
or
γ=
1
1 − v2 c2
461
462
CHAPTER 14
14-2. We introduce cosh α ≅ y , sinh α ≅ y v c and substitute these expressions into Eqs.
(14.14); then
x1′ = x1 cosh α − ct sinh α 


x1
t ′ = t cosh a − sinh α 
c


x2′ = x2 ;
x3′ = x3

(1)
Now, if we use cosh α = cos (iα) and i sinh α = sin (iα), we can rewrite (1) as
x1′ = x1 cos ( iα ) + ict sin ( iα )


ict ′ = − x1 sin ( iα ) + ict cos ( iα ) 
(2)
Comparing these equations with the relation between the rotated system and the original
system in ordinary three-dimensional space,
x1′ = x1 cos θ + x2 sin θ


x2′ = − x1 sin θ + x2 cos θ 


x3′ = x3

x2′
(3)
x2
x1′
θ
x1
We can see that (2) corresponds to a rotation of the x1 − ict plane through the angle iα.
14-3.
If the equation
∇ 2ψ ( x , ict ) −
1 ∂ 2 ψ ( x , ict )
=0
c2
∂t 2
(1)
is Lorentz invariant, then in the transformed system we must have
∇ ′ 2ψ ( x ′ , ict ′ ) −
1 ∂ 2 ψ ( x ′ , ict ′ )
=0
c2
∂t ′ 2
(2)
where
∇′2 =
∂2
∂2
∂2
+
+
2
2
∂x ′
∂y ′
∂z ′ 2
(3)
We can rewrite (2) as
∂ 2ψ ( x ′ , ict ′ )
∑ ∂x ′ 2 = 0
µ =1
µ
4
(4)
463
THE SPECIAL THEORY OF RELATIVITY
Now, we first determine how the operator
∂2
∑ ∂x ′ 2 is related to the original operator
µ
µ
∂2
∑ ∂x 2 .
µ
µ
We know the following relations:
xµ′ = ∑ λ µν xν
(5)
xν = ∑ λ µν xµ′
(6)
∑ λµν λµλ = δ νλ
(7)
∂
∂ ∂xν
∂
=∑
= ∑ λ µν
∂xµ′
∂xν
ν ∂xν ∂xµ′
ν
(8)
ν
µ
µ
Then,
∂2
∂
= ∑ λ µν
2
∂xµ′
∂xν
ν
∑ λ µλ
λ
∂
∂ ∂
= ∑∑ λ µν λ µλ
∂x λ
∂xν ∂xλ
ν λ
(9)
Therefore,
∂2
∂ ∂
∑ ∂x ′ 2 = ∑∑∑ λµν λµλ ∂x ∂x
µ
ν λ µ
µ
ν
λ
= ∑∑ δ νλ
ν
=∑
λ
∂ ∂
∂xν ∂xλ
∂2
∂xλ2
Since µ and λ are dummy indices, we see that the operator
(10)
∑∂
2
∂xµ2 is invariant under a
Lorentz transformation. So we have
∂ 2ψ ( x ′ , ict ′ )
∑ ∂x ′ 2 = 0
µ
µ
(11)
This equation means that the function ψ taken at the transformed point (x′,ict′) satisfies the
same equation as the original function ψ (x,ict) and therefore the equation is invariant. In a
Galilean transformation, the coordinates become
x ′ = x − vx t 

y ′ = y − vy t 


z ′ = z − vz t 

t′ = t

Using these relations, we have
(12)
464
CHAPTER 14
∂
∂ ∂x ∂ ∂ t
∂
1 ∂ 
=
+
=
−
∂ x ′ ∂x ∂ x ′ ∂t ∂ x ′ ∂x v x ∂ t 


1 ∂
∂
∂

=
−

∂ y ′ ∂y v y ∂t


1 ∂
∂
∂

=
−

∂z ′ ∂y v z ∂t


∂
∂

=
∂t ′ ∂t

(13)
Therefore,
 ∂2
1 ∂2
1 ∂2   1
1
1  ∂2
∂2
∂2
∂2
∂2
∂2
+
+
−
=
+
+
−
+
+
+


 
∂x ′ 2 ∂y ′ 2 ∂z ′ 2 c 2 ∂t ′ 2  ∂x 2 ∂y 2 ∂z 2 c 2 ∂t 2   vx2 vy2 vz2  ∂t 2
1 ∂
1 ∂
1 ∂ 
− 2
+
+

 vx ∂x∂t vy ∂y∂t vz ∂z∂t 
2
2
(14)
2
This means that the function ψ (x′,ict′) does not satisfy the same form of equation as does
ψ ( x , ict) , and the equation is not invariant under a Galilean transformation.
14-4. In the K system the rod is at rest with its ends at x1 and x2 . The K′ system moves with a
velocity v (along the x axis) relative to K.
K
K′
x1
x2
If the observer measures the time for the ends of the rod to pass over a fixed point in the K′
system, we have
v  

 t1 − 2 x1  
c
v

1− 2

c

v
1

 
t2′ =
 t − 2 x2  
2  2
c
v

1− 2

c
t1′ =
1
2
where t1′ and t2′ are measured in the K′ system. From (1), we have
(1)
465
THE SPECIAL THEORY OF RELATIVITY
t1′ − t2′ =
v


( t1 − t2 ) − c 2 ( x1 − x2 ) 
v
1− 2
c
1
2
(2)
We also have
x1 − x2 = A
(3)
v ( t1 − t2 ) = A
(4)
v ( t1′ − t2′ ) = A ′
(5)
Multiplying (2) by v and using (3), (4), and (5), we obtain the FitzGerald-Lorentz contraction:
A′ = A 1 −
v2
c2
(6)
14-5. The “apparent shape” of the cube is that shape which would be recorded at a certain
instant by the eye or by a camera (with an infinitesimally short shutter speed!). That is, we must
find the positions that the various points of the cube occupy such that light emitted from these
points arrives simultaneously at the eye of the observer. Those parts of the cube that are farther
from the observer must then emit light earlier than those parts that are closer to the observer. An
observer, looking directly at a cube at rest, would see just the front face, i.e., a square.
When in motion, the edges of the cube are distorted, as indicated in the figures below, where
the observer is assumed to be on the line passing through the center of the cube. We also note
that the face of the cube in (a) is actually bowed toward the observer (i.e., the face appears
convex), and conversely in (b).
(a)
Cube moving toward the
observer.
(a)
Cube moving away from the
observer.
466
CHAPTER 14
14-6.
K′
K
v
x2
x1
We transform the time t at the points x1 and x2 in the K system into the K′ system. Then,
 vx  
t1′ = γ t − 21  
c 



 vx 
t2′ = γ t − 22  
c  

(1)
From these equations, we have
∆t ′ = t1′ − t2′ = −γ v
( x1 − x2 ) = −γ v∆x
c
2
1
c2
(2)
14-7.
K
K′
v
x
Suppose the origin of the K′ system is at a distance x from the origin of the K system after a time
t measured in the K system. When the observer sees the clock in the K′ system at that time, he
actually sees the clock as it was located at an earlier time because it takes a certain time for a
light signal to travel to 0. Suppose we see the clock when it is a distance A from the origin of the
K system and the time is t1 in K and t′1 in K′. Then we have
vA  

t1′ = γ  t1 − 2  

c 


c ( t − t1 ) = A



tv = x


t1v = A

We eliminate A, t1 , and x from these equations and we find
(1)
467
THE SPECIAL THEORY OF RELATIVITY
 v
t1′ = γ 1 −  t
 c
(2)
This is the time the observer reads by means of a telescope.
14-8.
The velocity of a point on the surface of the Earth at the equator is
v=
2π Re
=
τ
(
2π × 6.38 × 108 cm
)
8.64 × 10 sec
4
(1)
= 4.65 × 10 4 cm/sec
which gives
β=
v 4.65 × 10 4 cm/sec
=
= 1.55 × 10 −6
c
3 × 1010 cm/sec
(2)
According to Eq. (14.20), the relationship between the polar and equatorial time intervals is
∆t ′ =
∆t
1 

≅ ∆t  1 + β 2 

2 
1− β
(3)
2
so that the accumulated time difference is
∆ = ∆t ′ − ∆t =
1 2
β ∆t
2
(4)
Supplying the values, we find
∆=
(
) (
) (
1
× 1.55 × 10 −6 × 3.156 × 107 sec/yr × 10 2 yr
2
)
(5)
Thus,
∆ = 0.0038 sec
(6)
14-9.
v + dv
w
dm′
m + dm
The unsurprising part of the solution to the problem of the relativistic rocket requires that we
apply conservation of momentum, as was done for the nonrelativistic case. The surprising, and
key, part of the solution is that we not assume the mass of the ejected fuel is the same as the
mass lost from the rocket. Hence
p = γ mv = (γ + dγ ) ( m + dm)( v + dv ) + γ w dm′ w
where –dm is the mass lost from the rocket, dm′ is the mass of the ejected fuel,
w ≡ ( v − V ) 1 − vV c 2 is the velocity of the exhaust with respect to the inertial frame, and
(
γw ≡1
)
1 − w 2 c 2 . One can easily calculate dγ = γ 3 β dβ , ad after some algebra one obtains
(1)
468
CHAPTER 14
γ 2 m dv + v dm +
γ w dm′
w
γ
(2)
where we of course keep infinitesimals only to first order. The additional unknown dm′ is
unalarming because of another conservation law
E = γ mc 2 = (γ + dγ ) ( m + dm) c 2 + γ w dm′ c 2
(3)
Subsequent substitution of dm′ into (2) gives, in one of its many intermediate forms


γ 2 m dv  1 −
βw 
 + dm ( v − w ) = 0
c 
(4)
and will finally come to its desired form after dividing by dt
m
(
)
dv
dm
+V
1− β2 = 0
dt
dt
(5)
The quantity dt can be measured in any inertial frame, but would presumably only make sense
for the particular one in which we measure v. Interestingly, it is not important for the ejected
fuel to have an especially large kinetic energy but rather that it be near light speed, a nontrivial
distinction. For such a case, a rocket can reach 0.6c by ejecting half its mass.
14-10.
From Eq. (14.14)
x1′ = γ ( x1 − vt )
(1)
v 

t ′ = γ  t − 2 x1 
 c

(2)
Solving (1) for x1 and substituting into (2) gives


v  x′
t ′ = γ t − 2  1 + vt  

 c γ
t′ +
v
v2
t
=
γ
−
γ
x
t
t=
′
1
2
2
γ
c
c
v 

t = γ t ′ + 2 x1′ 
c


Solving (2) for t and substituting into (1) gives

 t′ v  
x1′ = γ  x1 − v  + 2 x1  
γ c


or
x1 = γ ( x1′ + vt ′ )
469
THE SPECIAL THEORY OF RELATIVITY
14-11.
θ
x1
From example 14.1 we know that, to an observer in motion relative to an object, the dimensions
of objects are contracted by a factor of
component of the stick will be
1 − v 2 c 2 in the direction of motion. Thus, the x′1
A cos θ 1 − v 2 c 2
while the perpendicular component will be unchanged:
A sin θ
So, to the observer in K′, the length and orientation of the stick are
(
)
A ′ = A sin 2 θ + 1 − v 2 c 2 cos 2 θ 

12


 cos θ 1 − v 2 c 2 
θ ′ = tan −1 
sin θ
or

cos 2 θ 
A ′ = A  sin 2 θ +
γ 2 

12
tan θ ′ = γ tan θ
14-12.
The ground observer measures the speed to be
v=
100 m
= 2.5 × 108 m/s
.4 µsec
The length between the markers as measured by the racer is
A′ = A 1 − v2 c2
2
 2.5 
= 100 m 1 −   = 55.3 meters
 3 
The time measured in the racer’s frame is given by
470
CHAPTER 14
v 

t ′ = γ  t − 2 x1 
 c

(
=
)

2.5 × 10 8 m/s (100 m ) 
−
.4
sec
µ


c2


1 − ( 2.5 3)
2
= .22 µsec
The speed observed by the racer is
v=
A′ A
= = 2.5 × 108 m/s
t′ t
∆t ′ = γ ∆t
14-13.
∆t = 1.5 µs
γ = ( 1 − 0.9992 )
−1 2
22.4
Therefore ∆t ′ 34 µs .
14-14.
K
K′
v
source
receiver
In K, the energy and momentum of each photon emitted are
E = hν 0
and
p=
hν 0
c
Using Eq. (14.92) to transform to K′:
hν 

E′ = hν = γ ( E − vp1 ) ;  p1 = − 0 

c 
v


= γ  hν 0 + hν 0 


c
471
THE SPECIAL THEORY OF RELATIVITY
So
v
ν = ν 0 γ  1 + 

= ν0
c
1+ β
1− β
2
= ν0
1+ β
1− β
which agrees with Eq. (14.31).
14-15.
From Eq. (14.33)
ν=
1− β
1+ β
ν0
Since λ = c ν
1− β
λ0 =
1+ β
λ
or
λ=
With λ 0 = 656.3 nm and β =
1− β
1+ β
λ0
4 × 10 4
, λ = 656.4 nm.
3 × 10 8
So the shift is 0.1 nm toward the red (longer wavelength).
14-16.
K′
v
K
θ′
star
θ
Earth
Consider a photon sent from the star to the Earth. From Eq. (14.92)
E′ = γ ( E − vp1 )
also
E = γ ( E′ + vp1′ )
472
CHAPTER 14
Now
E = hν , E′ = hν 0 ,
p1 = −
hν
cos θ ,
c
p1′ = −
hν 0
cos θ ′
c
Substituting yields
ν 0 = ν γ (1 + β cos θ )
and
ν = γ ν 0 (1 − β cos θ ′ )
Thus
(1 + β cos θ )(1 − β cos θ ′ ) = γ −2
1 + β cos θ − β cos θ ′ − β 2 cos θ cos θ ′ = 1 − β 2
cos θ − cos θ ′ − β cos θ cos θ ′ = − β
Solving for cos θ yields
cos θ =
cos θ ′ − β
1 − β cos θ ′
where
β =v c
θ = angle in earth’s frame
θ ′ = angle in star’s frame
14-17.
From Eq. (14.33)
ν=
1− β
ν0
1+ β
Since
ν=c λ,
λ=
We have λ = 1.5 λ 0 . This gives β =
1+ β
λ0
1− β
5
13
or
v = 1.2 × 10 8 m/sec
473
THE SPECIAL THEORY OF RELATIVITY
14-18.
K′
v
source
θ
K
light
observer
Proceeding as in example 14.11, we treat the light as a photon of energy hν.
In K ′ : E′ = hν 0 , p ′ =
In K :
hν 0
c
E = hν = γ ( E′ + ν p1 )
For the source approaching the observer at an early time we have
p1 =
hν 0
c
v
c


Thus


ν = γ  ν0 + ν0  = ν0
1+ β
1− β
For the source receding from the observer (at a much later time) we have
p1 = −
hν 0
c
and
ν = ν0
1− β
1+ β
So
ν = ν0
1+ β
source approaching observer
1− β
ν = ν0
1− β
source receding from observer
1+ β
474
CHAPTER 14
14-19.
K
source
K′
θ
v
observer
Proceeding as in the previous problem, we have
In K ′ : E′ = hν
p1′ = −
In K :
hν
hν
cos θ = −
c
c
βr
β + βt2
2
r
E = γ ( E′ + ν p1′ ) = hν 0
So
hν 0 =

 hν β
r
 hν − c β r2 + βt2  
2
2
2


 c β r + βt2
1 − β r − βt 
1


 
or
ν0 =
ν (1 − β r )
1 − β r2 − βt2
1 − β r2 − β t2
ν λ0
=
=
ν0 λ
1 − βr
For λ > λ 0 , we have
(1 − β r )2 > 1 − βr2 − βt2
βt2 > 2β r − 2β r2
βt2 > 2β r (1 − β r )
14-20.
As measured by observers on Earth, the entire trip takes
 4 lightyears  80
2
 = 3 years
0.3 c

The people on earth age
the astronaut ages
80
years. The astronaut’s clock is ticking slower by a factor of γ. Thus,
3
475
THE SPECIAL THEORY OF RELATIVITY
80
 80 
1 − 0.32 = 0.95   years
3
3
So
Those on Earth age 26.7 years.
The astronaut ages 25.4 years.
14-21.


 1
−  ( −2β ) β 






m0 v
v
d
2
+v
F= 
 = m0 

32
2
dt  1 − β 2 
 1− β

1 − β2


(
)
(1)
 v
vββ 
= m0 
+
32
2


1 − β2
 1− β

(
)
If we take v = v1e1 (this does not mean v 2 = v 3 = 0 ), we have
v v 

v1 1 1 
 v1
m0
c c =
F1 = m0 
+

2
2 32
2
1− β
 1− β
 1− β


(
F2 =
F3 =
14-22.
)
m0
1 − β2
m0
1 − β2
(
)
32
v1 = mA v1
(2)
v 2 = mt v 2
(3)
v 3 = mt v 3
(4)
The total energy output of the sun is
(
)
dE
= 1.4 × 10 3 W ⋅ m −2 × 4π R 2
dt
(1)
where R = 1.50 × 1011 m is the mean radius of the Earth’s orbit around the sun. Therefore,
dE
3.96 × 10 26 W
dt
(2)
The corresponding rate of mass decrease is
dm 1 dE
=
4.4 × 109 kg ⋅ s −1
dt c 2 dt
(3)
The mass of the sun is approximately 1.99 × 10 30 kg , so this rate of mass decrease can continue
for a time
476
CHAPTER 14
T=
1.99 × 10 30 yr
1.4 × 1013 yr
9
−1
4.4 × 10 kg ⋅ s
(4)
Actually, the lifetime of the sun is limited by other factors and the sun is expected to expire
about 4.5 × 10 9 years from now.
14-23.
From Eq. (14.67)
p 2 c 2 = E2 − E02
= ( E0 + T ) − E02
2
= 2E0 T + T 2
p 2 c 2 = 2T mc 2 + T 2
14-24. The minimum energy will occur when the four particles are all at rest in the center of
the mass system after the collision.
Conservation of energy gives (in the CM system)
2Ep = 4 mp c 2
or
Ep ,CM = 2mp c 2 = 2E0
which implies γ = 2 or β = 3 2
To find the energy required in the lab system (one proton at rest initially), we transform back to
the lab
E = γ ( E′ + vp1′ )
(1)
The velocity of K′(CM) with respect to K(lab) is just the velocity of the proton in the K′ system.
So u = v.
Then
vp1′ = v ( pCM ) = v (γ mu) = γ mv 2 = γ mc 2 β 2
Since γ = 2, β = 3 2 ,
vp1′ =
Substituting into (1)
3
E0
2
477
THE SPECIAL THEORY OF RELATIVITY
3 

7 
Elab = γ  2E0 + E0  = 2  E0  = 7E0

2 
2 
The minimum proton energy in the lab system
is 7 mp c 2 , of which 6 mp c 2 is kinetic energy.
14-25.
Let B = B0 z
v = vx i + v y j
Then
i
j
k
qv × B = q v x
0
vy
0
0
B0
= q vy B0 i − vx B0 j
F = qv × B =
d
d
( p) = γ m ( v ) gives
dt
dt
(
)
qB
d
v = 0 v y i − vx j
dt
γm
Define ω ≡ q B0 γ m
Thus
v x = ω vy and v y = −ω vx
or
vx = ω v y = −ω 2 vx
and
vy = −ω v x = −ω 2 vy
So
vx = A cos ω t + B sin ω t
vy = C cos ω t + D sin ω t
Take vx ( 0 ) = v , vy ( 0 ) = 0 . Then A = v, C = 0. Then v x ( 0 ) = ω vy ( 0 ) = 0
v y ( 0 ) = −ω vx ( 0 ) = −ω v
→ B = 0, D = − v
478
CHAPTER 14
Thus
v = i v cos ω t − j v sin ω t
Then
r=i
The path is a circle of radius
v
ω
sin ω t + j
v
ω
cos ω t
v
ω
r=
v
q B0 γ m
=
γ mv
q B0
=
p
q B0
From problem 14-22

T2 
p =  2Tm + 2 
c 

12
So

T2 
2
Tm
+

c 2 
r=
q B0
14-26.
below
12
Suppose a photon traveling in the x-direction is converted into an e − and e + as shown
e+
θ
θ
e–
before
after
Cons. of energy gives
pp c = 2Ee
where
pp = momentum of the photon
Ee = energy of e + = energy of e −
Cons. of px gives
pp = 2 pe cos θ
Dividing gives
(p
e
= momentum of e + , e −
)
479
THE SPECIAL THEORY OF RELATIVITY
pp c
pp
=c=
Ee
pe cos θ
or
pe2 c 2 cos 2 θ = Ee2
(1)
But Ee2 > pe2 c 2 , so (1) cannot be satisfied for cos 2 θ ≤ 1 .
An isolated photon cannot be converted
into an electron-positron pair.
This result can also be seen by transforming to a frame where px = 0 after the collision. But,
before the collision, px = pp c ≠ 0 in any frame moving along the x-axis. So, without another
object nearby, momentum cannot be conserved; thus, the process cannot take place.
14-27. The minimum energy required occurs when the p and p are at rest after the collision.
By conservation of energy
2Ee = 2 ( 938 MeV )
Ee = 938 MeV = T + E0
Since Ee = 0.5 MeV ,
Te+ = Te− = 937.5 MeV
14-28.
Tclassical =
1
mv 2
2
Trel = (γ − 1) mc 2 ≥ Tclassical
We desire
Trel − Tclassical
≤ 0.01
Trel
1
mv 2
2
1−
≤ 0.01
(γ − 1) mc2
1 2
v
2
≥ 0.99
(γ − 1) c 2
β2
γ −1
(
Putting γ = 1 − β 2
)
−1 2
and solving gives
≥ 1.98
480
CHAPTER 14
v ≤ 0.115 c
The classical kinetic energy will be within 1% of the correct
value for 0 ≤ v ≤ 3.5 × 107 m/sec, independent of mass.
E = γ E0
14-29.
For
E = 30 × 109 eV
E0 0.51 × 106 eV,
γ 5.88 × 10 4
γ=
1
1 − β2
β 1−
1
2γ
2
(
or β = 1 − γ −2
)
12
= 1 − 1.4 × 10 −10
(
)
v = 1 − 1.4 × 10 −10 c
= 0.99999999986 c
14-30. A neutron at rest has an energy of 939.6 MeV. Subtracting the rest energies of the
proton (938.3 MeV) and the electron (0.5 MeV) leaves 0.8 MeV.
Other than rest energies 0.8 MeV is available.
14-31.
θ
0.98c
θ
Conservation of energy gives
Eπ = 2Ep
where Ep = energy of each photon (Cons. of py implies that the photons have the same energy).
481
THE SPECIAL THEORY OF RELATIVITY
Thus
γ E0 = 2Ep
Ep =
γ E0
2
=
135 MeV
2 1 − 0.98 2
= 339 MeV
The energy of each photon is 339 MeV.
Conservation of px gives
γ mv = 2 pp cos θ where pp = momentum of each photon
(135 Mev/c ) ( 0.98 c)
2
cos θ =
2 1 − 0.98 2 ( 339 MeV/c )
= 0.98
θ = cos −1 0.98 = 11.3°
14-32.
From Eq. (14.67) we have
E2 − E02 = p 2 c 2
With E = E0 + T , this reduces to
2E0 T + T 2 = p 2 c 2
Using the quadratic formula (taking the + root since T ≥ 0) gives
T = E02 + p 2 c 2 − E0
Substituting pc = 1000 MeV
E0 ( electron ) = 0.5 MeV
E0 ( proton ) = 938 MeV
gives
Telectron = 999.5 MeV
Tproton = 433 MeV
482
CHAPTER 14
14-33.
e
120˚
p
n
120˚
ν
before
after
Conservation of py gives
pe sin 60° = pν sin 60° or pe = pν
Conservation of px gives
pp = pe cos 60° + pν cos 60° = pe
So
pe = pp = pν ≡ p
Conservation of energy gives
E0 n = Ee + Ep + Eν
E0 n = E02e + p 2 c 2 + E02p + p 2 c 2 + pc
Substituting
E0 n = 939.6 MeV
E0 p = 938.3 MeV
E0 e = 0.5 MeV
and solving for pc gives
p = 0.554 MeV/c
pp = pe = pν = 0.554 MeV/c
Substituting into
T = E − E0
= E02 + p 2 c 2 − E0
gives ( E0ν = 0 )
Tν = 0.554 MeV
Tp = 2 × 10 −4 MeV, or 200 eV
Te = 0.25 MeV
(1)
483
THE SPECIAL THEORY OF RELATIVITY
∆s′ 2 = − c 2t ′ 2 + x1′ 2 + x2′ 2 + x3′ 2
14-34.
Using the Lorentz transformation this becomes
∆s′ 2 =
v 2 x12
+ 2 x1vt
x12 + v 2t 2 − 2x1vt
c2
+
+ x22 + x32
1 − v2 c2
1 − v2 c2
−c 2t 2 −
 2 v 2 x12  2  2 v 2 2 
 x1 − c 2  − c t − c 2 t 

 + x2 + x2

=
2
3
2
2
1− v c
= − c 2t 2 + x12 + x22 + x32
So
∆s′ 2 = ∆s2
14-35. Let the frame of Saturn be the unprimed frame, and let the frame of the first spacecraft
be the primed frame. From Eq. (14.17a) (switch primed and unprimed variables and change the
sign of v)
u1 =
u1′ + v
u ′v
1 + 12
c
Substituting v = 0.9 c
u1′ = 0.2 c
gives
u1 = 0.93 c
14-36.
Since
Fµ =
d
dτ
 dX µ 
 m dτ  and X µ = ( x1 , x2 , x3 , ict )
we have
F1 =
d
dτ
F2 = m
F4 =
d
dτ
d 2 x1
 dx1 
=
m
m
 dτ 
dτ 2
d 2 x2
dτ 2
F3 = m
d 2 x3
dτ 2
d 2t
 d ( ict ) 
=
m
icm

dτ 
dτ 2

484
CHAPTER 14
Thus
F1′ = m
d 2 x1
d2
γ ( x1 − vt ) 
m
−
dτ 2
dτ 2 
=γm
F2′ = m
d 2 x1′
d 2t
γ
mv
=
= γ ( F1 + iβ F4 )
dτ 2
dτ 2
d 2 x2′
d 2 x2
m
=
= F2 ; F3′ = F3
dτ 2
dτ 2
F4′ = icm
d
dτ 2
= γ icm
  vx1  
γ  t − c 2  


d 2t
d 2 x1
γ
β
−
i
m
dτ 2
dτ 2
= γ ( F4 − iβ F1 )
Thus the required transformation equations are shown.
14-37.
From the Lagrangian
(
)
L = mc 2 1 − 1 − β 2 −
1 2
kx
2
(1)
we compute
∂L
= − kx
∂x
(2)
∂ L ∂β ∂ L
β
=
= mc
∂v ∂ v ∂β
1 − β2
(3)
Then, from (2) and (3), the Lagrange equation of motion is
d
dt
 mcβ 

 + kx = 0
 1 − β 2 
(4)
from which
mcβ
(1 − β )
2 32
+ kx = 0
(5)
Using the relation
cβ =
we can rewrite (4) as
dv dv dx
dv
=
=v
dt dx dt
dx
(6)
485
THE SPECIAL THEORY OF RELATIVITY
mc 2 β
(1 − β )
2 32
dβ
+ kx = 0
dx
(7)
1 2
kx = E
2
(8)
This is easily integrated to give
mc 2
1− β
2
+
where E is the constant of integration.
The value of E is evaluated for some particular point in phase space, the easiest being x = a;
β = 0:
1 2
ka
2
(9)
1 2
1
kx = mc 2 + ka 2
2
2
(10)
E = mc 2 +
From (8) and (9),
mc 2
1− β
2
+
Eliminating β 2 from (10), we have
β2 = 1 −
m2 c 4
2
 2 1
2
2 
mc
+
k
a
−
x


2
(
(
= k a2 − x 2
)
)
 2 k 2
2 
 mc + 4 a − x 
2
 2 k 2
2 
mc
a
x
+
−


2
(
)
(
(11)
)
and, therefore,
(
k a2 − x 2
1 dx
β=
=
c dt
)
(
(
mc + k a − x
2
)
mc 2 + k a 2 − x 2 4
2
2
)
2
(12)
The period will then be four times the integral of dt = dt(x) from x = 0 to x = a:
m
τ=4
k
a
∫
0
k

2
2 
1 + 2mc 2 a − x 
dx
k
2
2
2
2
a − x 1+
a −x
4 mc 2
(
)
(
)
(13)
Since x varies between 0 and a, the variable x a takes on values in the interval 0 to 1, and
therefore, we can define
sin φ =
from which
x
a
(14)
486
CHAPTER 14
a2 − x 2
a
(15)
dx = a 2 − x 2 dφ
(16)
cos φ =
and
We also define the dimensionless parameter,
a
2
κ≡
k
mc 2
(17)
Using (14) – (17), (13) transforms into
τ=
2a
κc
π 2
∫
(1 + 2κ
2
cos 2 φ
1 + κ 2 cos 2 φ
0
) dφ
(18)
Since ka 2 mc 2 1 for the weakly relativistic case, we can expand the integrand of (18) in a
series of powers of κ :
(1 + 2κ cos φ ) ≅ 1 + 2κ
(
(1 + κ cos φ )
2
2
2
2
12
2
 κ2

cos 2 φ  1 −
cos 2 φ 
2


)
1

≅ 1 +  2 −  κ 2 cos 2 φ
2

3
= 1 + κ 2 cos 2 φ
2
(19)
Substitution of (19) into (18) yields
2a
τ≅
κc
=
π 2

3
∫ 1 + 2 κ
0
aπ 3κ a
+
κ c 2c
2

cos 2 φ  dφ

π 2
1


φ + 2 sin 2φ 
0
(20)
Evaluating (20) and substituting the expression for κ from (17), we obtain
τ = 2π
m 3π a 2
+
k
8c 2
k
m
(21)
or,

τ = τ 0 1 +

3 ka 2 
16 mc 2 
(22)
487
THE SPECIAL THEORY OF RELATIVITY
14-38.
F=
dp d
= (γ mu)
dt dt
=m
d
(γ u)
dt
=m

d 
u


dt  1 − u 2 c 2 


(for m = constant)

2
2
 1− u c
= m


(
(
= m 1 − u2 c 2
)
)
12
 u
− u  − 2  1 − u2 c 2
 c 
1 − u2 c2
(
)
(
−3 2
)
−1 2

 du

 dt

du
dt
Thus
F=m
14-39.
(
du
1 − u2 c 2
dt
)
−3 2
The kinetic energy is
T = p 2 c 2 + m02 c 4 − m0 c 2
(1)
Tproton = 10 4 + ( 931) − 931 ≅ 936 − 931 = 5 MeV
(2)
Telectron = 10 4 + ( 0.51) − 0.51 ≅ 100 − 0.5 = 99.5 MeV
(3)
For a momentum of 100 MeV/c,
2
2
In order to obtain γ and β, we use the relation
E = mc 2 = γ m0 c 2 =
m0 c 2
1 − β2
(4)
so that
γ=
E
m0 c 2
(5)
and
β = 1−
γ electron =
1
γ2
100
≅ 200
0.51
(6)
(7)
488
CHAPTER 14
2

βelectron = 1 − 
 ≅ 0.999988
200

1
(8)
This is a relativistic velocity.
γ proton =
936
≅ 1.0054
931
(9)
2

β proton = 1 − 
 ≅ 0.1
1.0053

1
(10)
This is a nonrelativistic velocity.
14-40.
If we write the velocity components of the center-of-mass system as v j , the
transformation of pα , j into the center-of-mass system becomes
v j Eα 

pα′ , j = γ  pα , j − 2 
c 

where γ =
1
1−
v 2j
. Since in the center-of-mass system,
∑ pα′
α
(1)
,j
= 0 must be satisfied, we have
c2
∑ pα′
α
,j
v j Eα 

= ∑ γ  pα , j − 2  = 0
c 
α

(2)
or,
∑ pα c
= α
∑ Eα
(3)
T1 E1 − m0 c 2
=
T0 E0 − m0 c 2
(1)
vj
c
,j
α
14-41.
We want to compute
where T and E represent the kinetic and total energy in the laboratory system, respectively, the
subscripts 0 and 1 indicate the initial and final states, and m0 is the rest mass of the incident
particle.
The expression for E0 in terms of γ 1 is
E0 − m0 c 2γ 1
(2)
E1 can be related to E′1 (total energy of particle 1 in the center of momentum reference frame
after the collision) through the Lorentz transformation [cf. Eq. (14.92)] (remembering that for the
inverse transformation we switch the primed and unprimed variables and change the sign of v):
489
THE SPECIAL THEORY OF RELATIVITY
E1 = γ 1′ ( E1′ + cβ1′p1′ cos θ )
(3)
where p1′ = m0 cβ1γ 1′ and E1′ = m0 c 2γ 1′ :
(
)
(4)
T1 γ 1′ 2 + γ 1′ 2 β1′ 2 cos θ − 1
=
T0
γ1 −1
(5)
E1 = m0 c 2γ 1′ 2 1 + β1′ 2 cos θ
Then, from (1), (2), and (4),
For the case of collision between two particles of equal mass, we have, from Eq. (14.127),
γ 1′ 2 =
1+ γ1
2
(6)
and, consequently,
γ 1′ 2 β1′ 2 = γ 1′ 2 − 1 =
γ1 −1
2
(7)
Thus, with the help of (6) and (7), (5) becomes
T1 γ 1 − 1 + (γ 1 − 1) cos θ
=
T0
2 (γ 1 − 1)
=
1 + cos θ
2
(8)
We must now relate the scattering angle θ in the center of momentum system to the angle ψ in
the lab system.
Squaring Eq. (14.128), which is valid only for m1 = m2 , we obtain an equation quadratic in cos θ.
Solving for cos θ in terms of tan 2 ψ , we obtain
cos θ =
−
γ1 +1
1+
2
tan 2 ψ ± 1
γ1 +1
2
tan ψ
(9)
2
One of the roots given in (9) corresponds to θ = π, i.e., the incident particle reverses its path and
is projected back along the incident direction. Substitution of the other root into (8) gives
2 cos 2 ψ
T1
1
=
=
T0 1 + γ 1 + 1 tan 2 ψ 2 cos 2 ψ + (γ 1 + 1) sin 2 ψ
2
An elementary manipulation with the denominator of (10), namely,
(10)
490
CHAPTER 14
(
)
2 cos 2 ψ + (γ 1 + 1) sin 2 ψ = 2 cos 2 ψ + γ 1 1 − cos 2 ψ + sin 2 ψ
= γ 1 + sin 2 ψ + cos 2 ψ − γ 1 cos 2 ψ + cos 2 ψ
= γ 1 + 1 − γ 1 cos 2 ψ + cos 2 ψ
= (γ 1 + 1) − (γ 1 − 1) cos 2 ψ
(11)
provides us with the desired result:
2 cos 2 ψ
T1
=
T0 (γ 1 + 1) − (γ 1 − 1) cos 2 ψ
(12)
Notice that the shape of the curve changes when T1 > m0 c 2 , i.e., when γ 1 > 2 .
T1
T0
1.0
T1 = 0.1 GeV
0.8
T1 = 1 GeV
0.6
T1 = 10 GeV
0.4
0.2
0
0˚
30˚
60˚
ψ
90˚
14-42.
y
γmec2
hν
x
φ
θ
hν′
From conservation of energy, we have
hν + me c 2 = γ me c 2 + hν ′
(1)
Momentum conservation along the x axis gives
hν hν ′
cos θ + γ me v cos φ
=
c
c
(2)
Momentum conservation along the y axis gives
γ me v sin φ =
hν ′
sin θ
c
(3)
491
THE SPECIAL THEORY OF RELATIVITY
In order to eliminate φ, we use (2) and (3) to obtain
1  hν hν ′

cos θ  
−

c
γ me v  c


hν ′

sin φ =
sin θ

γ me v
cos φ =
(4)
Then,
1
cos φ + sin φ = 1 = 2 2 2
γ me v
2
Since γ =
1
1−
2
v
c2
2
and v =
c
γ
  hν  2  hν ′  2

 hν   hν ′ 
2
cos
θ
−
+
  + 


 c   c 
 c   c 

(5)
γ 2 − 1 we have
γ 2 v 2 = c 2 ( γ 2 − 1)
(6)
Substituting γ from (1) into (6), we have
γ 2v2 =
2h
h2
( ν − ν ′ ) + 2 2 (ν − ν ′ ) 2
me
me c
(7)
From (5) and (7), we can find the equation for ν′:
h2
 hν   hν ′ 
 hν   hν ′ 
2
hm
2
cos
θ
2
ν
ν
ν − ν ′)
+
−
=
−
+
′
(
)
e
2 (
 c   c 
 c   c 
c
(8)
 2me c 2

2me c 2
2
1
cos
ν
θ
ν
ν
+
−
=
′
(
)
 h
h


(9)




1
ν
ν′ = 
 1 + hν (1 − cos θ ) 


me c 2
(10)
2
2
or,
Then,
or,


E
1 − cos θ ) 
E′ = E 1 +
2 (
 me c

The kinetic energy of the electron is
−1
(11)
492
CHAPTER 14
1


T = γ me c 2 − me c 2 = hν − hν ′ = E 1 −

E
1 − cos θ ) 
 1+
2 (
me c


T=
1 − cos θ
E2 

2 
me c 1 + E 1 − cos θ 
(
)

me c 2


(12)