Solutions manual for Classical mechanics

682 Pages·2006·2.488 MB·English

by  Gregory R.D.

CLASSICAL MECHANICS
SOLUTIONS MANUAL
R. Douglas Gregory
November 2006
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Contents
1 Thealgebraandcalculusofvectors 3
2 Velocity,accelerationandscalarangularvelocity 27
3 Newton’slawsofmotionandthelawofgravitation 62
4 Problemsinparticledynamics 76
5 Linearoscillationsandnormalmodes 139
6 Energyconservation 179
7 Orbitsinacentralfield 221
8 Non-linearoscillationsandphasespace 276
9 Theenergy principle 306
10 Thelinearmomentumprinciple 335
11 Theangularmomentumprinciple 381
12 Lagrange’sequationsandconservationprinciples 429
13 ThecalculusofvariationsandHamilton’sprinciple 473
14 Hamilton’sequationsandphasespace 505
15 Thegeneraltheoryofsmalloscillations 533
16 Vectorangularvelocity 577
17 Rotatingreference frames 590
18 Tensoralgebraandtheinertiatensor 615
19 Problemsinrigidbodydynamics 646
Chapter One
The algebra and calculus
of vectors
c Cambridge University Press,2006
Chapter1 Thealgebraandcalculusofvectors 4
Problem1.1
In terms of the standard basis set i;j;k , a 2i j 2k, b 3i 4k and
f g D (cid:0) (cid:0) D (cid:0)
c i 5j 3k.
D (cid:0) C
(i) Find3a 2b 4c and a b 2.
(ii) Find a ,Cb a(cid:0)nda
b. Djed(cid:0)ucejtheanglebetweenaandb.
j j j j
(iii) Findthecomponentofc inthedirectionofaandinthedirectionofb.
(iv) Finda b,b c and.a b/ .b c/.
(v) Finda
.b c/and.a b/
candverifythattheyareequal. Istheset a;b;c
  f g
right-orleft-handed?
(vi) Byevaluatingeach side,verifytheidentitya .b c/ .a
c/b .a
b/c.
  D (cid:0)
Solution
(i)
3a 2b 4c 3.2i j 2k/ 2.3i 4k/ 4.i 5j 3k/
C (cid:0) D (cid:0) (cid:0) C (cid:0) (cid:0) (cid:0) C
8i 17j 26k:
D C (cid:0)
a b 2 .a b/
.a b/
j (cid:0) j D (cid:0) (cid:0)
. i j 2k/
. i j 2k/
D (cid:0) (cid:0) C (cid:0) (cid:0) C
. 1/2 . 1/2 22 6:
D (cid:0) C (cid:0) C D
(ii)
a 2 a
a
j j D
.2i j 2k/
.2i j 2k/
D (cid:0) (cid:0) (cid:0) (cid:0)
22 . 1/2 . 2/2 9:
D C (cid:0) C (cid:0) D
Hence a 3.
j j D
b 2 b
b
j j D
.3i 4k/
.3i 4k/
D (cid:0) (cid:0)
32 . 4/2 25:
D C (cid:0) D
Hence b 5.
j j D
a
b .2i j 2k/
.3i 4k/
D (cid:0) (cid:0) (cid:0)
2 3 . 1/ 0 . 2/ . 4/
D  C (cid:0)  C (cid:0)  (cid:0)
14:
(cid:0)
(cid:0)
(cid:0)

D
c Cambridge University Press,2006
Chapter1 Thealgebraandcalculusofvectors 5
Theangle˛ betweenaandb isthengivenby
a
b
cos˛
D a b
j jj j
14 14
:
D 3 5 D 15

Thus˛ tan 1 14.
D (cid:0) 15
(iii) Thecomponentofc inthedirectionofais
a
c
a c

D a
j j
2i j 2k
b .i 5j 3k/
(cid:0) (cid:0)
D (cid:0) C 2i j 2k
j (cid:0) (cid:0) j
1 2 . 5/ . 1/ 3 . 2/
 C (cid:0)  (cid:0) C  (cid:0)
D 3
(cid:0)
(cid:0)
(cid:0)

1
:
D 3
Thecomponentofc inthedirectionofb is
b
c
b c

D b
j j
3i 4k
b .i 5j 3k/
(cid:0)
D (cid:0) C 3i 4k
j (cid:0) j
1 3 . 5/ 0 3 . 4/
 C (cid:0)  C  (cid:0)
D 5
(cid:0)
(cid:0)
(cid:0)

9
:
D (cid:0)5
(iv)
a b .2i j 2k/ .3i 4k/
 D (cid:0) (cid:0)  (cid:0)
i j k
2 1 2
D ˇ (cid:0) (cid:0) ˇ
ˇ3 0 4ˇ
ˇ (cid:0) ˇ
ˇ ˇ
4 0 i . 8/ . 6/ j 0 . 3/ k
ˇ ˇ
D ˇ (cid:0) (cid:0)ˇ (cid:0) (cid:0) (cid:0) C (cid:0) (cid:0)
4i 2j 3k:
(cid:0)
(cid:0)
(cid:0)

D C C
c Cambridge University Press,2006
Chapter1 Thealgebraandcalculusofvectors 6
b c .3i 4k/ .i 5j 3k/
 D (cid:0)  (cid:0) C
i j k
3 0 4
D ˇ (cid:0) ˇ
ˇ1 5 3ˇ
ˇ (cid:0) ˇ
ˇ ˇ
0 20 i 9 . 4/ j . 15/ 0 k
ˇ ˇ
D ˇ (cid:0) ˇ(cid:0) (cid:0) (cid:0) C (cid:0) (cid:0)
20i 13j 15k:
(cid:0)
(cid:0)
(cid:0)

D (cid:0) (cid:0) (cid:0)
Hence
.a b/ .b c/ .4i 2j 3k/ . 20i 13j 15k/
   D C C  (cid:0) (cid:0) (cid:0)
i j k
4 2 3
D ˇ ˇ
ˇ 20 13 15ˇ
ˇ(cid:0) (cid:0) (cid:0) ˇ
ˇ ˇ
. 30/ . 39/ i . 60/ . 60/ j . 52/ . 40/ k
ˇ ˇ
D ˇ (cid:0) (cid:0) (cid:0) ˇ (cid:0) (cid:0) (cid:0) (cid:0) C (cid:0) (cid:0) (cid:0)
9i 12k:
(cid:0)
(cid:0)
(cid:0)

D (cid:0)
(v)
a
.b c/ .2i j 2k/
. 20i 13j 15k/
 D (cid:0) (cid:0) (cid:0) (cid:0) (cid:0)
2 . 20/ . 1/ . 13/ . 2/ . 15/
D  (cid:0) C (cid:0)  (cid:0) C (cid:0)  (cid:0)
3:
(cid:0)
(cid:0)
(cid:0)

D
.a b/
c .4i 2j 3k/
.i 5j 3k/
 D C C (cid:0) C
4 1/ 2 . 5/ 3 3
D  C  (cid:0) C 
3:
(cid:0)
(cid:0)
(cid:0)

D
Thesevaluesareequalandthisverifiestheidentity
a
.b c/ .a b/
c:
 D 
Sincea
.b c/ispositive,theset a;b;c mustberight-handed.
 f g
(vi) Theleftsideoftheidentityis
a .b c/ .2i j 2k/ . 20i 13j 15k/
  D (cid:0) (cid:0)  (cid:0) (cid:0) (cid:0)
i j k
2 1 2
D ˇ (cid:0) (cid:0) ˇ
ˇ 20 13 15ˇ
ˇ(cid:0) (cid:0) (cid:0) ˇ
ˇ ˇ
15 26 i . 30/ 40 j . 26/ 20 k
ˇ ˇ
D ˇ (cid:0) (cid:0) ˇ(cid:0) (cid:0) C (cid:0) (cid:0)
11i 70j 46k:
(cid:0)
(cid:0)
(cid:0)

D (cid:0) C (cid:0)
c Cambridge University Press,2006
Chapter1 Thealgebraandcalculusofvectors 7
Since
.a
c/b 2 1 . 1/ . 5/ . 2/ 3 b
D  C (cid:0)  (cid:0) C (cid:0) 
b(cid:0)
(cid:0)
(cid:0)

D
3i 4k;
D (cid:0)
.a
b/c 2 3 . 1/ 0 . 2/ . 4/ c
D  C (cid:0)  C (cid:0)  (cid:0)
14(cid:0)c 1
4.i(cid:0) 5j 3
k/(cid:0)

D D (cid:0) C
14i 70j 42k;
D (cid:0) C
therightsideoftheidentityis
.a
c/b .a
b/c .3i 4k/ .14i 70j 42k/
(cid:0) D (cid:0) (cid:0) (cid:0) C
11i 70j 46k:
D (cid:0) C (cid:0)
Thustherightandleftsidesareequalandthisverifiestheidentity.
c Cambridge University Press,2006
Chapter1 Thealgebraandcalculusofvectors 8
Problem1.2
Findtheanglebetweenanytwodiagonalsofacube.
D
E
C
B
αααα
O
a
A
FIGURE1.1 Twodiagonalsofacube.
Solution
Figure 1.1 shows a cube of side a; OE and AD are two of its diagonals. Let O
be the origin of position vectors and suppose the points A, B and C have position
vectorsai,aj,akrespectively. ThenthelinesegmentO(cid:0)!E representsthevector
ai aj ak
C C
andthelinesegmentA(cid:0)!D representsthevector
.aj ak/ ai ai aj ak:
C (cid:0) D (cid:0) C C
Let˛ betheanglebetweenOE andAD. Then
.ai aj ak/
. ai aj ak/
cos˛ C C (cid:0) C C
D ai aj ak ai aj ak
j C C jj (cid:0) C C j
a2 a2 a2 1
(cid:0) C C :
D p3a p3a D 3
(cid:0)
(cid:0)

Hence the angle between the diagonalsis cos 1 1, which is approximately 70:5 .
(cid:0) 3 ı
c Cambridge University Press,2006
Chapter1 Thealgebraandcalculusofvectors 9
Problem1.3
ABCDEF is a regular hexagon with centre O which is also the origin of position
vectors. FindthepositionvectorsoftheverticesC,D,E,F intermsoftheposition
vectorsa,b ofAandB.
C B
b
D A
O a
FIGURE 1.2 ABCDEF is a regular
E F
hexagon.
Solution
(i) The position vector c is represented by the line segment O(cid:0)!C which has the
samemagnitudeanddirectionasthelinesegmentA(cid:0)!B. Hence
c b a:
D (cid:0)
(ii) The position vector d is represented by the line segment O(cid:0)!D which has the
samemagnitudeas,butoppositedirectionto,thelinesegmentO(cid:0)!A. Hence
d a:
D (cid:0)
(iii) The position vector e is represented by the line segment O(cid:0)!E which has the
samemagnitudeas,butoppositedirectionto,thelinesegmentO(cid:0)!B. Hence
e b:
D (cid:0)
(iv) The positionvector f is represented by the line segment O(cid:0)!F which has the
c Cambridge University Press,2006
Chapter1 Thealgebraandcalculusofvectors 10
samemagnitudeas,butoppositedirectionto,thelinesegmentA(cid:0)!B. Hence
e .b a/ a b:
D (cid:0) (cid:0) D (cid:0)
c Cambridge University Press,2006