cover and first pages NS - BGUbgu.ac.il/me/courses/pdf/36212221/Book.pdf · 2011. 11. 8. · vijk...
167
ψ ψ X Y Z ′ X ′ Y ′ Z λ ˙ φ ω 0 λ z z θ ˙ φ ˙ ψ ζ l
Transcript of cover and first pages NS - BGUbgu.ac.il/me/courses/pdf/36212221/Book.pdf · 2011. 11. 8. · vijk...
-
ψ
ψX
Y
Z′X
′Y
′Z
λ
φ̇
ω 0
λ
z
z
θ
φ̇
ψ̇ζ
l
-
© Reuven Segev, Beer-Sheva, 1993
-
y, fx, v, vβa,
F, v, ββββ, a
v|v|v
x, y, zk, j, i
x, y, z
v i j k= + +v v vx y z
θn, t, r,
v v̂v̂v = v
v
v
v
v
x
y
z
{ }
=
A, B, I, P, Q, R
}k,j,i{
[A] =
A A A
A A A
A A A
xx xy xz
yx yy yz
zx zy zz
© Reuven Segev, 2002
Au)u(A
-
A
A
a
c
F, f
ifi
ijfj i
H
I]I[
0K,0J,0I
K,J,I
M
AMA
n
p
P
r
r
cr
Ar
iri
′rii
R
0R
s
t
t
T
U
© Reuven Segev, 2002
v
-
W
X,Y,Z
0,Z0,Y0X
κ
θ
θ̂θ
ρ
ψ,θ,φ
ΩΩΩΩ, ωωωω
a b⋅b, a
a b×b, a
ḟf
f 'f
u̇XYZu
u̇ ′ ′ ′X Y Zu ′ ′ ′X Y Z, ,
© Reuven Segev, 2002
-
r
)t(r = r
t
tt)t(vt
v r( ) ˙( )t t=
© Reuven Segev, 2002
-
tt)t(v)t(at
a v r( ) ˙( ) ˙̇ ( )t t t= =
uk, j, i
u i j k= + +u u ux y z
x
y
z
i
k
j
u
uy
ux
uz
x
y
r
̂r ̂θθθθ
θ
u
u ruθ
© Reuven Segev, 2002
-
r i j k
v i j k
a i j k
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
. ( ) ( ) ( ) ( )
t x t y t z t
t v t v t v t
t a t a t a t
x y z
x y z
= + +
= + +
= + +
i, j, k
v x v y v z
a v x a v y a v z
x y z
x x y y z z
= = =
= = = = = =
˙ ˙ ˙
. ˙ ˙̇ ˙ ˙̇ ˙ ˙̇
r i j k= + +c t c t btcos sin
r c t c t b t
c b t
= + +
= +
2 2 2 2 2 2
2 2 2
cos sin
.
x,yc
v i j k= − + +
= + +
= +
c t c t b
v c t c t b
c b
sin cos
sin cos
.
2 2 2 2 2
2 2
a i j= − −c t c tcos sin
a c t c t
c
= +
=
2 2 2 2cos sin
.
a i j k0 0 0 0= + +a a ax y z
© Reuven Segev, 2002
= 0t
-
v i j k
r i j k0 0 0 0
0 0 0 0
= + +
= + +
v v v
x y z
x y z
.
x x v t a t
y y v t a t
z z v t a t
x x
y y
z z
= + +
= + +
= + +
0 012 0
2
0 012 0
2
0 012 0
2.
r r v a− = +0 012 0
2t t
0r - r
0a0v2tt
v v a− =0 0t
z
r̂rθ̂θr̂90˚ur
ur̂uruuθ̂θ
θu u
u r= +u ur ˆ ˆθθθ
r̂θxx y θcosθsinθ̂θθyx
y θsinθcos
ˆ cos sin
. ˆ sin cos
r i j
i j
= +
= − +
θ θ
θ θθθ
© Reuven Segev, 2002
-
θθ
d
d
d
d
ˆsin cos
,ˆ
cos sin
ri j
i j
θθ θ
θθ θ
= − +
= − −θθ
d
d
d
d
ˆ ˆ ,ˆ
ˆr rθ θ
= = −θθθθ
rθr r= rˆ
r r( ) ( )ˆ( ( ))t r t t= θ
vr
rr
rr
=
= +
= +
d
dt
dr
dtr
d
dt
dr
dtr
d
d
d
dt
ˆˆ
, ˆˆ
θθ
v r= +˙ˆ ˙ ˆr rθθθ
v r v rr = =˙ ˙, θ θ
© Reuven Segev, 2002
-
av
rr
rr
r
=
= + + + +
= + + + +
= + + + +
d
dt
dr
dtr
d
dt
dr
dtr
d
dtr
d
dt
r rd
d
d
dtr r r
d
d
d
dt
r r r r r
˙ˆ ˙
ˆ ˙ ˆ˙
ˆ ˙ˆ
˙̇ ˆ ˙ˆ
˙ ˙ ˆ ˙̇ ˆ ˙ˆ
. ˙̇ ˆ ˙ ˙ ˆ ˙ ˙ ˆ ˙̇ ˆ ˙
θθ
θ
θθ
θ θ θθ
θ
θ θ θ θ
θθ θθθθ
θθ θθθθ
θθ θθ θθ 22 ( ˆ)−r
a r= − + +
= − = +
(˙̇ ˙ )ˆ ( ˙̇ ˙ ˙)ˆ
. ˙̇ ˙ , ˙̇ ˙ ˙
r r r r
a r r a r rr
θ θ θ
θ θ θθ
2
2
2
2
θθ
2 m3r t t( ) , ( ) , ,θ v a
x
y
rθ
θ
r
x
y
) = 2 mt(rθ̇
θ̇π
π=⋅
=2 3
16
rad
s
rad
s
= 0) = 0t (θθ π( )t t= 6 rad˙ ˙̇ , ˙̇r r= = =0 0θ
© Reuven Segev, 2002
-
v
a r r
= ⋅ =
= − ⋅ = −
2 6 12
2 6 722 2
π π
π π
ˆ ˆ
. ( ) ˆ ˆ
θθ θθms
m
s2
˙̇rra
2m/s 2π-72
)θ(r = r
)t(θ = θ
˙ ' ˙
˙̇' ˙ ' ˙̇
' ˙ ' ˙̇
. " ˙ ' ˙̇
rdr
d
d
dtr
rdr
dtr
dr
d
d
dtr
r r
= =
= +
= +
= +
θθ
θ
θ θ
θθθ θ
θ θ2
v r
a r
= +
= + − + +
r r
r r r r r
' ˙ ˆ ˙ ˆ
. ( " ˙ ' ˙̇ ˙ )ˆ ( ' ˙ ˙̇ )ˆ
θ θ
θ θ θ θ θ
θθ
θθ2 2 22
θ̇ ω=
v r
a r
= +
= − +
r r
r r r
' ˆ ˆ
. ( " ) ˆ ' ˆ
ω ω
ω ω
θθ
θθ2 22
= 0θar2θ̇ = const.
= 0θaθaa r rθ θ θ= + =2 0˙ ˙ ˙̇r
2 02 2rr rd
dtr˙ ˙ ˙̇ ( ˙)θ θ θ+ = =
r2θ̇ = const.
© Reuven Segev, 2002
-
RR= 1.5e φ̇ ω= = const.
˙, ˙̇ , ˙, ˙̇r r θ θ = 45˚φ
θ
r
x
y
e
φR
θφ
̂θθθθ
̂rr
x
y
e
R
R̂
φ̂φφφ
rR θφ
ωRR φ
θr,
tansin
cos
sin
sinˆ cos sin
ˆ sin cos
ˆ cos sin
. ˆ sin cos
θφ
φ
φθ
φ φ
φ φ
θ θ
θ θ
=+
=
= +
= − +
= +
= − +
R
R e
rR
R i j
i j
r i j
i j
φφ
θθ
eφ
R = 2.32r
© Reuven Segev, 2002
= 17.76˚θ
-
R φ
v R
i j
a R
R
i j
= +
− +
= − + +
= −
= − +
˙ ˆ ˙ˆ
ˆ
( sin cos )
( ˙̇ ˙ ) ˆ ( ˙̇ ˙ ˙)ˆ
ˆ
. (cos sin )
R R
R
R
R R R R
R
R
φ
ω
ω φ φ
φ φ φ
ω
ω φ φ
φφ
== φφ
==
φφ2
2
2
2
θr,
r v
R
R
R
r
r v
R
R
r
˙
ˆ
(sin sin cos cos )
cos( )
˙ cos( ) ,
˙
ˆ
( sin cos cos sin )
. sin( )
θ
ω φ θ φ θ
ω φ θ
θ ω φ θ
ω φ θ φ θ
ω θ φ
θ=
= ⋅
= +
= −
= −
=
= ⋅
= − +
= −
v
v r
θθ
rθ
˙ .
. ˙ .
r R= −
=
0 458
0 383
ω
θ ω
˙̇ ˙
ˆ
(cos cos sin sin )
cos( ),
r r a
R
R
r− =
= ⋅
= − +
= − −
θ
ω φ θ φ θ
ω φ θ
2
2
2
a r
© Reuven Segev, 2002
-
r r a
R
R
˙̇ ˙ ˙
ˆ
( cos sin sin cos )
. sin( )
θ θ
ω φ θ φ θ
ω φ θ
θ+ =
= ⋅
= − − +
= − −
2
2
2
a θθ
˙̇ ˙ cos( )
. ˙̇ ˙ ˙ sin( )
r r R
rr R
= − −
= − + −[ ]θ ω φ θ
θ θ ω φ θ
2 2
21 2
˙̇ .
. ˙̇ .
r R= −
= −
0 549
0 0461
2
2
ω
θ ω
)t(r = r )0t(r = 0r
r r r r r( ) ( ) ˙( ) ˙̇ ( ) ˙̇ (̇ )
!t t t t t t
tt
t0 0 0 0
2
0
3
2 3+ = + + + +∆ ∆
∆ ∆K
0rv r0 0= ˙( )t
r r r
r r v
( ) ( ) ˙( )
. ( )
t t t t t
t t t0 0 0
0 0 0
+ ≅ +
+ ≅ +
∆ ∆
∆ ∆
0r
0r
r r r r
r r v a
( ) ( ) ˙( ) ˙̇ ( )
, ( )
t t t t t tt
t t tt
0 0 0 0
2
0 0 0 0
2
2
2
+ ≅ + +
+ ≅ + +
∆ ∆∆
∆ ∆∆
© Reuven Segev, 2002
a r0 0= ˙̇ ( )t
-
0r0r
x
y
z
va
)t(s = st∆t∆ ∆ ∆s t≅ ≅| | | |r v
∆t∆t → 0
| |vr
= =d
dt
ds
dt
˙ | |s v= =v
© Reuven Segev, 2002
-
x
y
z
v
v∆t
∆ r ≅v∆t
∆s ≅ |∆r|≅ |v∆t|
1t2t
s v t dtt
t
= ∫ ( )1
2
0rv r0 0= ˙( )tt0r
tv v
= =v ṡ
t
rr
= =
ddtdsdt
d
ds
)t(u = uu)t(uut
© Reuven Segev, 2002
u
-
ut
d
dtu
d
dt( ) ( )2 0= ⋅ =u u
0
2
= ⋅
= ⋅ + ⋅
= ⋅
d
dt( )
˙ ˙
, ˙
u u
u u u u
u u
0 = ⋅u u̇
u
tsts)s(t = t
ss's'r = t
t)s(t = t)s(t'tt
't0r0rn't0rκ
κ1/ρ
r t n n" '= = =κρ1
)s(r = r0r
r r r r
r r t n
≅ + +
≅ + +
0
2
0
2
2
2
' "
.
∆∆
∆∆
ss
ss
κ
0r)s(r = r
© Reuven Segev, 2002
tn
-
κρ
t(s)
t(s + ∆s) ∆α
ρ
n(s)
n(s + ∆s)
t(s)
t(s + ∆s) ∆α
t(s +
∆s) −
t(s)
∆s
∆αt( )st( )s s+ ∆
∆∆
∆
∆
α
κ
≅+ −
≅
≅
t tt
t
( ) ( )
,
s s s
d
dss
s
t
∆
∆
ακ
s≅
κρ
∆∆
αρ
≅s
κ = 1/ρ
nt t( ) ( )s s s+ −∆n
© Reuven Segev, 2002
-
˙̇˙
˙
˙̇ ˙
˙̇ ˙
, ˙̇ ˙
rr
t
tt
tt
t n
=
= ( )
= +
= +
= +
d
dtd
dts
s sd
dt
s sd
ds
ds
dt
s sκ 2
ṡ v=
˙̇ ˙̇ ˙̇r t n t n= + = +s v s v1 2 2ρ
κ
a st = ˙̇
a v vn = =κ ρ2 1 2
)t(r = r
tnt
˙̇ ˙̇ ˙̇˙
sv
= ⋅ = ⋅r t rr
κv s
v v
v
2
2
n r t
r rr r
rr r r
= −
= − ⋅
= −⋅
˙̇ ˙̇
˙̇ ˙̇˙ ˙
. ˙̇(˙̇ ˙)˙
nκv2n
© Reuven Segev, 2002
-
κρ
κ
= = −⋅
= −⋅
1 1
1
2 2
2 2
v v
v v
˙̇(˙̇ ˙)˙
. ˙̇(˙̇ ˙)˙
rr r r
n rr r r
tttn1
˙̇ ˙̇r t t t n t× = × + ×s vκ 2
˙̇r t n t× = ×κv2
˙̇r t× = κv2
κ =×˙̇ ˙r rv3
ṙ t= vntn t×
t n t× ×( )tt n t× ×( )nn
t
t r t t n t
n
× × = × ×
=
(˙̇ ) ( )
,
κ
κ
v
v
2
2
nt r t
rr
r
r r
=× ×
=
× ×
×
(˙̇ )
,
˙˙̇
˙
˙̇ ˙
κv
v v
vv
2
32
κ
nr r r
r r=
× ×
×
˙ (˙̇ ˙)˙̇ ˙v
t
© Reuven Segev, 2002
-
)x(y = y
xt = x)t(y) = x(yy y' ˙=
r i j
r i j
r j
= +
= +
=
t y t
y
y
( )
˙ ˙
, ˙̇ ˙̇
v y
y
= +( )× = −
1 21
2˙
, ˙̇ ˙ ˙̇r r k
κ =×
=+( )
=+( )
˙̇ ˙ | ˙̇ |
˙
| "|
'
r rv
y
y
y
y3 2
32 2
321 1
r i j k= + +c t c t btcos sin
= 0t
˙ sin cos
, ˙̇ cos sin
r i j k
r i j
= − + +
= − −
c t c t b
c t c t = 0t
˙ ,
˙̇ ,
,
,
r j k
r i
tj k
= +
= −
= +
=+
+
c b
c
v b c
c b
b c
2 2
2 2
© Reuven Segev, 2002
˙̇ ˙ ,r r
i j k
j k× = − = −c
c b
bc c0 0
0
2
-
˙ (˙̇ ˙) ( )r r r
i j k
i× × =
−
= − +0
0 2
3 2c b
bc c
c cb
tnκ
κ
ρκ
=×
=+
+( )=
+
= =+
=× ×
×=
− +
+ += −
˙̇ ˙
,˙ (˙̇ ˙)
| ˙̇ ˙ |
( )
r r
nr r r
r ri
i
v
c b c
b c
c
b c
b c
c
v
c cb
b c c b c
3
2 2
2 23
22 2
2 2
3 2
2 2 2 2
1
a s
a ct
n
= = ⋅ =
= ⋅ =
˙̇ ˙̇
˙̇
r t
r n
0
ata an tn r t= −˙̇an
˙̇r t− atn˙̇r t− ata vn = κ
2
θ0v˙̇s
x
y
θ
xyjg = -ag
© Reuven Segev, 2002
-
t i j
a t
= +
= = ⋅ = −
= − = − =
= =
cos sin
˙̇ sin
sin cos
.cos
θ θ
θ
θ θ
ρθ
s a g
a a a g g g
v
a
v
g
t
n t
n
2 2 2 2 2
2 2
v i
t i
a t
=
=
= = ⋅ =
= − =
= =
v
s a
a a a g
v
g
v
g
t
n t
cos
˙̇
.( cos ) cos
θ
ρθ θ
0
2 2
2 2 2
Philosophiae naturalis""principia matematica
© Reuven Segev, 2002
-
mp
vm = p
H
H r p= ×
© Reuven Segev, 2002
-
∑ = =f p a˙ m
∑ fpa
∑ = =
∑ = =
∑ = =
f ma mx
f ma my
f ma mz
x x
y y
z z
˙̇
˙̇
, ˙̇
© Reuven Segev, 2002
-
∑ = = −
∑ = = +
f ma m r r
f ma m r r
r r (˙̇ ˙ )
( ˙̇ ˙ ˙)
θ
θ θθ θ
2
2
∑ = =
∑ = = =
f ma ms
f mamv
m v
t t
n n
˙̇
.2
2
ρκ
f A t
f A t
f B
x
y
z
= −
= −
=
sin
cos
.
ω
ω
ωA, B, = 0t(0,1,0)
0vz
˙̇ sin
˙̇ cos
. ˙̇
xA
mt
yA
mt
zB
m
= −
= −
=
ω
ω
˙ cos
˙ sin
. ˙
xA
mt c
yA
mt d
zB
mt e
= +
= − +
= +
ωω
ωω
1
1
1
© Reuven Segev, 2002
-
1, e1, d1c˙( ) , ˙( ) , ˙( )x y z v0 0 0 0 0 0= = =
cA
md e v1 1 1 00= − = =ω
, ,
˙ (cos )
˙ sin
. ˙
xA
mt
yA
mt
zB
mt v
= −
= −
= +
ωω
ωω
1
0
xA
mt t c= − +
ωω ω
2 2(sin )
yA
mt d= +
ωω
2 2cos
zB
mt v t e= + +
22
0 2
c dA
me2 2 2 20 1 0= = − =, ,ω
xA
mt t
yA
mt
zB
mt v t
= −
= + −
= +
ωω ω
ωω
2
2
20
1 1
2
(sin )
(cos )
.
˙̇ ( )xm
f v=1
)v(fdv
dt˙̇x
© Reuven Segev, 2002
-
dv
dt mf v
mdv
f vdt
=
=
1( ) ,
( ),
mdv
f vdt c t c
( )∫ ∫= + = +
c1
f v( ))v(F
F vt c
m( ) =
+
vGF
v Gt c
m=
+
mkvk
x
mg
kv
© Reuven Segev, 2002
-
mx mg kv
dv
dt mmg kv
˙̇ ,
. ( )
= −
= −1
f v mg kv( ) = −F
1 1
f v mg kv( )=
−
F vk
mg kv( ) ln( )= − −1
F vt c
m( ) =
+
t c
m kmg kv
+= − −
1ln( )
= 0) = 0t(v
cm
kmg= − ln
tm
k
mg kv
mg= −
−
ln
v
vmg
ke
k
mt= − −( )1
kv
© Reuven Segev, 2002
-
= 0x
xmg
kt
m
ke
k
mt= + −
−( )1
)x(f
˙̇ ( )xm
f x=1
˙̇˙
˙
˙˙
, ˙
xdx
dtdx
dx
dx
dt
xdx
dxd
dxx
=
=
=
= ( )12 2
d
dxx
mf x
d xm
f x dx
12
2
12
2
1
1
˙ ( ) ,
, ˙ ( )
( ) =
( ) =
12
2 1˙ ( )xm
f x dx c= +∫
˙ ( ) , ( ) ( )x G x G xm
f x dx c= = +∫2
© Reuven Segev, 2002
-
� � �������������� ����� ������������������dxdt
G x= ( )������ ������� ������� ���� ��� �������� �
������
dxG x
dt
t dxG x
d H x
( )
.( )
( )
=
= + =���������x�������������������������������������������H�������������������������������������c
��d���������������������������������� = 0t��
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m
k
x
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12
2
2
1
2
( )
,
xm
f x dx c
km
xdx c
kmx c
= +
= � +
= � +
�
�
������
G x c kmx( ) = �2 2��
����
������© Reuven Segev, 2002
G(x) =±
2c− kmx2
AdministratorRectangle
-
t =∫ dx
G(x)+d,
=±∫ dx√
2c− kmx2+d,
=±√
mk
sin−1(√
k2cm
x
)+d,
.x =±√
2cmk
sin
[√km(t−d)
]נקבל גזירה ידי ועל
, ẋ =±√
2ccos
[√km(t−d)
]v(0) = 0 ההתחלה תנאי ומהצבת
d =√
mk
π2,
.x =±√
2cmk
sin
[√km
t− π2
]התנאי) את לקיים כדי המינוס בסימן לבחור (צריך השני ההתחלה תנאי בהצבת
, x0 =−√
2cmk
sin[−π
2
]המבוקשת התוצאה את לבסוף נקבל
c =x20k2m
,
.x =−x0 sin
[√km
t− π2
]= x0 cos
√km
t
דוגמא 2.2.9מסה. חסר מוט של לאורכו חיכוך ללא להחליק חופשי מסה בעל חלקיק
של ,R המרחק ,θ̇ = ω המוט של הזוויתית המהירות נתונה ההתחלתי במצבאת לחשב דרוש .v0 למוט, המסה בין היחסית והמהירות הסיבוב, מציר החלקיקשל הערכים ואת החלקיק יגיע אליו המוט של הסיבוב מציר המינימלי המרחק
זה. במצב r̈, θ̈ , ṙ, θ̇
־ 31 ־
C© Reuven Segev 2002
-
ω
m
v0
R
v0
v R
L
α
v
Rω
v r
r
= +
= − +
v v
v R
rˆ ˆ
. ˆ ˆθ
ω
θθ
θθ0
L
L R RR
v R
R
v R= =
+=
+sinα
ω
ω
ω
ω02 2 2
2
02 2 2
˙ ,
, | | ˙
r v
v v R L
r= =
= = + =
0
02 2 2
θ ω θv
© Reuven Segev, 2002
-
θ̇ω ω
ω=
+=
+v R
L
v R
R02 2 2
02 2 2
2
a r r a r rr = − = = + =˙̇ ˙ , ˙ ˙ ˙̇θ θ θθ2 0 2 0
˙̇ ˙ ,
. ˙̇˙ ˙
r Lv R
R
r
L
= =+( )
= − =
θω
ω
θθ
2 02 2 2
32
2
20
ṙ v= − = constω0
R
fθ = 0aθ = 0
r2θ̇r R2 2 0θ̇ ω=tr R vt= −
˙( )
θω
=−
R
R vt
20
2
a r rθ θ θ= + =2 0˙ ˙ ˙̇
˙̇( )
θω
=−
2 2 03
vR
R vt
f m r r m R vtR
R vtm
R
R vtr= − = − −
−
= − −
(˙̇ ˙ ) ( )( ) ( )
θω ω2
402
4
402
30
rr
θ̇ ω= = constR
© Reuven Segev, 2002
0v
-
a r r r rr = − = − =˙̇ ˙ ˙̇θ ω2 2 0
˙̇r r= ω 2
˙̇ ˙rd
drr= ( )12 2
12
2 2
12
2 2
12
2 2
12
22 2
2
d
drr r
d r rdr
r rdr c
rr
c
˙ ,
( ˙ ) ,
˙ ,
. ˙
( ) =
=
= +
= +
∫
ω
ω
ω
ω
˙( )r r R v= = − 0
c v R
r v R r
r v R r
= −( )= − −
= − −
12 0
2 2 2
202 2 2 2
02 2 2 2
ω
ω
ω
,
˙ ( ) ,
. ˙ ( )
ṙ = 0v R r0
2 2 2 2 0− − =ω ( )
rR v
min =−ω
ω
2 202
= 00vR = minrRω = 0vRω > 0v
> 0r
θ̇˙̇θ = 0
a r r v R r
f f m v R r
θ
θ
θ θ ω ω
ω ω
= + = − −
= = − −
2 2
2
02 2 2 2
02 2 2 2
˙ ˙ ˙̇ ( ) ,
, ( )
θ
© Reuven Segev, 2002
-
v
y bx
l= sin
2π
x
y
l
v
yNN mg my− = ˙̇N
y –g
mg N
v θ
˙̇s = 0a n= κv2ny-
sin2
1πxl
=cos ,2
0πxl
=
)x(y = y
κ
π π
π π=
+
4 2
12 2
2
2
22
32
lb
x
l
b
l
x
l
sin
cos
© Reuven Segev, 2002
-
κπ
max =4 2
2lb
n
˙̇ ( )min miny v nb v
ly= = −κ
π22 2
2
4
g-
vl g
bmax=
2π
m = 0 5. kgr = +10 6sin θr
θ̇ = 2 rad/sθ = °10100N
r = 10 + sin 6θ
m
θ
© Reuven Segev, 2002
v r
a r
= +
= + − + +
r r
r r r r r
' ˙ ˆ ˙ ˆ
. ( " ˙ ' ˙̇ ˙ )ˆ ( ' ˙ ˙̇ )ˆ
θ θ
θ θ θ θ θ
θθ
θθ2 2 22
-
r
r
' cos
, " sin
=
= −
6 6
36 6
θ
θ
r
r
r
=
=
= −
= +
= − + + +
10 87
3
31 18
6 21 17
168 16 3 24 10 87
.
'
" . /
ˆ . ˆ
. ( . ˙̇ )ˆ ( . ˙̇ )ˆ
m
m/s
m s2
v r
a r
θθ
θθθ θ
fGN
100
̂θθθθ ̂r
n N
f G
t
Nˆ, ˆr θθnnt
t
© Reuven Segev, 2002
tv
r
n r
= = +
= − +
v0 266 0 964
0 964 0 266
. ˆ . ˆ
. . ˆ . ˆ
θθ
θθ
-
nN r= −0 964 0 266. ˆ . ˆN NθθN f r a+ − =G m100ˆ
0 964 0 266 100 168 16 3 0 5 24 10 87 0 5. ˆ . ˆ ˆ ˆ ( . ˙̇ ) . ˆ ( . ˙̇ ) . ˆN N fGr r r− + − = − + + +θθ θθ θθθ θ
0 964 100 84 08 1 5
0 266 12 5 44
. . . ˙̇
. . . ˙̇
N
N fG
− = − +
− + = +
θ
θ
˙̇θ = 0N fG= =16 51 16 39. .N , N
H r p r v= × = × m
˙ ˙
˙ ˙ ˙̇
.
H r r
r r r r
r f
= ×( )
= × + ×
= × ∑( )
d
dtm
m m
M r f= × ∑( )
M H= ˙
z
H r r
k
= × +
=
m v v
mv rr( ˆ ˆ )
,θ
θ
θθ
v rθ θ= ˙
H mr= 2θ̇
© Reuven Segev, 2002
-
M r r
k
= × +( )=
f f
rf
rˆ ˆ
,
θ
θ
θθ
rfd
dtmrθ θ= ( )2 ˙
fθ = 0r2θ̇
KG
KG
m s2
kgslug
lb
slug =lb
ft s2
kg
N = kgm
s2
lbm
lbmft
s2
Poundal
© Reuven Segev, 2002
-
g1 lb = 0.454 KG
g = =9 81 32 2. .m
s
ft
s2 2
fvP
P = ⋅f v
fP f v f st t= = ˙tf
1t2tf( )tW
W P t dt t t dtt
t
t
t
= = ⋅∫ ∫( ) ( ) ( )1
2
1
2
f v
PdW
dt=
f
vr
rdtd
dtdt d= =
W d= ⋅∫ f rr
r
1
2
r r r r1 1 2 2= =( ) , ( )t t
© Reuven Segev, 2002
-
α1r2rβ2r3r
αWβWα-α
2r1r
2r1rrdW W− = −α αα1r2r
β2r3r1r3rαβαβα β+αβWα β+
W W Wα β α β+ = +
r1
r2
r3
α
β
W f dx f dy f dzxx
x
y
y
y
z
z
z
= + +∫ ∫ ∫1
2
1
2
1
2
W f dsts
s
= ∫1
2
© Reuven Segev, 2002
-
r i j= +c t c tcos sincπ2
f i j=−
++
+
ye
x y
xe
x y2 2 2 2
x c t y c t= =cos , sinsin cos2 2 1t t+ =f i j= − +e t e tsin cos
v i j= − +c t c tsin cosP ec t t ec= ⋅ = + =f v (sin cos )2 2
W cedt cet
t
= ==
=
∫0
2
2
π
π
Tv m
T mv m= = ⋅122 1
2 v v
dT
dtm
m
= ⋅ + ⋅
= ⋅
= ∑( ) ⋅
12 ( ˙ ˙ )
,
v v v v
a v
f v
PdW
dt=
dT
dt
dW
dt=
T W c= +c
1r2rc T mv= =112 1
2
© Reuven Segev, 2002
-
T T W2 1− =
12 2
2 12 1
2
1
2
1
2
mv mv d f dsts
s
− = ∑( ) ⋅ = ∑∫ ∫f rr
r
12 2
2 12 1
2
1
2
mx mx f dxxx
x
˙ ˙− = ∑∫
12 2
2 12 1
2
12 2
2 12 1
2
1
2
1
2
my my f dy
mz mz f dz
y
y
y
z
z
z
˙ ˙
. ˙ ˙
− = ∑
− = ∑
∫
∫
∑ =f mst ˙̇
my bx= 2
k = 0y
0y
© Reuven Segev, 2002
-
x
m
k
y
f j= −ky
12
2
0
12 0
2
1
2
1
2
0
mv f dx f dy
kydy
ky
x
x
x
y
y
y
y
= +
= −
=
∫ ∫
∫.
vk
my= 0
f r⋅ =∫ d 0
© Reuven Segev, 2002
-
1r2r
1r2rαβα β+ −( )α
β
W W W W
W
α β α β
α β
− = +
= =
−
+ −, ( ) 0
r1
r2
α r1
r2
α
β −β
ABABW
OOAAW
O
W WA OA=
OAW AUA
−WA
© Reuven Segev, 2002
U U A W W WA A OA AO= ( ) = − = − =
-
AA O
ABW W WAB AO OB= +
W U UAB A B= −
mmgg = 9 81. m/s2zf k= −mg
1r2r
W d
mgdz
mg z z
= ⋅
= −
= −
∫
∫
f rr
r
r
r
1
2
1
2
1 2, ( )
1z2zz
U A W mg z zAO A O( ) ( )= = −
U A mgzA( ) =
f r= −kkr
1r
2r
W d
k xdx ydy zdz
k x x y y z z
kr kr
= ⋅
= − + +
= − −( ) + −( ) + −( )[ ]= −
∫
∫
f rr
r
r
r
1
2
1
2
12 2
212
22
12
22
12
12 1
2 12 2
2
( )
,
© Reuven Segev, 2002
-
U A W kr krAO A O( ) = = −12
2 12
2rO = 0
U A krA( ) =12
2
W T TAB B A= −T T U UB A A B− = −
U T U TA A B B+ = +
© Reuven Segev, 2002
-
r1
r2
r3
ri rn
O
mi mn
m1
m2 m3 F1
F2
F3
Fi
Fn
© Reuven Segev, 2002
-
nnin = 1,2,...i
iiriv rj j= ˙j kFk
Ff
ifiijfji
F f fi i ijj
n
= +=∑
1
iif ji
ijji
f fij ji= −
ijij
ijfr ri j−
( )r r f 0i j ij− × =
© Reuven Segev, 2002
-
O
mi
m j
f i f ij
f ji = −f ij
f j
ri
rj
ri − rj
ijfjif imim
r f r f r f r f
r r f
0
i ij j ji i ij j ij
i j ij
× + × = × − ×
= − ×
=
( )
.
ijmijfm r fij i ij= ×
m m 0ij ji+ =
mi
m j
© Reuven Segev, 2002
-
ijf f 0ji ij+ =
f fijj
n
i
n
ij
i j
n
== =∑∑ ∑
=
11 1,
jii
f f fiji j
n
ij ji
j
i
i
n
, = ==∑ ∑∑= +( )
1 11
j > i
jif
F f f
f f
f
i
i
n
i ij
j
n
i
n
i
i
n
ij
i j
n
i
i
n
= ==
= =
=
∑ ∑∑
∑ ∑
∑
= +
= +
=
1 11
1 1
1
,
© Reuven Segev, 2002
-
m mijj
n
i
n
ij
i j
n
== =∑∑ ∑
=
11 1,
jii
m m miji j
n
ij ji
j
i
i
n
, = ==∑ ∑∑= +( )
1 11
r F r f f
r f r f
r f m
r f
i i
i
n
i i ij
j
n
i
n
i i
i
n
i ij
i j
n
i i
i
n
ij
i j
n
i i
i
n
× = × +
= × + ×
= × +
= ×
= ==
= =
= =
=
∑ ∑∑
∑ ∑
∑ ∑
∑
1 11
1 1
1 1
1
,
,
cr
r
r
c
i i
i
n
i
i
n
m
m
= =
=
∑
∑1
1
m cṙ12
2m cṙ
© Reuven Segev, 2002
-
m mii
n
==∑
1
iiww
r w r wc i ii
n
× = ×=∑
1
zkgim = iw
r k r k
i j k i j k
i j i j
c i i
i
n
c c c i i i i
i
n
c c i i i
i
n
mg m g
mg x y z m g x y z
mg y x m g y x
× = ×
=
− = −
=
=
=
∑
∑
∑
1
1
1
0 0 1 0 0 1
. ( ) ( )
g
x
m x
my
m y
mc
i i
i
n
c
i i
i
n
= == =∑ ∑
1 1,
y
cx
z
m z
mc
i i
i
n
= =∑
1
© Reuven Segev, 2002
-
p
p v r= == =∑ ∑m mi i
i
n
i i
i
n
1 1
˙
m mc i ii
n
r r==∑
1
m mc i ii
n
˙ ˙r r p= ==∑
1
m mc i ii
n
˙̇ ˙̇ ˙r r p= ==∑
1
mi ii
n
i
i
n
i
i
n
˙̇r F f= = =∑ ∑ ∑= =
1 1 1
m c ii
n
˙̇ ˙r p f= ==∑
1
m
H
H r v r r= × = ×= =∑ ∑i i i
i
n
i i i
i
n
m m1 1
˙
© Reuven Segev, 2002
-
˙ ˙ ˙ ˙̇
.
H r r r r
r F
= × + ×
= ×
= =
=
∑ ∑
∑
i i i
i
n
i i i
i
n
i i
i
n
m m1 1
1
ṙi
if
Ḣ r f= ×=∑ i i
i
n
1
'
i'ri˙' 'r vi i=
cH
H r rc i i ii
n
m= ×=∑ ' ˙'
1
© Reuven Segev, 2002
-
r1
r2
ri rn
O
mi
mn
m1
m2
v i
vn
v1
v2
C. M.
r '2
r '1
r 'i
r 'n
rc
r r ri c i= + '˙ ˙ ˙'r r ri c i= +
H r r r
r r r r
c i i i c
i
n
i i i
i
n
i i c
i
n
m
m m
= × −
= × − ×
=
= =
∑
∑ ∑
' (˙ ˙ )
. ' ˙ ' ˙
1
1 1
ṙc
mi ii
n
r'=∑
1
m mi ii
n
cr r' '=∑ =
1
© Reuven Segev, 2002
-
r' c
mi ii
n
r 0'=∑ =
1
H r rc i i ii
n
m= ×=∑ ' ˙
1
˙'r iṙi
O
H r r
r r r
r r r r
= ×
= + ×
= × + ×
=
=
= =
∑
∑
∑ ∑
i i i
i
n
i c i i
i
n
i i i
i
n
c i i
i
n
m
m
m m
˙
( ' ) ˙
. ' ˙ ˙
1
1
1 1
O
H H r p H r v= + × = + ×c c c c cm
H r rc i i ii
n
m= ×=∑ ' ˙
1
© Reuven Segev, 2002
-
˙ ˙' ˙ ' ˙̇
(˙ ˙ ) ˙ '
˙ ˙ ˙ ˙ '
, ˙
H r r r r
r r r r F
r r r r r F
r
c i i i
i
n
i i i
i
n
i c i i
i
n
i i
i
n
i i i
i
n
c i i
i
n
i i
i
n
c
m m
m
m m
= × + ×
= − × + ×
= × − × + ×
= − ×
= =
= =
= = =
∑ ∑
∑ ∑
∑ ∑ ∑
1 1
1 1
1 1 1
mmi ii
n
i i
i
n
˙ 'r r F= =∑ ∑+ ×
1 1
˙'r iir˙ ˙r ri i×
m mi ii
n
c˙ ˙r r
=∑ =
1
ṙci'rir
∑Mc
∑ = × ==∑M r f Hc i i
i
n
c' ˙
1
a u b× =a u b× =
i j k
i j ka a a
u u u
b b bx y z
x y z
x y z= + +
abu
r f M× =r
© Reuven Segev, 2002
-
u u ux y z, ,
− + =
− =
− + =
a u a u b
a u a u b
a u a u b
z y y z x
z x x z y
y x x y z.
a u b× =
0uuu u a= +0 cc
a u a u a a u a a a u b× = × + = × + × = × =( )0 0 0c c
u0u
0
0
0
0
−
−
−
= − =
a a
a a
a a
a a a a a az y
z x
y x
x y z x y z
u u
baba
200 m/s37˚ = 100 kg3m = 60 kg, 2m = 40 kg, 1m11
3mr i j3 1800 500= +v i j k3 150 100 120= + −2mr i j k2 2000 110 400= + +
y x0.1
1m2m, 1m
© Reuven Segev, 2002
-
x
y
z
37˚
200 m / s
t = 11 s1m2m , 1m
02m , 1m
0vθxmgz-
x x v t z z v tgt
v v v v gtx z= + = + − = = −0 0 0 02
0 02( cos ) , ( sin ) , cos , sinθ θ θ θ
t = 11 s
© Reuven Segev, 2002
-
x y z
v v vc c c
cx cy cz
= = =
= = =
1760 0 727 1
160 0 12 45
m , m
m/s , , m/s
, .
. .
m m m mcr r r r= + +1 1 2 2 3 31r
r i j k1 1300 1415 3036= − +
m m m mcv v v v= + +1 1 2 2 3 3
1v , 2v
r r r i j k
r r r i j k
r r r i j k
'
'
. '
1 1
2 2
3 3
460 1415 2308
240 110 327
40 500 727
= − = − − +
= − = + −
= − = + −
c
c
c
2v , 1v
r v r v r v 0' ' '1 1 1 2 2 2 3 3 3× + × + × =m m m
v v v v11
2 2 3 3
1= − −( )
mm m mc
r v v v r v r v 0' ' '1 11
2 2 3 3 2 2 2 3 3 3
1× − −( ) + × + × =m
mm m m m mc
2v
m m m c2 2 1 2 3 1 3 3 1( ' ' ) ( ' ' ) 'r r v r r v r v− × = − × − ×
a u b× =aba r r b r r v r v= − = − × − ×m m m c2 2 1 3 1 3 3 1( ' ' ) , ( ' ' ) '
2v
© Reuven Segev, 2002
v vy x2 20 1= .
-
m2 2 1( ' ' )r r−
3v
60
i j k
240− (−460) 110− (−1415) −327− 2308
v2x v2y v2z
=100
i j k
−460− 40 −1415− 500 2308− (−727)
150 100 −120
−200
i j k
−460 −1415 2308
160 0 12.45
,
60[(110+1415)v2z+ (327+ 2308)v2y ] =100[(1415+ 500)120− (2308+ 727)100]
− 200(−1415)12.45
60[(−327− 2308)v2x − (240+ 460)v2z] =100[(2308+ 727)− (−460− 40)(−120)]
−200[2308 ⋅160− (−460)12.45].
v vy x2 20 1= .2v
v v vx y z2 2 2246 24 6 84= = = −m s , m s , m s.
v v v v11
2 2 3 3
1= − −( )
mm m mc1v
v v vx y z1 1 135 7 296 498= − = − =. m s , m s , m s
0rt
3u, 2u, 1u
r r i j k
r r i j k
r r i j k
1 0 1 1 112
2
2 0 2 2 212
2
3 0 3 3 312
2
= + + + −
= + + + −
= + + + −
u t u t u t gt
u t u t u t gt
u t u t u t gt
x y z
x y z
x y z
( )
( )
. ( )
© Reuven Segev, 2002
-
r r i j k3 2 3 2 3 2 3 2− = − + − + −( ) ( ) ( )u u t u u t u u tx x y y z z
zxy
u v u v u v u vx x y y x x y y2 2 2 2 3 3 3 3= = = =, , ,
v v v vx y x y2 2 3 3, , , 1r 2r
1800 200 150 246
500 110 100 24 6
− = −
− = −
( )
. ( . )
t
t
t
T T mii
n
i i i
i
n
= = ⋅= =∑ ∑
1
12
1
v v
v v vi c i= + 'v' i
T m
m m m
m m m
i c i c i
i
n
i c c
i
n
i i i
i
n
i i c
i
n
c c i i i
i
n
c i i
i
n
= + ⋅ +
= ⋅ + ⋅ + ⋅
= ⋅ + ⋅ + ⋅
=
= = =
= =
∑
∑ ∑ ∑
∑ ∑
12
1
12
1
12
1 1
12
12
1 1
( ' ) ( ' )
' ' '
. ' ' '
v v v v
v v v v v v
v v v v v v
© Reuven Segev, 2002
-
T mv mc i i ii
n
= + ⋅=∑12 2 12
1
v v' '
i
F vi it
t
i idt T T⋅ = −∫1
2
2 1
F vi it
t
i
n
i i
i
n
dt T T⋅ = −∫∑ ∑= =
1
2
2 1
1 1
( )f f vij
ij i
t
t
i
n
i i
i
n
dt T T+ ⋅ = −∫∑ ∑= =
Σ
1
2
2 1
1 1
( ),
Σ Σi
i ii j
ij i
t
t
dt T Tf v f v⋅ + ⋅ = −∫1
2
2 1
Σ i j ij, f
iv
Σi j
ij i,
f v⋅
© Reuven Segev, 2002
-
f12 f 21
v1 v2
−kim
f ki im g= −
W dt m g dt
g m dt
ii i
t
t
ii i
t
t
ii i
t
t
= ⋅ = − ⋅
= − ⋅
∫ ∫
∫
Σ Σ
Σ
f v k v
k v
1
2
1
2
1
2
W g m dtct
t
= − ⋅ ∫k v1
2
kg-kg-
W mg dtct
t
= − ⋅∫ k v1
2
© Reuven Segev, 2002
-
ij r ri jt t( ) ( )−
0
2
2= −[ ]= − ⋅ −[ ]= − ⋅ −
d
dtt t
d
dtt t t t
t t t t
i j
i j i j
i j i j
r r
r r r r
r r r r
( ) ( )
( ( ) ( )) ( ( ) ( ))
. (˙ ( ) ˙ ( )) ( ( ) ( ))
0 = − ⋅ −( ) ( )v v r ri j i j
f v f vij i ji j⋅ + ⋅
f v f v f f v f v vij i ji j ij ji i ji j i⋅ + ⋅ = + ⋅ + ⋅ −
=
( ) ( )
. 0
f vji i⋅( )f f 0ij ji+ =jif ir - jr
Σ Σi i it
t
ii idt d T T
i
i
f v f rr
r
⋅ = ⋅ = −∫ ∫1
2
1
2
2 1
( ) ( )Σi
i c c c
c
c
d m v vf rr
r
⋅ = −∫1
2
2 1
12
2 2
© Reuven Segev, 2002
-
x
y
θAB
xA yA
A
x¨ A
yθ
© Reuven Segev, 2002
-
rB
000 )Z,Y,X(
)z,y,x(
I0
J0
K0
i
j
k
k,j,iz,y,xI J K0 0 0, ,
X Y Z0 0 0, ,
R I J K0 0 0 0 0 0 0= + +X Y Z
r r R= ( )0
© Reuven Segev, 2002
-
r r R= ( , )0 t
ρ
m dVV
= ∫ ρ
r r1 2( ) , ( )t tt
| ( ) ( ) | | ( ) ( ) |r r r r1 2 1 2 0t td
dtt t− = − =constant ,
t
0 2 1 2 1= − ⋅ −( ) ( )v v r r
( , , )X Y Z0 0 0
tAt
Ar0XAr
ArA
© Reuven Segev, 2002
0Y0Z
-
Z,Y,XI J K( ), ( ), ( )t t t
Z,Y,X
B(X0,Y0, Z0)
I0 J0
K0
ij
k
ΑI(t)
J(t)
K(t)
rA
X0
Y0
Z0
x
y
z
R0
R
(x,y,z)
r
B Bt X
YZ
0R BR R= ( )tBtAr r= ( )tBi j k, ,
z,y,x
Ri j k, ,I J K( ), ( ), ( )t t t
BZ,Y,X
X Y Z0 0 0, ,RI J K( ), ( ), ( )t t t
R I J K= + +X Y Z0 0 0
r r R r I J K= + = + + +A A X Y Z0 0 0
r I J KA , , ,X Y Z0 0 0, ,
ArI J K, ,
© Reuven Segev, 2002
-
ArAI J K, ,
I J K, ,
X Y Z0 0 0, ,z,y,xz,y,x
X Y Z0 0 0, ,z,y,x
Z,Y,X
I J K, ,
AI J K, ,
xXAx IyZAyK
A
A
A
xX
yX
xZ
= ⋅ =
= ⋅ =
= ⋅ =
I i I i
I j I j
K i K i
cos( , )
cos( , )
cos( , )
M M M
cos( , )I jIj
[ ]A =
A A A
A A A
A A A
xX xY xZ
yX yY yZ
zX zY zZ
I i j k
J i j k
K i j k
= + +
= + +
= + +
A A A
A A A
A A A
xX yX zX
xY yY zY
xZ yZ zZ.
IJK
© Reuven Segev, 2002
-
]A[
I0 J0
K0
ij
k
X0
Y0
Z0
x
y
z
X
Y
Z I
J
KA
B
A
B
I k i j k
J j i j k
K i i j k
= − = + +
= = + +
= = + +
A A A
A A A
A A A
xX yX zX
xY yY zY
xZ yZ zZ.
A A AxX yX zX= = = −0 0 1, ,
[ ]A =
=
−
A A A
A A A
A A A
xX xY xZ
yX yY yZ
zX zY zZ
0 0 1
0 1 0
1 0 0
ψk'K', J', Iθ'I"K", J", I
φ"KK, J, I
© Reuven Segev, 2002
-
I0 = i
J0 = j
K0 = k
θ'I ')X(
ψ kz
i
j
I'
J '
K '= k
ψ
ψ
ψ
i
j
J '
K '= k
ψ
ψ
K"J"
θ
I"= I'
θ
θ
φ"K"Z
i
j
J '
K '= k
J"
I"= I'
θK = K"
I
J
φ
φ
φ
φ, θ, ψψθφ
© Reuven Segev, 2002
-
i I j J, ' , , 'J J K k K K' , " , ' , "= ='I
I I I J J" ' , , " ,=K K= "
i
j
J '
K '= k
J"
I"= I'
θ
K = K"
I
J
φ
φ
ψ
θ
i
j
k
I
J
K = K"
ψ
θ
φ I'
© Reuven Segev, 2002
-
I i j
J i j
K k
I I
J J K
K J K
I I J
J I J
K K
' cos sin
' sin cos
'
" '
" cos ' sin '
" sin ' cos '
cos " sin "
sin " cos "
, "
= +
= − +
=
=
= +
= − +
= +
= − +
=
ψ ψ
ψ ψ
θ θ
θ θ
φ φ
φ φ
y,x'Z', Y
I i j
J i j k
K i j k
I i j i j k
J i j i j k
" cos sin
" cos ( sin cos ) sin
" sin ( sin cos ) cos
cos (cos sin ) sin [cos ( sin cos ) sin ]
sin (cos sin ) cos [cos ( sin cos ) sin ]
= +
= − + +
= − − + +
= + + − + +
= − + + − + +
ψ ψ
θ ψ ψ θ
θ ψ ψ θ
φ ψ ψ φ θ ψ ψ θ
φ ψ ψ φ θ ψ ψ θ
.. sin ( sin cos ) cosK i j k= − − + +θ ψ ψ θ
I i j k
J i j k
K i j k
= − + + +
= − − + − + +
= − +
(cos cos sin cos sin ) (cos sin sin cos cos ) sin sin
( sin cos cos cos sin ) ( sin sin cos cos cos ) cos sin
. sin sin sin cos cos
φ ψ φ θ ψ φ ψ φ θ ψ φ θ
φ ψ φ θ ψ φ ψ φ θ ψ φ θ
θ ψ θ ψ θ
I i j k
J i j k
K i j k
= + +
= + +
= + +
A A A
A A A
A A A
xX yX zX
xY yY zY
xZ yZ zZ,
© Reuven Segev, 2002
-
[ ]A =
A A A
A A A
A A A
xX xY xZ
yX yY yZ
zX zY zZ
[ ]
cos cos sin cos sin sin cos cos cos sin sin sin
cos sin sin cos cos sin sin cos cos cos sin cos
sin sin cos sin cos
A =
− − −
+ − + −
φ ψ φ θ ψ φ ψ φ θ ψ θ ψ
φ ψ φ θ ψ φ ψ φ θ ψ θ ψ
φ θ φ θ θ
x
y
z
x
y
z
x
y
z
90˚y 90˚z
x
y
z
90˚z
x
y
z
x
y
z
90˚y
© Reuven Segev, 2002
-
X Y Z0 0 0, ,
r r R= +A
R I J K= + +X Y Z0 0 0I J K, ,A
R i j k i j k i j k
R i j k
= + + + + + + + +
= + + + + + + + +
X Y Z
X Y Z X Y Z X Y Z
xX yX zX xY yY zY xZ yZ zZ
xX xY xZ yX yY yZ zX zY zZ
0 0 0
0 0 0 0 0 0 0 0 0
( ) ( ) ( ) ,
, ( ) ( ) ( )
A A A A A A A A A
A A A A A A A A A
R X Y Z
R X Y Z
R X Y Z
x xX xY xZ
y yX yY yZ
z zX zY zZ
= + +
= + +
= + +
A A A
A A A
A A A
0 0 0
0 0 0
0 0 0.
R
R
R
X
Y
Z
x
y
z
xX xY xZ
yX yY yZ
zX zY zZ
=
A A A
A A A
A A A
0
0
0
}0R]{A} = [R{
r
x
y
z
x
y
z
X
Y
Z
A
A
A
xX xY xZ
yX yY yZ
zX zY zZ
=
+
A A A
A A A
A A A
0
0
0
{ } { } [ ]{ }r r RA= + A 0
A
© Reuven Segev, 2002
-
c d e= = =0 5 0 2 0 1. , . . m m , m
A
r i j kA = + +0 5. m
B
ij
k
x
y
z
X
Y
Z I
J
K
A
B
I0 J0
K0
X0
Y0
Z0
AB
c
d
e
}0R{]A[}Ar{
{ }
.
.
.
R0
0 5
0 2
0 1
=
x
y
z
=
+
−
=
0 5
1
1
0 0 1
0 1 0
1 0 0
0 5
0 2
0 1
0 6
1 2
0 5
. .
.
.
.
.
.
r i j k= + +0 6 1 2 0 5. . . m
© Reuven Segev, 2002
-
]A[K, J, I
I I J J K K
I J J K K I
⋅ = ⋅ = ⋅ =
⋅ = ⋅ = ⋅ =
1 1 1
0 0 0
, ,
. , ,
]A[
A A A
A A A
A A A
A A A A A A
A A A A A A
A A A A A A
xX yX zX
xY yY zY
xZ yZ zZ
xX xY yX yY zX zY
xY xZ yY yZ zY zZ
xZ xX yZ yX zZ zX
2 2 2
2 2 2
2 2 2
1
1
1
0
0
0
+ + =
+ + =
+ + =
+ + =
+ + =
+ + =.
[ ] [ ] [ ]A AT = 1
[ ]AT]A[ ]A[
[1]
J K I× =J K I× = −I
© Reuven Segev, 2002
-
I J K⋅ × =( ) 1
I J K⋅ × = −( ) 1
u v w⋅ ×( )
u v w⋅ × =( )
u u u
v v v
w w w
x y z
x y z
x y z
I J K⋅ × =
=
( )
.
A A A
A A A
A A A
A A A
A A A
A A A
xX yX zX
xY yY zY
xZ yZ zZ
xX xY xZ
yX yY yZ
zX zY zZ
I J K⋅ × =( ) A
A = 1 A = −1
y
© Reuven Segev, 2002
-
]A[AûA
]A[û [ ]{ˆ} {ˆ} {ˆ}A u u u= = 1û]A[
ArK, J, IAr
K, J, IAr
nn n nx y z
2 2 2 1+ + =IJI
I J⋅ = 0JIJ
K
© Reuven Segev, 2002
-
X Y Z0 0 0, ,
r r R r I J K= + = + + +A A X Y Z0 0 0
r I J KA , , ,t
r r R r I J K( ) ( ) ( ) ( ) ( ) ( ) ( )t t t t X t Y t Z tA A= + = + + +0 0 0
˙( ) ˙ ( ) ˙ ( ) ˙ ( ) ˙( ) ˙( ) ˙ ( )r r R r I J Kt t t t X t Y t Z tA A= + = + + +0 0 0
˙ , ˙ , ˙ , ˙r I J KAX Y Z0 0 0, ,
ṙA
İI J K, ,t˙ , ˙ , ˙I I I J I K⋅ ⋅ ⋅
˙ (˙ ) (˙ ) (˙ )I I I I I J J I K K= ⋅ + ⋅ + ⋅
vv̇ v⋅ = 0I
J K I K J I= × = − ×,
˙ (˙ ) (˙ )
. [(˙ ) (˙ ) ]
I I J K I I K J I
I J K I K J I
= ⋅ × − ⋅ ×
= ⋅ − ⋅ ×
I K⋅ = 0˙ ˙I K I K⋅ + ⋅ = 0˙ ˙I K K I⋅ = − ⋅
˙ [(˙ ) ( ˙ ) ]I I J K K I J I= ⋅ + ⋅ ×
( ˙ )J K I I⋅ ×
© Reuven Segev, 2002
˙ [(˙ ) ( ˙ ) ( ˙ ) ]I I J K J K I K I J I= ⋅ + ⋅ + ⋅ ×
-
ωωωω
ωω = ⋅ + ⋅ + ⋅(˙ ) ( ˙ ) ( ˙ )I J K J K I K I J
ωωωω
İ I= ×ωω
ωωωωKJJIIK
J̇K̇J̇J'IK'JI'K
'İ
˙ [(˙ ) ( ˙ ) ( ˙ ) ]′ = ′ ⋅ ′ ′ + ′ ⋅ ′ ′ + ′ ⋅ ′ ′ × ′I I J K J K I K I J I
˙′IJ̇'IJωωωω
J̇ J= ×ωω
İ
K̇ K= ×ωω
ωωωω
˙ ˙ ˙ ˙
, ( )
R I J K I J K
I J K
= + + = × + × + ×
= × + +
X Y Z X Y Z
X Y Z0 0 0 0 0 0
0 0 0
ωω ωω ωω
ωω
R
˙
. ˙ ˙
R R
r r R
= ×
= + ×
ωω
ωωA
ωωωωAṙ R= ×ωωṙ
ωωωωR
© Reuven Segev, 2002
ωωωω
-
x
y
z
ωωωω
̇r
R
Rsin
α
α θ
̇r = ρθ̇θ̂θθθ
ρ=
Rsi
n α
̇r = ωRsinα ̇r = θ̇Rsinα
ωωωω
ωωωωωωωωρ α= Rsinωωωω
Rθ̇ ω=
B, CR RC B−tωωωω
BC
R BRC
I
JK
ṙA
ṙBṙC
x
y
zωωωω
© Reuven Segev, 2002
-
˙ ˙
˙ ( )
˙ ( )
˙
, ˙
r r R
r R R R
r R R R
r R
r
C A C
A B C B
A B C B
A B
B
= + ×
= + × + −
= + × + × −
= + ×
=
ωω
ωω
ωω ωω
ωω
ωωωωR RC B−
ωωωω
tt t+ ∆∆ttBCωωωω
∆t∆tR RC Bt t t t( ) ( )+ − +∆ ∆R RC Bt t( ) ( )−
BCttt t+ ∆ωωωω
Aωωωω
ACωωωω
CC
C
˙ ˙r r R 0C A C= + × =ωω
ṙ RA C= − ×ωω
© Reuven Segev, 2002
-
x
y
z
BC
ωωωω
ṙB∆t ̇rC∆t
t
t + ∆t
ṙAωωωωṙAωωωω
RC
ṙAωωωωṙA
˙ ||rAωωωωṙA⊥ωωωω
˙ (˙ ˆ ) ˆ(˙ )
| |||r r
rA A
A= ⋅ =⋅
ωω ωωωω ωω
ωω 2
˙ ˙ ˙ ||r r rA A A⊥ = −
˙ ||rAωω × RC
ṙ RA C⊥ = − ×ωω
ṙA⊥ωωωωωωωω
© Reuven Segev, 2002
-
˙ ˙
˙ ˙
˙
, ˙
||
||
||
r r R
r r R
r R R
r
= + ×
= + + ×
= − × + ×
=
⊥
A C
A A C
A C C
A
ωω
ωω
ωω ωω
CRωωωω
ACωωωω
˙ ˙r r R= + ×C ωω
ωωωω ωωωωωωωω
ωωωω
x, yOn
OlαωωωωD
x
y
z
A
CO
l
α
D ρ
© Reuven Segev, 2002
-
OA
OAωωωω
Cθ̇ π= 2 n rad/sρ
ρ α= OC cosOC l= cosαρ α= l cos2C
v nlC = =ρθ π α˙ cos22
v v R
RC O C
OC
= + ×
= ×
ωω
ωω,
OAx, yIOAJ
KzIK
x
y
z
A
CO
l
α
D
I
K
J
ωωωω
vC
l
α
A
C
DK
I l
© Reuven Segev, 2002
-
ωω =
= +
ω
α α α
I
R I K
,
cos sin cos ,OC l l2
v J JC Cv nl= = 22π αcos
C
2 0 02
2
π α ω
α α α
ω α α
nl
l l
l
cos
cos sin cos
, sin cos
J
I J K
J
=
= −
ω π α
π α
= −
−
2
2
n
n
cot ,
. cotωω == I
D
v v R
I I K
J
D O OD
n l l
nl
= + ×
= − × +
=
ωω
( cot ) ( cos sin )
, cos
2 2 2
4 2π α α α
π α
R I KOD l l= +cos sin2 2α ααsin2
ABA
R RB AB=B
v v RB A AB= + ×ωω
ωωωω
v v RB A AB− = ×ωω
a u b× =ωωωω
ωωωωRABRABAB
© Reuven Segev, 2002
ωω⊥ωωωωRAB
-
ωω⊥ ⋅ =RAB 0ωωωω
A
B
ABBuAB
u j R i j k v i= = − + − =, ,AB A4 2m m/sBωωωωAB
A
B
u
vA
ABB Bv
Buv jB Bv=BA
v v RB A AB− = ×ωω
ωωωωBBv
© Reuven Segev, 2002
v vB A−ABR
-
ABR
( )v v RB A AB− ⋅ = 0
( ) ( )
.
v v
vB B
B
j i i j k− ⋅ − + − = + =
= −
2 4 2 4 0
0 5. m/s
v v i jB A− = − −2 0 5. m/s
v v RB A AB− = ×ωω
− −
− −
= − − + − + +
2 0 5
1 4 1
4 4
i j
i j k
i j k
.
, ( ) ( ) ( )
= xω ω ω
ω ω ω ω ω ω
y z
y z x z x y
− =
− =
=
− −
−
+
2
0 5
0
4
4
.
.
ω ω
ω ω
ω ω
y z
x z
x y
B
u
vB
N
A h
ωωωω
© Reuven Segev, 2002
-
ABuωωωωuAB
Nωωωω NωωωωuN
huNωωωωh
ωω⋅ =h 0
h uN
h N u= ×
N
N R u
i j k j
N i k
= ×
= − + − ×
= −
AB
( ) ,
.
4
h
h N u
i k j
i k
= ×
= − ×
= +
( )
,
ωω⋅ = + + ⋅ +
= + =
h i j k i k( ) ( )
.
ω ω ω
ω ωx y z
x z
,
0
v v RB A AB− = ×ωω
0
0 5
0 4
=
− =
=
+
−
+
.
,
ω ω
ω ω
ω ω
x z
x z
x y
ω ω ωx y z= − = =0 25 1 0 25. . rad/s , rad/s , rad/s
© Reuven Segev, 2002
-
AC0.1 m zx
ABBDDC
A = 30 rad/sAωCωCωωωωBD
B(−0.2,0,0)
A
C
D
0. 6
0. 3x
y
z
37°
ABz
ABCωωωωCωD
CωDBDBD
ωωωωωωωω
© Reuven Segev, 2002
-
BvA
A
v v RB A A AB= + ×ωω
R i kAB A= − =0 2 30. ,m rad/sωωA
v k i jB = × − = −30 0 2 6( . ) m/s
DCC
CR j k iCD C C= − + =0 08 0 06. . ,m rad/sωω ωC
v v R
i j k
j k
D C C CD
C
C C
= + ×
= × − +
= − −
ωω
ω
ω ω
( . . )
. . .
0 08 0 06
0 06 0 08
DBDBDB
R i j k i j kBD x y z= + − = + +0 2 0 6 0 3. . . m , ωω ω ω ω
v v R
v
i j kD B BD
B x y z
= + ×
= +
−
ωω
.
. . .
ω ω ω
0 2 0 6 0 3
DD
v v R
j k j
i j kD B BD
C C x y z
− = ×
− − + =
−
ωω ,
, . .
. . .
0 06 0 08 6
0 2 0 6 0 3
ω ω ω ω ω
0
6 0 06
0 08
0 3 0 6
0 3 0 2
0 6 0 2
−
−
=
=
=
− −
+
−
.
. .
. .
. .
. .
ω
ω
ω ω
ω ω
ω ωC
C
y z
x z
x y
D
© Reuven Segev, 2002
ωωωω
-
Cω
CωDv v RD B BD− = ×ωωv vD B−BDR
( )v v RD B BD− ⋅ = 0
( . ) . ( . )( . )6 0 06 0 6 0 3 0 08 0− + − − =ω ωC C
CωDvωωωω
ωC D= = − −300 18 24rad/s , m/sv j k
ωωωωωωωω
ω ω
ω ωz x
y x
= − −
= +
1 5 60
3 120
.
.
ωωωωAB
ABABBDBωωωωk
uωωωωN
ωω⋅ =N 0
Nu
kBD
u k R
N k u k k R
= ×
= × = × ×BD
BD
,
. ( )
© Reuven Segev, 2002
-
D
A
B
ωωωω
N
k
u
BDR
N i j= − −0 2 0 6. .
− − =0 2 0 6 0. .ω ωx y
ωωωω
ωω = − + −36 12 6i j k rad/s
UU t U t U tX Y Z( ) , ( ) , ( )
U I J K( ) ( ) ( ) ( ) ( ) ( ) ( )t U t t U t t U t tX Y Z= + +
© Reuven Segev, 2002
-
x
y
z
U(t)
I(t)
J(t)K(t)XU (t)
U Y(t)
UZ (t )
˙ ˙ ˙ ˙ ˙ ˙ ˙
˙ ˙ ˙
, ˙ ˙ ˙ ( )
U I I J J K K
I J K I J K
I J K I J K
= + + + + +
= + + + × + × + ×
= + + + × + +
U U U U U U
U U U U U U
U U U U U U
X X Y Y Z Z
X Y Z X Y Z
X Y Z X Y Z
ωω ωω ωω
ωω
ωωωωU
UU
U
˙ ˙ ˙ ˙U I J KXYZ X Y ZU U U= + +
X,Y,Z UXYZ
© Reuven Segev, 2002
-
˙ ˙U U U= + ×XYZ ωω
ΩΩΩΩ′ ′ ′X Y Z, ,′ ′ ′I J K, ,
˙ ˙U U U= + ×′ ′ ′X Y Z ΩΩ
OzΩΩΩΩ 'I
C'Z', XKz
ωωωω'Kz
x
y
z
A
CO α
D
ωωωω
vC
ΩΩΩΩ
I = I'
J = J'
K = K '
C
I'
K '
ΩΩΩΩ
ωωωω
© Reuven Segev, 2002
-
AAIx, yωωωω'K', I'I
x, yωωωωROC˙ ' 'U 0X Y Z =Ṙ ROC OC= ×ΩΩ
ω̇ωωωωωωωωωωω == −2π αncot IωωωωI
AωωωωOCω̇ω
ωωωωωωωω ω̇ωXYZ
ωωωω
ωω == −2π αncot 'I
˙' ' 'ωωX Y Z = 0
ωωωωI J K' , ' , '
˙ ˙U U U= + ×′ ′ ′X Y Z ΩΩ
ωωωωI J K' , ' , 'ΩΩΩΩ'K
˙ ˙
.' ' 'ωω ωω ΩΩ ωω
ΩΩ ωω
= + ×
= ×X Y Z
ΩΩΩΩθ̇ π= 2 n rad/sΩΩ = 2πnK' rad/s
˙ ' ( cot ' )
. ˙ cot '
ωω
ωω
= × −
= −
2 2
4 2 2π π α
π α
n n
n
K I
J
ωωωωωωωω
© Reuven Segev, 2002
-
X Y Z0 0 0, ,R
R I J K= + +X Y Z
˙ ˙R R R= + ×XYZ ωω
˙ ˙ ˙r r R= +A
˙ ˙ ˙r r R R= + + ×A XYZ ωω
ṘXYZ
˙ ˙ωω == ωω ωω ωωXYZ + ×
ωωωω
˙ ˙ωω == ωωXYZ
ωωωω
ωωωωω̇ωXYZ
˙ cot ' cotωωXYZ n n= − = −4 42 2 2 2π α π αJ J
© Reuven Segev, 2002
-
R I J K= + +X Y Z
X, Y, Z
VṘV
˙ ˙̇ ˙V R V V= = + ×XYZ ωω
˙ ˙R V R R= = + ×XYZ ωω
˙ ( ˙ )
, ˙̇ ˙ ˙
V R R
R R R
XYZ XYZ XYZ
XYZ XYZ XYZ
d
dt= + ×
= + × + ×
ωω
ωω ωω
˙̇RXYZR
˙̇RXYZ˙ ˙ωω ωωXYZ =
˙ ˙̇ ˙ ˙V R R RXYZ XYZ XYZ= + × + ×ωω ωω
˙̇ ˙R V=
˙̇ ˙̇ ˙ ˙ ( ˙ )
. ˙̇ ˙ ˙ ( )
R R R R R R
R R R R
= + × + × + × + ×
= + × + × + × ×
XYZ XYZ XYZ
XYZ XYZ
ωω ωω ωω ωω
ωω ωω ωω ωω2
r r R= +A˙̇ ˙̇ ˙̇r r R= +A
˙̇ ˙̇ ˙̇ ˙ ˙ ( )r r R R R R= + + × + × + × ×A XYZ XYZωω ωω ωω ωω2
˙̇rAA
© Reuven Segev, 2002
˙̇ ˙r r 0A A= =
-
˙̇RXYZ
ω̇ωr ˙̇θ
0 = ωωωωω̇ω × Rωωωωω̇ωωωωω
R
ωωωω ==== 0000ω̇ωωω ≠ 0000
˙̇r
2ωω × ṘXYZ
ωωωω ωωωω
ωω × ṘXYZ
© Reuven Segev, 2002
-
ω̇ωωω ==== 0000
ωωωω ≠ 0000
R( t)
ω̇ωωω ==== 0000
ωωωω ≠ 0000
Ṙ XYZ
ωωωω × R(t )ωωωω × R(t + ∆t)
R(t + ∆t)
(t)
Ṙ XYZ(t + t∆ )
ωω ωω× ×( )Rωω × Rωωωω
ωω ωω× ×( )Rωωωωωω × R
ωωωω
v( t) v( t + ∆t)
ωωωωωω × RRωωωω
90˚-R180˚R
© Reuven Segev, 2002
-
ωω ωω× × = −( )R Rω 2
− ˙ ˆθ 2rrR
zA
x
y
z
A
CO
l
α
D
I
K
J
ωωωω
vC
d
y
x
z
A
CO
l
α
D
I
K
J
ωωωω
vC
d
OD Dd
ed
© Reuven Segev, 2002
D
-
A
˙̇ ˙̇ ˙̇ ˙ ˙ ( )r r R R R R= + + × + × + × ×A XYZ XYZωω ωω ωω ωω2
˙̇rAωω == −2π αncot I
˙ cotωω = −4 2 2π αn J
AA
R I R 0 R 0= = =l XYZ XYZ, ˙ , ˙̇
˙̇ ˙ ( )r R R= × + × ×ωω ωω ωω
Rωωωω
˙̇ ( cot )
. ˙̇ cot
r J I
r K
= − ×
=
4
4
2 2
2 2
π α
π α
n l
n l
A
R I K
R I K
R I K
= +
= − −
= − −
l l
d d
e e
XYZ
XYZ
cos sin
˙ cos sin
. ˙̇ cos sin
2 2
2 2
2 2
α α
α α
α α
l
α
A
C
DK
I l
d
© Reuven Segev, 2002
-
˙̇ cos sin ( cot ) ( cos sin )
( cot ) ( cos sin )
( cot ) [( cot ) ( cos sin )]
r I K J I K
I I K
I I I K
= − − + − × + +
+ − × − − +
+ − × − × +
e e n l l
n d d
n n l l
2 2 4 2 2
2 2 2 2
2 2 2 2
2 2α α π α α α
π α α α
π α π α α α
.˙ ̇ r = −(e cos2α+ 8π 2n2l cos2α)I− 8πnd cos2αJ + [4π 2n2l cotα− e sin 2α]K
av d
ln= =
2 2
ρ αsin
˙̇sin
(sin cos )cot
R I K I KXYZd
l
d
l
d
l= − = −
2 2 2
αα α
α
l
α
A
C
DK
I l
α
an =d2
l sinα
Ṙ JXYZ d= −
˙̇cot
( cot ) ( cos sin ) ( cot ) ( )
( cot ) [( cot ) ( cos sin )]
r I K J I K I J
I I I K
= − + − × + + − × − +
+ − × − × +
d
l
d
ln l l n d
n n l l
2 22 24 2 2 2 2
2 2 2 2
απ α α α π α
π α π α α α
.˙ ̇ r =d2
l− 8π 2n2l cos2α
I+ cotα
d − 2πnl( )2
lK
© Reuven Segev, 2002
-
Aω̇ A = 25 rad/s
2Dω̇ωBD
3
B(−0.2,0,0)
A
C
D
0. 6
0.
x
y
z
37°
BDBDC
BDCC
BaBA
a a R R R R
0 0 k i 0 k k i
i j
B A AB XYZ A AB A AB XYZ A A AB= + ( ) + × + × ( ) + × ×= + + × − + + × × −
= −
˙̇ ˙ ˙ ( )
( . ) [ ( . )]
.
ωω ωω ωω ωω2
25 0 2 30 30 0 2
180 5
DC
© Reuven Segev, 2002
Cωωωω
-
a a R R R R
0 0 i j k 0 i i j k
j k
D C CD XYZ C CD C CD XYZ C C CD
C
C C
= + ( ) + × + × ( ) + × ×= + + × − + + + × × − +
= − − +
˙̇ ˙ ˙ ( )
˙ ( . . ) [ ( . . )]
. ( . ˙ ) ( . ˙ )
ωω ωω ωω ωω2
0 08 0 06 300 300 0 08 0 06
7200 0 06 5400 0 08
ω
ω ω
DBD
a a R R R R
a R R
D B BD XYZ BD BD XYZ BD
B BD BD
= + ( ) + × + × ( ) + × ×= + × + × ×
˙̇ ˙ ˙ ( )
. ˙ ( )
ωω ωω ωω ωω
ωω ωω ωω
2
Daω̇ω
a i j i j k
i j k i j k i j kD
y z
x z
x
= − + × + − +
+ − + − × − + − × + −
= −
−
− −
+ +
+ −
180 5 0 2 0 6 0 3
36 12 6 36 12 6 0 2 0 6 0 3
180
869
342
0 3 0 6
0 3 0 2
0 6
˙ ( . . . )
( ) [( ) ( . . . )]
.
(
(
(
. ˙ . ˙
. ˙ . ˙
. ˙
ωω
ω ω
ω ω
ω 00 2. ˙
)
)
)ω y
i
j
k
ωω × RBDv vD B−
DD
˙ ( ) ˙ ( )ωω ωω ωω ωω ωω ωωC CD C C CD B BD BD× + × × = + × + × ×R R a R R
˙ ( ) ( ) ˙ωω ωω ωω ωω ωω ωωC CD C C CD B BD BD× + × × − − × × = ×R R a R R
ω̇ωC
180
8069 0 06
5832 0 08
0 3 0 6
0 3 0 2
0 6 0 2
−
− −
=
=
=
− −
+
−
. ˙
. . ˙
. ˙ . ˙
. ˙ . ˙
. ˙ . ˙
ω
ω
ω ω
ω ω
ω ωC
C
y z
x z
x y
ω̇ωCD
˙ ( ) ( ) ˙ωω ωω ωω ωω ωω ωωC CD C C CD B BD BD× + × × − − × × = ×R R a R R
© Reuven Segev, 2002
-
[ ˙ ( ) ( )]ωω ωω ωω ωω ωωC CD C C CD B BD BD× + × × − − × × ⋅ =R R a R R 0
BDR
0 180 0 2 8069 0 06 0 6 5832 0 08 0 3= + − + − − −( )( . ) ( . ˙ )( . ) ( . ˙ )( . )ω ωC C
˙ , /ωC = 552 250 rad s2
Da
a j kD = − −25935 49580 m/s2
180
25066
50012
0 3 0 6
0 3 0 2
0 6 0 2
−
−
=
=
=
− −
+
−.
. ˙ . ˙
. ˙ . ˙
. ˙ . ˙
ω ω
ω ω
ω ω
y z
x z
x y
ωω ωω⋅ = ⋅ × × =N k k R( ( ))BD 0
ω̇ω
˙ ( ( )) ( ( ˙ ))ωω ωω⋅ × × + ⋅ × × =k k R k k RBD BD 0
kN
Ṙ RBD BD= ×ωω
0 0 2 0 6
36 12 6 36 12 6 0 2 0 6 0 3
= + + ⋅ − + +
+ − + − ⋅ × × − + − × + −
( ˙ ˙ ˙ ) ( . . )
( ) { [ (( ) ( . . . ))]}
.
ω ω ωx y zi j k i j
i j k k k i j k i j k
© Reuven Segev, 2002
˙ ˙ω ωx y= −720 3
-
˙ , ˙ , ˙
. ˙
ω ω ωx y z= − = = −
= − + −
74946 25222 12911
74946 25222 12911
rad/s rad/s rad/s
rad/s
2 2 2
2ωω i j k
l
X X ' Y Y '
Z Z'
α
b
x
y
z
OA
P φ C
ωωωω0
0ωωωωbCCAlα
CAPd0v0a
φ φ̇˙̇φP
© Reuven Segev, 2002
0ωωωω
-
X Y Z, ,a vXYZ XYZ,A
ZCAX
R I J
v I J
a I J
= +
= +
= +
d d
v v
a aXYZ
XYZ
cos sin
cos sin
. cos sin
φ φ
φ φ
φ φ0 0
0 0
′ ′ ′X Y Z, ,X Y Z, ,
˙ ˙ ˙
, ˙̇ ˙̇ ˙̇ ˙ ˙ ( )
r r R R
r r R R R R
= + + ×
= + + × + × + × ×
A XYZ
A XYZ XYZ
ωω
ωω ωω ωω ωω2
˙ , ˙̇r 0 r 0A A= =˙ , ˙̇R v R aXYZ XYZ XYZ XYZ= =ωω = φ̇K
X Y Z, ,′ ′ ′X Y Z, ,˙ ˙̇ωω = φK ′ ′ ′X Y Z, ,a v′ ′ ′ ′ ′ ′X Y Z X Y Z,
ωωωωRωω ωω× × = −( )R Rω 2
v I J K I J
I J
a I J K I J
K I
′ ′ ′
′ ′ ′
= + + × +
= − + +
= + + × + +
+ × +
X Y Z
X Y Z
v v d d
v d v d
a a d d
v v
0 0
0 0
0 0
0 02
cos sin ˙ ( cos sin )
( cos ˙ sin ) ( sin ˙ cos )
cos sin ˙̇ ( cos sin )
˙ ( cos sin
φ φ φ φ φ
φ φ φ φ φ φ
φ φ φ φ φ
φ φ φφ φ φ φ
φ φ φ φ φ φ φ
φ φ φ φ φ φ φ
J I J
I
J
) ˙ (cos sin )]
. ( cos ˙ sin ˙ cos ˙̇ sin )
( sin ˙ cos ˙ sin ˙̇ cos )
− +
= − − − +
+ + − +
2
0 02
0 02
2
2
d
a v d d
a v d d
′ ′ ′X Y Z, ,ωω0 0= ω kx, y, z
zxCṘXYZ˙̇RXYZa v′ ′ ′ ′ ′ ′X Y Z X Y Z,
ṙA˙̇rAA′ ′ ′X Y Z, ,
A0ωωωω
© Reuven Segev, 2002
˙
. ˙̇ ( )
r R
r RA OA
A OA
= ×
= × ×
ωω
ωω ωω0
0 0
-
RR I JAP d d= +cos sinφ φ
˙
. ˙̇ ( ) ( )
r R v R
r R a v R
= × + + ×
= × × + + × + × ×′ ′ ′
′ ′ ′ ′ ′ ′
ωω ωω
ωω ωω ωω ωω ωω0 0
0 0 0 0 02OA X Y Z AP
OA X Y Z X Y Z AP
a v′ ′ ′ ′ ′ ′X Y Z X Y Z,X, Y, Z
ωω0 0 0= − +ω α ω αcos sinI KR I KOA b b l= + +sin ( cos )α α
x, y, z
˙ ( cos ˙ sin sin sin )
, ( cos sin ˙ cos sin cos ) cos sin
r I
J K
= − − +
+ + + + − −
v d d
b l v d d d
0 0
0 0 0 0 0
φ φ φ ω α φ
ω ω α φ φ φ ω α φ ω α φ
˙̇ ( sin sin cos cos ˙ sin ˙ cos ˙̇ sin
sin sin ˙ sin cos sin cos )
( sin ˙ cos ˙ sin ˙̇ cos
r
I
= − + + − − −
− − + +
+ + − +
+
ω α ω α α φ φ φ φ φ φ φ
ω α φ ω φ α φ ω α φ
φ φ φ φ φ φ φ
ω
02
02
0 02
0 0 0 02 2
0 02
0
2
2 2
2
2
b l a v d d
v d d
a v d d
v 00 0 02
02
02 2
0 0 0
02
2
2 2
sin cos ˙ sin sin sin )
( cos cos cos sin ˙ cos cos
. sin cos cos )
α φ ω φ α φ ω φ
ω α ω α ω α φ ω φ α φ
ω α α φ
− −
+ − − − −
+
d d
b l v d
d
J
K
© Reuven Segev, 2002
-
u v w u w v u v w× × = ⋅ − ⋅( ) ( ) ( )
v w
i j k
i j k× = = − + − + −v v v
w w w
v w v w v w v w v w v wx y z
x y z
y z z y z x x z x y y x( ) ( ) ( )
u v w
i j k
i
j
× × =
− − −
= − − + +
+ − − + +
+ −
( )
( )
( )
(
u u u
v w v w v w v w v w v w
u v w u v w u v w u v w
u v w u v w u v w u v w
u v w
x y z
y z z y z x x z x y y x
y x y y y x z z x z x z
z y z z z y x x y x y x
x z x uu v w u v w u v wx x z z y z z z y− + )k
( ) ( ) ( ) ( )
( ) ( )
, ( ) (
u w v u v w i i
j j
k
⋅ − ⋅ = + + − + +
+ + + − + +
+ + + −
u w u w u w v u v u v u v w
u w u w u w v u v u v u v w
u w u w u w v u
x x y y z z x x x y y z z x
x x y y z z y x x y y z z y
x x y y z z z xx x y y z z zv u v u v w+ + ) k
© Reuven Segev, 2002
-
xy
z
′X
′Y
′Z
X
Y
Z
R
x y z, ,′ ′ ′X Y Z, ,ωω1
X Y Z, ,ωω21′ ′ ′X Y Z, ,ωω2x y z, ,
x y z, ,
ωω2R
X Y Z, ,′ ′ ′X Y Z, ,ωω21
X Y Z, ,
Ṙ R′ ′ ′ = ×X Y Z ωω21
Rx y z, ,ωω1R
˙ ˙
,
. ˙ ( )
R R R
R R
R R
= + ×
= × + ×
= + ×
′ ′ ′X Y Z ωω
ωω ωω
ωω ωω
1
21 1
1 21
x y z, ,ωω2
© Reuven Segev, 2002
-
Ṙ R= ×ωω2
( )ωω ωω ωω1 21 2+ × = ×R R
Rωω ωω ωω2 1 21= +R( )ωω ωω ωω1 21 2+ − × =R 0R
ωω ωω ωω1 21 2+ −ωω ωω ωω1 21 2+ − = 0
ωω ωω ωω2 1 21= +
ω̇ω2ω̇ω21ω̇ω1
R
˙̇Rω̇ω2
˙̇R ′ ′ ′X Y Z˙̇R
˙̇ ( ) ˙ ˙̇ ˙
. ( ) ˙
R R R R R
R R′ ′ ′ = × × + × + + ×
= × × + ×X Y Z XYZ XYZωω ωω ωω ωω
ωω ωω ωω21 21 21 21
21 21 21
2
˙̇ ( ) ˙ ˙̇ ˙
. ( ) ˙ ( ) ˙ ( )
R R R R R
R R R R R
= × × + × + + ×
= × × + × + × × + × + × ×′ ′ ′ ′ ′ ′ωω ωω ωω ωω
ωω ωω ωω ωω ωω ωω ωω ωω1 1 1 1
1 1 1 21 21 21 1 21
2
2X Y Z X Y Z
ωω ωω ωω2 1 21= +ω̇ω2
˙̇ ( ) ˙ ˙̇ ˙
( ) [( ) ] ˙
. ( ) ( ) ( ) ( ) ˙
R R R R R
R R
R R R R R
= × × + × + + ×
= + × + × + ×
= × × + × × + × × + × × + ×
ωω ωω ωω ωω
ωω ωω ωω ωω ωω
ωω ωω ωω ωω ωω ωω ωω ωω ωω
2 2 2 2
1 21 1 21 2
1 1 21 1 1 21 21 21 2
2XYZ XYZ
© Reuven Segev, 2002
-
ωω ωω ωω ωω ωω ωω ωω ωω ωω
ωω ωω ωω ωω ωω ωω ωω ωω1 1 21 1 1 21 21 21 2
1 1 1 21 21 21 1 212
× × + × × + × × + × × + × =
= × × + × + × × + × + × ×
( ) ( ) ( ) ( ) ˙
, ( ) ˙ ( ) ˙ ( )
R R R R R
R R R R R
˙ ˙ ˙ ( ) ( )ωω ωω ωω ωω ωω ωω ωω2 1 21 1 21 21 1× = × + × + × × − × ×R R R R R
u v w u w v u v w× × = ⋅ − ⋅( ) ( ) ( )
ωω ωω ωω ωω ωω ωω ωω ωω ωω ωω ωω ωω
ωω ωω ωω ωω
ωω ωω
ωω ωω
1 21 21 1 1 21 1 21 21 1 21 1
1 21 21 1
21 1
1 21
× × − × × = ⋅ − ⋅ − ⋅ − ⋅
= ⋅ − ⋅
= × ×
= × ×
( ) ( ) ( ) ( ) [( ) ( ) ]
( ) ( )
( )
. ( )
R R R R R R
R R
R
R
˙ ˙ ˙ ( )
. ˙ [ ˙ ˙ ( )]
ωω ωω ωω ωω ωω
ωω ωω ωω ωω ωω2 1 21 1 21
2 1 21 1 21
× = × + × + × ×
× = + + × ×
R R R R
R R
R
˙ ˙ ˙ ( )ωω ωω ωω ωω ωω2 1 21 1 21= + + ×
ω̇ω21ωω21′ ′ ′X Y Z, ,ωω21
ω̇ω21 ′ ′ ′X Y Z
′ ′ ′X Y Z, ,x y z, ,
˙ ˙ωω ωω ωω ωω21 21 1 21xyz X Y Z= + ×′ ′ ′
ω̇ω21xyzωω21x y z, ,ω̇ω21 ′ ′ ′X Y Z
˙ ˙ωω ωω ωω ωω21 21 1 21xyz = + ×
© Reuven Segev, 2002
˙ ˙ ˙ωω ωω ωω2 1 21= + xyz
-
ω̇ω2ωω ωω ωω2 1 21= +
˙̇R
ωω ωω ωω ωω1 1 3 3, ˙ , , ˙
ωω21v̂ωω32
ûω̇ω21ω̇ω32
ωωωω
N
û
̂v
ωω ωω ωω ωω3 1 21 32= + +
ωω ωω21 21 32 32= =ω ωˆ , ˆv u
ωω ωω
ωω ωω3 1 21 32
3 1 21 32
= + +
− = +
ω ω
ω ω
ˆ ˆ ,
. ˆ ˆ
v u
v u
v̂ûNv̂ûN
© Reuven Segev, 2002
-
( ) ˆ , ( ) ˆ , ( )ωω ωω ωω ωω ωω ωω3 1 21 3 1 32 3 1 0− ⋅ = − ⋅ = − ⋅ =v u Nω ω
ωω ωω3 1⋅ = ⋅N N
1ωωωωNkωω3 0⋅ =N
˙ ˙ ˙ , ˙ ˙ ˙ωω ωω ωω ωω ωω ωω ωω ωω ωω ωω2 1 21 1 21 3 2 32 2 32= + + × = + + ×
ω̇ω2ωω ωω ωω2 1 21= +ω̇ω3
˙ ˙ ˙ ˙ ( )ωω ωω ωω ωω ωω ωω ωω ωω ωω3 1 21 1 21 32 1 21 32= + + × + + + ×
ω̇ω32ω̇ω21˙ ˙ ˆ , ˙ ˙ ˆωω ωω21 21 32 32= =ω ωv u
˙ ˙ ( ) ˙ ˆ ˙ ˆωω ωω ωω ωω ωω ωω ωω3 1 1 21 1 21 32 21 32− − × − + × = +ω ωv u
v̂û
˙ [ ˙ ˙ ( ) ] ˆ
. ˙ [ ˙ ˙ ( ) ] ˆ
ω
ω21 3 1 1 21 1 21 32
32 3 1 1 21 1 21 32
= − − × − + × ⋅
= − − × − + × ⋅
ωω ωω ωω ωω ωω ωω ωω
ωω ωω ωω ωω ωω ωω ωω
v
u
© Reuven Segev, 2002
-
θ
© Reuven Segev, 2002
-
Σ i im( )∫ ( )m dm
rr
c
mdm
m=∫
m c˙̇r F= ΣΣ
FΣΣΣΣ
H r v= ×∫ dmm
H r v r vcm m
dm dm= ′ × ′ = ′ ×∫ ∫
H H r v= + ×c c cm
ΣΣM H= ˙
ΣΣM Hc c= ˙
T dmm
= ⋅∫12 v v
T dmcm
= ′ ⋅ ′∫12 v v
© Reuven Segev, 2002
-
T T mvc c= +12
2
W T T12 2 1= −
12W
H r v= ×∫ dmm′rr'vv
AR = r
v R R= = ×˙ ωω
H R R= × ×∫ ( )ωω dmm
dm
A
dm
rcc.m.
O
R = r
R = ′r
A
x y
z
X
YZ
x
y
z
X
YZ
© Reuven Segev, 2002
-
R r= ′′ = = ×v R R˙ ωω
H r v R Rcm m
dm dm= ′ × ′ = × ×∫ ∫ ( )ωω
c
H R R= × ×∫ ( )ωω dmmRr
R
R R R R R R× × = ⋅ − ⋅( ) ( ) ( )ωω ωω ωω
R R I I
J J
K K
× × = + + − + + +
+ + + − + + +
+ + + − + +
=
+
+
+ −
( ) ( ) ( )
( ) ( )
( ) ( )
.
[
[
[
( )
ωω X Y Z X Y Z X
X Y Z X Y Z Y
X Y Z X Y Z Z
Y Z XY
X X Y Z
Y X Y Z
Z X Y Z
X
2 2 2
2 2 2
2 2 2
2 2
ω ω ω ω
ω ω ω ω
ω ω ω ω
ω ωω ω
ω ω ω
ω ω ω
Y Z
X Y Z
X Y Z
XZ
YX X Z YZ
ZX ZY X Y
−
− + + −
− − + +
+
+
]
( ) ]
( ) ]
2 2
2 2
I
J
K
ωωωω
H R R
I
J
K
= × ×
= + − − +
+ − + + − +
+ − − + +
∫∫ ∫ ∫
∫ ∫ ∫∫ ∫ ∫
( )
( )
( )
. ( )
[ ][ ][ ]
ωω dm
Y Z dm XYdm XZdm
XY X Z dm YZdm
ZXdm ZYdm X Y dm
m
mX
mY
mZ
mX
mY
mZ
mX
mY
mZ
2 2
2 2
2 2
ω ω ω
ω ω ω
ω ω ω
© Reuven Segev, 2002
-
H Y Z dm XYdm XZdm
H XYdm X Z dm YZdm
H ZXdm ZYdm X Y dm
Xm
Xm
Ym
Z
Ym
Xm
Ym
Z
Zm
Xm
Ym
Z
= + − −
= − + + −
= − − + +
∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫
( )
( )
. ( )
2 2
2 2
2 2
ω ω ω
ω ω ω
ω ω ω
H
H
H
X
Y
Z
XX XY XZ
YX YY YZ
ZX ZY ZZ
X
Y
Z
=
I I I
I I I
I I I
ω
ω
ω
[ ]
( )
( )
( )
I =
=
+ − −
− + −
− − + +
∫ ∫ ∫∫ ∫ ∫∫ ∫ ∫
I I I
I I I
I I I
XX XY XZ
YX YY YZ
ZX ZY ZZ
m m m
m m m
m m m
Y Z dm XYdm XZdm
XYdm X Z dm YZdm
ZXdm ZYdm X Y dm
2 2
2 2
2 2
]I[
H
H
H
X XX X XY Y XZ Z
Y YX X YY Y YZ Z
Z ZX X ZY Y ZZ Z
= + +
= + +
= + +
I I I
I I I
I I I
ω ω ω
ω ω ω
ω ω ω,
{ } [ ]{ ]H = I ω
I]I[
H = I( )ωω
H
© Reuven Segev, 2002
-
H = I( )ωω
I I I I I IYX XY ZX XZ YZ ZY= = =, ,
x, yz
zωzH
zzI
X
Y
Z
X, Y
© Reuven Segev, 2002
)X,Y,Z()Z-X,Y,(
-
)Z-X,Y,(YZdm)X,Y,Z(
IYZ = 0
XZI
IXYYXωω = I
H
H
H
X XX
Y YX
Z ZX
= ⋅ =
= ⋅ =
= ⋅ =
H I
H J
H K
I
I
I.
H I= =I I( ) ( )ωω
I
I
I
XX
YX
ZX
= ⋅
= ⋅
= ⋅
I I
J I
K I
I
I
I
( )
( )
. ( )
JK
I
I
I
I
I
I
XY
YY
ZY
XZ
YZ
ZZ
= ⋅
= ⋅
= ⋅
= ⋅
= ⋅
= ⋅
I J
J J
K J
I K
J K
K K
I
I
I
I
I
I
( )
( )
. ( )
( )
( )
( )
ωωωωXYZ
HY YX x= I ω HZ ZX x= I ωωωωωHωωωωnnωωωωnωωωωH
1I H = I1ωω
H
I( )ωω ωω= I1
n
1Inωωωω
© Reuven Segev, 2002
n
-
ωωωω
H
ωωωω = ωn
n
H = I1ωωωω
n n n1 2 3, ,
I I I1 2 3, ,
I
I
. I
( )
( )
( )
ω ω
ω ω
ω ω
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
n n
n n
n n
=
=
=
I
I
I
X, Y, Z K, J, In n n1 2 3, ,
I
I
. I
( )
( )
( )
ω ω
ω ω
ω ω
X X
Y Y
Z Z
I I
J J
K K
=
=
=
I1
2
3
I
I
© Reuven Segev, 2002
H
H
H
X X
Y Y
Z Z
=
=
=
I1
2
3
ω
ω
ω
I
I,
-
H
H
H
X XX X XY Y XZ Z X
Y YX X YY Y YZ Z Y
Z ZX X ZY Y ZZ Z Z
= + + =
= + + =
= + + =
I I I
I I I I
I I I I
ω ω ω ω
ω ω ω ω
ω ω ω ω
I1
2
3.
ωωωω
I I I I I
I I I I I IXX YY ZZ
XY YX XZ ZX YZ ZY
= = =
= = = = = =
I1 2 30
, , ,
,
[ ]I =
I1
2
3
0 0
0 0
0 0
I
I
X, Y, ZX Y Zc c c, ,
X, Y, Zc
I I
I I
I I
I I
I I
I I
XX X X c c
YY Y Y c c
ZZ Z Z c c
XY X Y c c
XZ X Z c c
YZ Y Z c c
c c
c c
c c
c c
c c
c c
m Y Z
m X Z
m X Y
mX Y
mX Z
mY Z
= + +
= + +
= + +
= −
= −
= −
( )
( )
( )
.
2 2
2 2
2 2
© Reuven Segev, 2002
-
a
c
bX
Y
Z
IXX =112 m (a
2 + c2 )
IYY =1
12 m ( b2 + c2 )
X
Y
Z
R
I XX = I YY = IZZ =25 mR
2
X
Y
Z
R
l
IXX =12 mR
2
IYY =14 mR
2 + 112 ml2
I XX =3
10 mR2
IYY = 320 mR2 + 35 mh
2
X
Y
Z
h
© Reuven Segev, 2002
-
m
X
Y
Z
b c
m ω 0
[ ]
( )
I =
−
−
+
mc mbc
mbc mb
m b c
2
2
2 2
0
0
0 0
ωω = ω0I
{ } [ ]{ }
(
H
mc mbc
mbc mb
m b c
mc
mbc= =
−
−
+
= −
I ω
ω ω
ω
2
2
2 2
0 02
0
0
0
0 0
0
0 0
IYX mbc= −Y
H= r× mv = mr× (ωωωω×R )
= m(bI+ cJ)× [(ω0I)× (bI+ cJ)
. = ω0mc2I−ω0mbcJ
© Reuven Segev, 2002
-
Z10 rad/s14 kg
0.2
0.1
0.1
0.1
0
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