Kinematics in 3-D

taupeselectionMechanics

Nov 14, 2013 (3 years and 10 months ago)

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Kinematics in 3
-
D


1. Position

a) Rectangular Coordinates: x(t), y(t), and z(t)

r

= x
x

+ y
y

+ z
z
, where
x
,
y

and
z

are constant unit vectors.


b) Spherical Coordinates: r(t),

(t) and

(t)

Here r is the radial coordinate, a distance measured from the
origin to the location of the
point. The angle


is the angle
r

makes with the z
-
axis. The angle


is the angle the
projection of
r

on the x
-
y plane makes with the x
-
axis.


We need to determine the unit vectors for this system, and we need to determine t
he
transformation equations to the rectangular system:


r

= sin(

)cos(

x

+ sin(

)sin(

)
y

+ cos(

)
z

=
r
(

,

)



=
cos(

)cos(

x

+ cos(

)sin(

)
y

-

sin(

)
z

=

(

,

)




=
-
sin(

)
x

+ cos(

)
y

=

(

)


Notice that the


摩牥c瑩潮⁨o猠s⁤楲 c瑩潮‹o
o

di
fferent from the
r

direction (replace


with

+90
o
). Notice that the


摩dec瑩潮⁨o猠a⁤楲 c瑩潮‹o
o

different from the projection
direction (replace


with

+90
o

in the normal projection direction of cos(

)
x

+ sin(

)
y

.


In spherical coordinates, like po
lar, the position vector is simply:
r

= r
r

. While the r
dependence is explicit, both the


and the


dependence are "hidden" in the unit vector,
r
.



2. Velocity

a) Rectangular Coordinates:

v

= d
r
/dt = d[x
x

+ y
y

+ z
z
]/dt = (dx/dt)
x

+ (dy/dt)
y

+ (dz/dt)
z

= v
x
x

+ v
y
y

+ v
z
z
.


b) Spherical Coordinates:

v

= d
r
/dt = d[r
r
]/dt = (dr/dt)
r

+ r(d
r
/dt) .


We note that the unit vector,
r
, is not a constant. As we did with polar, we must
determine what the derivatives of the unit vectors in spherical coordinates are. Note that
d
r
/dt = (

r
/


)(d

/dt) + (

r
/


)(d

/dt).



r
/



=

[sin(

)cos(

x

+ sin(

)sin(

)
y

+ cos(

)
z
]/



=

cos(

)cos(

)
x

+ cos(

)sin(

)
y

-

sin(

)
z

=

.


渠獩浩污l⁦a獨楯sⰠ睥⁣a渠晩湤⁴桥n摥物癡瑩癥猠潦sa汬⁴桲 e⁵湩 ⁶ c瑯牳⁷楴t⁲ 獰sc琠t漠
扯瑨b


and

. We summarize these on the next page:



r
/



=




r
/



= sin(

)




/



=
-
r



/



= cos(

)




/



= 0



/



=
-
sin(

)
r

-

cos(

)





We潷⁨ 癥⁦潲⁴ e⁶ l潣楴y⁩渠獰桥物ca氠捯潲摩da瑥猺


v

= d
r
/dt = d[r
r
]/dt = (dr/dt)
r

+ r(d
r
/dt) =

r'
r

+ r[(

r
/


)(d

/dt) + (

r
/


)(d

/dt)] =

r'
r

+ r

'


+⁲

'sin(

)


= †
v

.


We recognize the first term as the regular radial velocity, the second term is the circular
velocity around an axis perpendicular to the z axis (

r), and the third term is the circular
velocity around the z axis
-

but this has a rad
ius not of r, but of [r sin(

)]. On the earth
these three velocities would correspond to: 1) velocity up or down (away from or toward
the earth; 2) velocity North or South; and 3) velocity East or West.



3. Acceleration

a) Rectangular Coordinates:

a

= d
v
/dt = d[v
x
x

+ v
y
y
+ v
z
z
]/dt = (dv
x
/dt)
x

+ (dv
y
/dt)
y
+ (dv
z
/dt)
z

= a
x
x

+ a
y
y
+ a
z
z

.


b) Spherical Coordinates:

a

= d
v
/dt = d[r'
r

+ r

'


+⁲

'sin(

)

]⽤/


=†⡤ '⽤/)
r

+ r'(d
r
/dt) + (dr/dt)

'


+⁲⡤

'/dt)


+⁲

'(d

⽤/⤠)
摲⽤/)

'sin(

)



爨r

'
/dt)sin(

)


+⁲

'(d[sin(

)]/dt)


+⁲

'sin(

)(d

/摴d


=⁲✧
r

+ r'[

'




' sin(

)

]‫⁲

'


+⁲

''


+⁲

'[

'(
-
r
) +

' cos(

)

] +⁲'

'sin(

)



r

''sin(

)


+
r

'[cos(

)

']


+⁲

'sin(

)

'[
-
sin(

)
r

-

cos(

)

]


=⁛爧'
-

r

'
2

-

rsin
2
(

)

'
2
]
r


+ [2r'

' + r

''
-

r
sin(

)cos(

)

'
2
]



+⁛㉲2

'sin(

) + 2r

'

'cos(

) + rsin(

)

'']




Analysis of each term:

For the
r

component: 1) the r'' is just the regular straight line acceleration in the radial
direction (up
-
down); 2) the
-
r

'
2

term is the centripetal acceleration d
ue to rotation
around an axis perpendicular to the z
-
axis (due to a North
-
South motion on the earth); 3)
the
-
rsin
2
(

)

'
2

term is the r
-
component of the centripetal acceleration due to rotation
around the z
-
axis, with r sin(

) the radius of the circular m
otion (due to an East
-
West
motion on the earth).


For the


c潭灯湥湴㨠‱⤠瑨攠㉲'

' is the Coriolis term due to the radius changing affecting
the


rotational motion (N
-
S); 2) the r

'' term is the regular tangential acceleration
causing the tangential (N
-
S) speed to change; 3) the
-
rsin(

)cos(

)

'
2

term is the

-
component of the centripetal acceleration due to rotation around the z
-
axis (E
-
W), with r
sin(

) the radius of the circular motion.


For the


c潭灯湥湴㨠‱⤠瑨攠㉲'

'sin(

) is a Coriolis term du
e to the radius changing
affecting the


rotational motion (E
-
W); 2) the 2r

'

'cos(

) is a Coriolis term due to the

-
radius (rsin(

) changing due to the r

' motion; 3) the rsin(

)

'' is the regular tangential
acceleration causing the tangential (E
-
W) sp
eed to change.


Note that each term has one r (distance) and two '' (per time squared) are required by an
acceleration.


Homework Problem

#16: Given that
A

is a vector function, find d
2
A
/dt
2

in cylindrical
coordinates (a 3
-
D vector problem).



4. The del

operator

a) Rectangular Coordinates:



=
(


/

x)

x

+ (


/

y)

y

+ (


/

z)

z

d
r

= dx
x

+ dy
y

+ dz
z

df = (

f/

x) dx + (

f/

y) dy + (

f/

z) dz


f


d
r

=
df = d
r



f

b) Spherical Coordinates:

d
r

= dr
r

+

rd




+⁲獩 (

)d




(since a small dista
nce in the radial direction is simply dr, a small distance in the


direction is rd

, and a
small distance in the


direction is r sin


d

)


df = (

f/

r)dr + (

f/


)d


+ (

f/


)d


d
r




映f
=
df

To make this last statement work, we need for the del opera
tor:



= (


/

r)
r

+ (1/r)(


/


)


+
ㄯ

sin(

))(


/


)




5. Divergence (


A
)

a) Rectangular Coordinates



=
(


/

x)

x

+ (


/

y)

y

+ (


/

z)

z

A

= A
x
x

+ A
y
y

+ A
z
z



A

= (

A
x
/

x) + (

A
y
/

y) + (

A
z
/

z)


b) Spherical Coordinates



=†


/

r)
r

+ (
1/r)(


/


)


+⁛ㄯ1

sin(

)](


/


)


A

= A
r
r

+ A



+ A





A

=
r




A
/

r +

⽲/



A
/



+

⽛爠獩渨

)]



A
/



=

r



[(

A
r
/

r)
r

+ A
r
(

r
/

r) + (

A

/

r)


+⁁

(


/

r) + (

A

/

r)


+ A

(


/

r)] +



/r


[(

A
r
/


)
r

+ A
r
(

r
/


) + (

A

/


)


+⁁

(


/


) + (

A

/


)


+⁁

(


/


)] +



/[r sin(

)]


[(

A
r
/


)
r

+ A
r
(

r
/


) + (

A

/


)


+⁁

(


/


) + (

A

/


)


+⁁

(


/


)]


We can eliminate all terms that have (

r
/

r), (


/

r) and (


/

r) since all these unit
vectors do not depend on r; we can also eliminate the term

that has (


/


) since


摯d猠
湯琠摥灥湤渠

. Recall that (

r
/


)=

㬠;

r
/


)=sin(

)

㬠(


/


)=
-
r
; (


/


)=cos(

)


a湤



/


)= [
-
sin(

)
r

-

cos(

)






A

=

r



[(

A
r
/

r)
r

+ 0 + (

A

/

r)


+‰‫


A

/

r)


+‰ ‫



/r


[(

A
r
/


)
r

+ A
r


+ (

A

/


)


+⁁

(
-
r
) + (

A

/


)


+‰ ‫



/[r sin(

)]


[(

A
r
/


)
r

+ A
r

sin(

)


⬠+

A

/


)


⬠A


cos(

)


⬠+

A

/


)


⬠+

(
-
sin(

)
r

-

cos(

)




Now we can take the dot products which will further reduce the number of terms:



A

= (

A
r
/

r) + A
r
/r + (

A

/


)/r + A
r
/r + A


cos(

)/r sin(

) + (

A

/


)/r sin(

) =

(

A
r
/

r) + 2A
r
/r + (

[A


sin(

)]/


)/r sin(

) + (

A

/


)/r sin(

) =


A

where we have combined the second and fourth terms into the 2A
r
/r term, and we have
combined the third and fifth terms into the (

[A


sin(

)]/


)/r sin(

) term.