CHAPTER 14 Vectors in three space Team 6: Bhanu Kuncharam Tony Rocha-Valadez Wei Lu
Feb 06, 2016
CHAPTER 14 Vectors in three space
Team 6:Bhanu Kuncharam
Tony Rocha-ValadezWei Lu
The position vector R from the origin of Cartesian coordinate system to the point (x(t), y(t), z(t)) is given by the expression
k)t(zj)t(yi)t(x)t(R
k)t(zj)t(yi)t(x)t(R)t(v
i
j
A Cartesian coordinate system (by MIT OCW)
14.6 Non-Cartesian Coordinates
k)t(zj)t(yi)t(x)t(R)t(a
The vector expression for velocity is given by
The vector expression for acceleration is given by
http://www.wepapers.com/Papers/4521/1_Newton's_Laws,_Cartesian_and_Polar_Coordinates,_Dynamics_of_a_Single_Particle
14.6.1 Plane polar coordinate
To define the Polar Coordinates of a plane we need first to fix a point which will be called the Pole (or the origin) and a half-line starting from the pole. This half-line is called the Polar Axis.
Polar Angles: The Polar Angle θ of a point P, P ≠ pole, is the angle between the Polar Axis and the line connecting the point P to the pole. Positive values of the angle indicate angles measured in the counterclockwise direction from the Polar Axis.
The Polar Coordinates (r,θ) of the point P, P ≠ pole, consist of the distance r of the point P from the Pole and of the Polar Angle θ of the point P. Every (0, θ) represents the pole.
θ
Polar Axis
rP(r, θ) Definitio
ns:
http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node5.html
Plane polar coordinate
More than one coordinate pair can refer to the same point.
150o
30o210o
2
2,30o
2,210o
2, 150o
All of the polar coordinates of this point are: 2,30 360
2, 150 360 0, 1, 2 ...
o o
o o
n
n n
http://www.geom.uiuc.edu/docs/reference/CRC-formulas/node5.html
Plane polar coordinate
Difference quotient method to get rr e
dedande
ded ˆˆˆˆ
))((ˆ)( tetrR r
rr ererRtv ˆˆ)(
ded
dtd
ded
tedtde rr
rr
ˆˆ))((ˆˆ
)(ˆ)(ˆlim
ˆ0
rrr eeded
e
eded r ˆ
ˆ)1(lim
ˆ0
erertv r ˆˆ)( ererererer)t(v)t(a rr
What is ? d
ed rˆ
Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd ed.): Prentice Hall.
Plane polar coordinate
Difference quotient method to get rr e
dedande
ded ˆˆˆˆ
What is ?
ded ˆ
d
eddtd
ded
tedtde
ˆˆ)(ˆˆ
)(ˆ)(ˆlim
ˆ0
eeded
rr ee
ded ˆ)ˆ)(1(
limˆ
0
ree ˆˆ
errerrta r ˆ)2(ˆ)()( 2 Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd ed.): Prentice Hall.
Transform method to get rr e
dedande
ded ˆˆˆˆ
Plane polar coordinate
jided
jided
jie
jie
r
r
ˆsinˆcosˆ
ˆcosˆsinˆ
ˆcosˆsinˆ
ˆsinˆcosˆ
eejeei
r
r
ˆcosˆsinˆsinˆcos
r22
rr
22rr
r
e)sin(cos)e cose (sinsin)e sine (coscosded
ee)cos(sin)e cose (sincos)e sine (cossinded
e)r2r(e)rr()t(a
ererR)t(v
))t((e)t(rR
r2r
r
xyyxr
ryrx
1
22
tan
sincos
ktzjtyitxtRtvta
ktzjtyitxtRtv
ktzjtyitxtR
ˆ)(ˆ)(ˆ)()()()(
ˆ)(ˆ)(ˆ)()()(
ˆ)(ˆ)(ˆ)()(
e re
A polar coordinate system (by MIT OCW)
rr
rr
r
erererererdtdva
ererdtedre
dtdr
dtdRv
ˆˆˆˆˆ
ˆˆˆˆ
2
The expressions of R, v, a in polar coordinates
http://www.wepapers.com/Papers/4521/1_Newton's_Laws,_Cartesian_and_Polar_Coordinates,_Dynamics_of_a_Single_Particle
r
r
(r,,z)
Cylindrical coordinates are a generalization of two-dimensional polar coordinates to three dimensions by superposing a height z axis.
14.6.2 Cylindrical coordinates
A cylindrical coordinate system
)2(
http://mathworld.wolfram.com/CylindricalCoordinates.html
Cylindrical coordinates
xyyxr
ryrx
1tan
22
sincos
The relations between cylindrical coordinates and Cartesian coordinates.
2 2 2
tan( )
r x yyx
z z
Definitions:
zr
zr
zr
ezerrerrtRta
ezerertRtv
ezerR
ˆˆ)2(ˆ)()()(
ˆˆˆ)()(
ˆˆ
2
The expressions of position R, velocity v, and acceleration a in Cylindrical coordinates are
given by
Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd ed.): Prentice Hall.
Cylindrical coordinates
Example1:
Find the cylindrical coordinates of the point whose Cartesian coordinates are (1, 2, 3)
Answer:3
1071.15
z
r
Example2:Find the Cartesian coordinates of the point whose cylindrical coordinates are (2, Pi/4, 3)
Answer:3
22
22
zy
x
http://mathworld.wolfram.com/CylindricalCoordinates.html
14.6.3 Spherical coordinates
(x,y,z) r
z
Spherical coordinates, also called spherical polar coordinates (Walton 1967, Arfken 1985), are a system of curvilinear coordinates that are natural for describing positions on a sphere or spheroid. Define to be the azimuthal angle in the -plane from the x-axis with (denoted when referred to as the longitude), to be the polar angle (also known as the zenith angle and colatitude, with where is the latitude) from the positive z-axis with , and to be distance (radius) from a point to the origin.
Spherical coordinates
The expressions of Spherical coordinates for velocity and acceleration
eeeee
ee
eee
ee
eee
ˆcosˆsinˆ
,0ˆ
,0ˆ
ˆcosˆ
,ˆˆ
,0ˆ
ˆsinˆ
,ˆˆ
,0ˆ
eR ˆ ))(),((ˆˆ ttee
)ˆsinˆ(ˆ
)ˆˆ
(ˆ
eee
dtde
dtde
eRv
e
eeta
ˆ)cos2sin2sin(
ˆ)cossin2(ˆ)sin()( 2222
The expressions of R, v, a in Spherical coordinates
ktzjtyitxtRtvta
ktzjtyitxtRtv
ktzjtyitxtR
ˆ)(ˆ)(ˆ)()()()(
ˆ)(ˆ)(ˆ)()()(
ˆ)(ˆ)(ˆ)()(
e
eeta
eeetv
eR
ˆ)cos2sin2sin(
ˆ)cossin2(ˆ)sin()(
ˆsinˆˆ)(
ˆ
2222
cossinsincossin
zyx
Figure taken from reference: http://mathworld.wolfram.com/SphericalCoordinates.html
Example 3 Calculate the three components of the position, velocity and
acceleration vectors at t=3. The position of the point R is given by R=(t, exp(t), 3t ). Do this for the in Cartesian coordinates,
Cylindrical coordinates, and Spherical coordinates
Examples: The expressions of R, v, a in Non-Cartesian coordinates
Solution:In Cartesian Coordinates:
ktzjtyitxtR ˆ)(ˆ)(ˆ)()(
ktzjtyitxtRtv ˆ)(ˆ)(ˆ)()()(
ktzjtyitxtRtvta ˆ)(ˆ)(ˆ)()()()( jeta
kjeitv
ktjeittR
t
t
t
ˆ)(
ˆ3ˆˆ)(
ˆ3ˆˆ)(
0,08.20,0
3,08.20,1
9,08.20,3,ˆ)(,ˆ3ˆˆ)(,ˆ9ˆˆ3)( 333
zyx
zyx
zyx
aaa
vvv
RRRorjetakjeitvkjeitR
Solution: In Cylindrical Coordinates:
eejeei
r
r
ˆcosˆsinˆsinˆcos
put into
jeta
kjeitv
ktittR
t
t
ˆ)(
ˆ3ˆˆ)(
ˆ3ˆ)(
1468.0sin1cos,989.03
sin 2
62
3
22
e
e
yx
y
0ˆˆ
RezerR zr
)ˆcosˆ(sin)(
ˆ3)ˆcosˆ(sin)ˆsinˆ(cos)(
ˆ3ˆcos)(
eeeta
eeeeeetv
etettR
rt
zrt
r
zr
get
0,948.2,86.193,959.1,00.209,0,4404.0
zr
zr
zr
aaavvvRRR
The expressions of R, v, a in Non-Cartesian coordinates
In Spherical Coordinates:
eek
eeej
eeei
ˆsinˆcosˆ
ˆcosˆsincosˆsinsinˆ
ˆsinˆcoscosˆcossinˆ
put
jeta
kjeitv
ittR
t
t
ˆ)(
ˆ3ˆˆ)(
ˆ)(
into
Solution: 0.0
ˆ
zRR
eR
get
)ˆcosˆsincosˆsin(sin
)ˆsinˆ(cos
)ˆcosˆsincosˆsin(sin
)ˆsinˆcoscosˆcos(sin
ˆcossin
eeeea
eet
eeee
eeev
etR
t
t
9142.0cos1sin,4051.093
9cos
1468.0sin1cos,989.03
sin
2
262222
2
62
3
22
ezyx
ze
e
yx
y
045.8,948.2,15.18
361.5,936.3,49.19
0,0,4026.0
aaa
vvv
RRR
The expressions of R, v, a in Non-Cartesian coordinates
Using the omega method derive the space derivatives of base vectors
0
02
tan2
AA
AAAAAA
tconsAAA
Consider a rigid body B undergoing an arbitrary motion through 3-space. And let A be any fixed vector with B, that is, A is a vector from one material point in B to another so is constant with time, because b is rigid. Thus A=A(t)Fixed vector in B
There exists a vector such that AA 1
1
There exists a vector such that 2 BB 1
Greenberg, M. D. (1998). Advanced Engineering Mathematics (2nd ed.): Prentice Hall.
A
14.6.4 Omega Method
Omega method
Since B is arbitrary:
0)( 21 ASince A is arbitrary:
021
0)(0
0)()(0
cos
21
21
21
BASoBABA
BABABABA
BABA
So we get AA
Omega Method
Omega method In cylindrical coordinates: ze
Let A be :re eeeedted
rzrr ˆˆˆˆ
ˆ
Using chain differentiation to write:
ze
ze
re
r
dtdz
ze
dtde
dtdr
re
tzttredtd
rrr
rrrr
ˆˆˆ
ˆˆˆ))(),(),((ˆ
zeze
rerzer rrr
ˆˆˆ
0ˆ0
0ˆ
,ˆˆ,0
ˆ
zeee
re rrr
Similarly, let A be :e 0ˆ
,ˆˆ,0
ˆ
zeee
re
r
Let A be :ze 0ˆ
,0ˆ
,0ˆ
zee
re zzz
Omega Method
End of Chapter 14