path
DESCRIPTION
Summary: Three Coordinates (Tool). Velocity. Acceleration. Reference Frame. Reference Frame. Path. Path. x. x. r. y. Observer’s measuring tool. r. y. O. Observer. Observer. (x,y) coord. (n,t) coord velocity meter. (r, q ) coord. r. q. Choice of Coordinates. - PowerPoint PPT PresentationTRANSCRIPT
1
O
PathReference Frame
(x,y) coord
rq
(r,q) coord
xv
yv
x
y
xv xyv y
nv
0nv tv v
r
rvvq
v rq q rv r
PathReference Frame
xa
ya
x
y
ta
na
r
ra
aq
tv
xa x ya y2
nva
ta v
2a r rq q q 2ra r rq
(n,t) coordvelocity meter
Summary: Three Coordinates (Tool)Velocity Acceleration
Observer
Observer’smeasuringtool
Observer
2
O
PathReference Frame
(x,y) coord
rq
(r,q) coord
xv
yv
x
y
xv xyv y
nv
0nv tv v
r
rvvq
v rq q rv r
PathReference Frame
xa
ya
x
y
ta
na
r
ra
aq
tv
xa x ya y2
nva
ta v
2a r rq q q 2ra r rq
(n,t) coordvelocity meter
Choice of CoordinatesVelocity Acceleration
Observer
Observer’smeasuringtool
Observer
4
Path
(x,y) coord
rq
(r,q) coord
(n,t) coordvelocity meter
Translating Observer
Two observers (moving and not moving) see the particle moving the same way?
Observer O(non-moving)
Observer’sMeasuring tool
Observer (non-rotating)
Two observers (rotating and non rotating) see the particle moving the same way?
Observer B (moving)
Rotating
No!
No!
“Translating-only Frame” will be studied today
Which observer sees the “true” velocity?
both! It’s matter of viewpoint.
“Rotating axis” will be studied later.
Point: if O understand B’s motion, he can describe the velocity which B sees.
This particle path, depends on specific observer’s viewpoint
“relative” “absolute”
A
“translating” “rotating”
5
2/8 Relative Motion (Translating axises)
A = a particle to be studied
Ar
A
Reference frame O
frame work O is considered as fixed (non-moving)
Br
If motions of the reference axis is known, then “absolute motion” of the particle can also be found.
O
Motions of A measured using framework O is called the “absolute motions”
For most engineering problems, O attached to the earth surface may be assumed “fixed”; i.e. non-moving.
Sometimes it is convenient to describe motions of a particle “relative” to a moving “reference frame” (reference observer B)
B
Reference frame B B = a “(moving) observer”
BAr /
Motions of A measured by the observer at B is called the “relative motions of A with respect to B”
6
Relative positionIf the observer at B use the x-y **
coordinate system to describe the position vector of A we have
jyixr BAˆˆ
/
where
= position vector of A relative to B (or with respect to B),
and are the unit vectors along x and y axes
(x, y) is the coordinate of A measured in x-y framei j
BAr /
** other coordinates systems can be used; e.g. n-t.
Br
B
ArBAr /
A
X
Y
x
y
O
j
i
Here we will consider only the case where the x-y axis is not rotating (translate only)
J
I
7
ˆ ˆ ˆ ˆ( ) ( )A Br r xi yj xi yj
ˆ ˆ ˆ ˆ( )A Br r xi yj xi yj
Br
B
Ar/A Br
A
X
Y
x
y
O
j
i
x-y frame is not rotating (translate only)
Relative Motion (Translating Only)
Direction of frame’s unit vectors do not change
ˆ 0i
ˆ 0j
0
/A Bv
/A Ba
0
/A B A Br r r
ˆ ˆxi yjNotation using when
B is a translating frame.
BABA vvv /
BABA aaa /
Note: Any 3 coords can be applied toBoth 2 frames.
8
Understanding the equation
BABA vvv /
Translation-only Frame!Path
Observer O
Observer B
This particle path, depends on specific observer’s viewpoint
Ar
A
reference
framework O
Br
O
B
reference
frame work B
BAr /
A
/A Ov /B Ov
Observer O Observer O
Observer B (translation-onlyRelative velocity with O)
This is an equation of adding vectors of different viewpoint (world) !!!
O & B has a “relative” translation-only motion
9
The passenger aircraft B is flying with a linear motion to theeast with velocity vB = 800 km/h. A jet is traveling south with velocity vA = 1200 km/h. What velocity does A appear to a passenger in B ?
A B A Bv v v Solution
800Bv
1200Av
x
y
A Bvq
j1200i800v BA
2 2800 1200A Bv
1200800tan q
ˆ1200Av j ˆ800Bv i
10
A B A Bv a
A B A Bv v v
A B A Ba a a
18 ˆ ˆ5 /3.6Av i i m s
2ˆ3 /Aa i m s
12 3 rad/s60 10
q
0q
Translational-only relative velocity
You can find v and a of B
11
vA
vB vA/B
Velocity Diagramx
y
aAaBaA/B
Acceleration Diagramxy
9 ˆ ˆ( ) cos45 sin 45 2 210
o oBv i j i j
/ˆ ˆ3 2 /A B A Bv v v i j m s
2
ˆ ˆ ˆ ˆcos45 sin 45 0.628 0.628o oBB
va i j i jR
/ˆ ˆ3.628 0.628 /A B A Ba a a i j m s
v rad/s
10q
910Bv rq
0q
2B
BvaR
ˆ5 /Av i m s 2ˆ3 /Aa i m s
0ta rq 2
2n
va rr
q
: /B A rel B Av v v r
?
?/A B A Bv v v
?/B A B Av v v
B
?/A B A Bv v v
?/B A B Av v v
Yes
Yes
Yes
No
O
Is observer B a translating-only observer
relative with O
13
50 : obserber B, translating?A Bv
/ : obserber A, translating?B Av
BAv
To increase his speed, the water skier A cuts across the wake of the tow boat B, which has velocity of 60 km/h. At the instant when q = 30°, the actual path of the skier makes an angle = 50° with the tow rope. For this position determine the velocity vA of the skier and the value of q
Relative Motion:(Cicular Motion)
m10A
B
A
B
10A Bv rq q
sm67.166.3
60vB
Av
60 120 20
40
120sinv
40sin67.16 A
sm5.22vA sin 2016.67 10sin 40A Bv q
0.887 rad sq
o30
D
M ? ?O.K.
Point: Most 2 unknowns canbe solved with 1 vector (2D) equation.
A B A Bv v v
20
2060
60
30
30q
Consider at point A and B as r-q coordinate system
14
2/206 A skydriver B has reached a terminal speed . The airplane has the constant speed and is just beginning to follow the circular path shown of curvature radius = 2000 mDetermine (a) the vel. and acc. of the airplane relative to skydriver. (b) the time rate of change of the speed of the airplane and the radius of curvature of its path, both observed by the nonrotating skydriver.
0Ba
ˆ50Av i
0Aa
/ /, B A B Ar q
50 /Bv m s
ˆ50Bv j
0 ( )A A txa a
2
( ) AA A ny
A
va a
2ˆ ˆ( ) 1.250 /A ya a j j m s
/ /= - , - A B A B A B A Bv v v a a a
50 m/sAv
rrv
50 m/sBv
/ˆ ˆ50 50A Bv i j
/
ˆ1.250A Ba j
15
/ A Bv/ A Ba
(b) the time rate of change of the speed of the airplane and the radius of curvature of its path, both observed by the nonrotating skydriver.
rrv
n
t
/ /( ) sin 45or A B t A Bv a a
2/
/ /( ) cos 45oA BA B n A B
r
v a a
/ / , A B A Bv a
/ˆ ˆ50 50A Bv i j
/ˆ1.250A Ba j
0Ba
ˆ50Av i
ˆ50Bv j
2ˆ1.250 /Aa j m s
coordn t
rv r45o
45o
16
1000 ˆ m /3.6Av i s
2ˆ1.2 /Aa i m s
1500 ˆ /3.6Bv i m s
20 /Ba m s
, : relative worldr q
/ /, B A B Ar q
coord r q
/ /, B A B Av a
17
/500 ˆ3.6B Av i
/
ˆ1.2B Aa i
va
q
/( )B A rv r cos 120.3r v q
/( )B Av rq q 0.00579q
2/( )B A ra r rq
/( ) 2B Aa r rq q q
0.637r
30.166 10q
rq
1000 ˆ m /3.6Av i s
2ˆ1.2 /Aa i m s
1500 ˆ /3.6Bv i m s
20 /Ba m s
cosv q
sinv q
cosa q
sina q
30o
coord r q
1800 1200 1200sin 30or