1
ENGR 220
–
Lecture 3: Path Coordinates
and Relative Motion
2D Path Coordinates (n

t, or normal

tangential coordinates)
Explanation of what path coordinates mean
:
Take a known curved path in space. Examples could be cloud chamber tracks,
planetary orbits, a projectile, droplets injected into a piston, etc.
From the particle point of view there are two components to motion:
1.
Tangential motion
to path (moving straight ahead)
2.
Normal motion
to path (changing direction)
Note: This is how we think when giving directions to people because we think
from the perspective of the person traveling.
Definitions
s = path, ds = infinitesimal arc
r
C
= radius of curvature of arc
C = center of curvature of arc
e
t
= tangential component of motion
e
n
= normal component of motion
Diagram
2
2D Path Coordinates Equations
Diagram showing components of motion
Quantity
Vector Components
Notes
Position
Particle always at origin, but
Velocity
where
Acceleration
where
t
P
e
v
v
ˆ
and
C
r
v
n
n
t
t
P
e
a
e
a
a
ˆ
ˆ
and
v
a
t
C
n
r
v
a
2
2
2
so
n
t
a
a
a
2
2
2
2
3
1
dx
y
d
dx
dy
r
C
s
d
s
ds
s
3
Relative Motion
•
Two particles move independently
Diagram
Position
Velocity
Acceleration
A
B
O
A
O
B
r
r
r
/
/
/
A
B
A
B
v
v
v
/
A
B
A
B
a
a
a
/
dt
r
d
dt
r
d
dt
r
d
A
B
O
A
O
B
/
/
/
2
/
2
2
/
2
2
/
2
dt
r
d
dt
r
d
dt
r
d
A
B
O
A
O
B
dt
v
d
dt
v
d
dt
v
d
A
B
A
B
/
General Analysis Methodology
1.
Find motion equations for A and
B separately
2.
Apply relative motion definition
to get B/A motion
4
Motion with Constraints (Dependent Motion)
•
Absolute Dependent Motion Concept
•
Two particles are connected: by cords, inclined planes, sliding sleeves, etc.
•
Properties of particles for absolute dependent motion:
–
They have a fixed total distance along the cord between them
–
Each can move in a separate direction
–
Each undergoes rectilinear motion
Example
Example
5
Motion with Constraints (Dependent Motion)
Absolute Dependent Motion Equations
1.
position equations (x): relate segments to total cord length
a.
some cord sections may change length
b.
some cord sections may be constant length
c.
total length is constant
2.
velocity equations (v)
a.
take derivative of cord lengths (position) equations
3.
acceleration equations (a)
a.
take derivatives of cord velocity equations
General Analysis Methodology for Dependent Motion
1.
Find fixed points (and datum lines through them) from which to measure
distances
2.
Find all lengths of cord sections using datum points
a.
use given geometry and trig for each section
b.
usually can ignore distances wrapping around pulleys
c.
usually can ignore constant length pieces
3.
Write equation for total cord length in terms of cord section pieces
4.
Find
v
,
a
equations from derivatives of
s
equation
6
Example: Ch 2.4 #7
Skier starts at A with a=28m/s
2
to B where curve starts.
Find magnitude of total acceleration felt by skier at B.
Assume tangential accel doesn’t change right away.
7
Example: Ch 2.4 #12
Rocket has overall acceleration
a
= (5.675
i
–
3.843
j
) m/s
2
due to thrust and
g=9.5
m/s
2
gravity at an instant in time.
Also,
v
= (5000
i
+2000
j
) m/s at that instant.
a) Find acceleration due to engine
b) Find a
t
and a
n
c) Find r
C
8
Example: Ch 2.5 #7
You’re paddling at 4m/s at 45
°
angle
and are 11m from either shore at center
of river with 3m/s current. How far
from F will you land?
11m
11m
9
Example: Ch 2.5 #15
Free end at B pulled at v
B
=
–
4
j
ft/s. Find v
A
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