and Relative Motion

doutfanaticalMechanics

Nov 14, 2013 (3 years and 9 months ago)

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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