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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

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

Translating

Observer

Rotating

Two observers (moving and not moving) see the particle moving the same way?

“Translating-only Frame”

will be studied today

No!

Observer’s

Measuring tool

Path

Which observer sees the “true” velocity?

Observer B

(moving)

(x,y)

coord

A

both! It’s matter of viewpoint.

This particle path, depends on specific observer’s viewpoint

“relative” “absolute”

(n,t) coord

velocity meter

Observer O

(non-moving)

Point: if O understand B’s motion, he can describe the velocity which B sees.

r

(r,q)

coord

q

Two observers (rotating and non rotating) see the particle moving the same way?

No!

“translating”

“rotating”

“Rotating axis”

will be studied later.

Observer

(non-rotating)

A

B

O

- Sometimes it is convenient to describe motions of a particle “relative” to a moving “reference frame” (reference observer B)

- If motions of the reference axis is known, then “absolute motion” of the particle can also be found.

- A = a particle to be studied

- B = a “(moving) observer”

Reference frame O

Reference frame B

- Motions of A measured by the observer at B is called the “relative motions of A with respect to B”

- 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.

frame work O is considered as fixed (non-moving)

y

A

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 frame

x

B

O

- If the observer at B use the x-y ** coordinate system to describe the position vector of A we have

Y

- Here we will consider only the case wherethe x-y axis is not rotating (translate only)

X

**other coordinates systems can be used; e.g. n-t.

A

B

O

- x-y frame is not rotating(translate only)

y

Y

Direction of frame’s

unit vectors do not change

x

0

X

Notation using when

B is a translating frame.

Note: Any 3 coords can be applied to

Both 2 frames.

0

A

B

O

Path

Translation-only Frame!

Observer B

A

O & B has a “relative” translation-only motion

This particle path, depends on specific observer’s viewpoint

Observer O

reference

framework O

reference

frame work B

Observer B

(translation-only

Relative velocity with O)

Observer O

Observer O

This is an equation of adding vectors of different viewpoint (world) !!!

Solution

y

x

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 ?

Translational-only relative velocity

You can find v and a of B

v

y

y

x

x

vA/B

vA

vB

Velocity Diagram

aA

aB

aA/B

Acceleration Diagram

Is observer B a translating-only observer

B

relative with O

Yes

Yes

O

Yes

No

?

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 = 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

10

m

o

30

Relative Motion:

(Cicular Motion)

Consider at point A and B

as r- coordinate system

30

A

A

o

o

B

B

30

D

?

O.K.

?

M

Point: Most 2 unknowns can

be solved with

1 vector (2D) equation.

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 m

Determine

(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.

t

n

(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.

r

v

a