L2:1

Kine

mat

ics

of a

Par

ticle

•Th

e st

udy

of D

ynam

ics

incl

udes

–Ki

nem

atic

s—

the

desc

ript

ion

of m

otio

n wi

thou

t re

gard

to

the

forc

es c

ausi

ng it

–Ki

neti

cs—

the

rela

tion

ship

bet

ween

mot

ion

and

appl

ied

forc

es

•In

thi

s le

ctur

e we

foc

us o

n th

e ki

nem

atic

s of

pa

rtic

les

(bod

ies

of n

eglig

ible

dim

ensi

ons)

•La

ter

stud

ies:

kin

emat

ics

of r

igid

bod

ies

•Re

fere

nce:

–

Engi

neer

ing

Mec

hani

cs ,

Volu

me

2, D

ynam

ics,

5th

Edit

ion

by J

. L.M

eria

m, L

. G. K

raig

e, (W

iley,

200

2)

L2:2

To m

otivat

e th

is s

tudy

…

•Im

agin

e we

hav

e be

en a

ssig

ned

to

inve

stig

ate

a wi

ng-f

lutt

er p

robl

em o

n a

stun

t ai

rcra

ft in

a s

pira

l man

oeuv

re•

We

will

fit

an a

ccel

erom

eter

to

the

wing

ti

p to

mea

sure

the

flu

tter

vib

rati

ons

•W

e ne

ed t

o kn

ow h

ow t

he s

pira

l stu

nt

man

oeuv

re w

ill a

ffec

t th

e ac

cele

rom

eter

re

adin

gs

L2:3

•D

iffe

rent

obs

erve

rs o

f th

e sa

me

'par

ticl

e'

will

see

diff

eren

t th

ings

Refe

renc

e fr

ames

Refe

renc

e fr

ames

Acc

eler

omet

eron

win

g tip

O

L2:4

•D

iffe

rent

obs

erve

rs o

f th

e sa

me

'par

ticl

e' w

ill s

ee d

iffe

rent

thi

ngs

Refe

renc

e fr

ames

Refe

renc

e fr

ames

O

L2:5

Iner

tial r

efer

ence

fra

mes

•N

ewto

n's

1st

law

defi

nes

a re

fere

nce

fram

e in

wh

ich

the

'nat

ural

' sta

te o

f ob

ject

s is

res

t, o

r un

ifor

m m

otio

n in

a s

trai

ght

line

•Su

ch a

fra

me

is c

alle

d an

Ine

rtia

l(or

Gal

ilean

, or

Abs

olut

e) r

efer

ence

fra

me

•A

n in

erti

al f

ram

e m

ust

be a

bsol

utel

y un

acce

lera

ted

and

non-

rota

ting

•O

nly

iner

tial

obs

erve

rs w

ill a

gree

tha

t N

ewto

n's

2nd

law

(F=

mA

)des

crib

es t

heir

ob

serv

atio

ns

L2:6

V =

V0

= c

onst

ant

m

V =

V0

= c

onst

ant

V ≡

0

Yep,

V =

con

st.,

so A

= 0

, and

F =

mA

,no

wor

ries

!

Forc

e sc

ale

read

ing

=W

T=W

W =

mg

Free

-bod

y di

agra

m

Yep,

V ≡

0,

so A

= 0

, an

dF

= m

A,

no w

orri

es!

Two

iner

tial o

bser

vers

…

F=

mA

and

don'

t yo

u fo

rget

it

!

Σ F

= 0

L2:7

V ≡

0

Yep,

A =

-g/

√3,

and

F =

-mg/

√3,

so F

= m

A,

no w

orri

es!

m

Now

, V ≡

0,

so A

= 0

, bu

t F

≠0.

Wha

t th

e …

!

A n

on-i

nert

ial ob

serv

er …

F=

mA

and

don'

t yo

u fo

rget

it

!

3dV

gA

dt=

=−

2 3W

Free

-bod

y di

agra

m

3WFΣ

=−

W =

mg

2 3T

W=

L2:8

Abs

olut

e sp

ace?

Abs

olut

e sp

ace?

•D

oes

the

eart

h's

surf

ace

prov

ide

an

iner

tial

ref

eren

ce f

ram

e?–

OK

for

'loca

l' ev

ents

•D

oes

the

eart

h's

surf

ace

prov

ide

an

iner

tial

ref

eren

ce f

ram

e?–

OK

for

'loca

l' ev

ents

Iner

tial p

ositi

on v

ecto

rR

Iner

tial p

ositi

on v

ecto

rROO

XXYY

ZZ

L2:9

Abs

olut

e sp

ace?

•

Doe

s th

e ea

rth'

s su

rfac

e pr

ovid

e an

iner

tial

re

fere

nce

fram

e?–

not

for

‘glo

bal'

even

ts

–U

se a

ny n

on-r

otat

ing

fram

ewh

ich

is u

nacc

eler

ated

rela

tive

to

eart

h’s

cent

re o

f gr

avit

y (C

G)–

Non

-rot

atin

g w.

r.t.

the

‘fixe

d st

ars’

•D

oes

the

eart

h's

surf

ace

prov

ide

an in

erti

al

refe

renc

e fr

ame?

–no

t fo

r ‘g

loba

l' ev

ents

–

Use

any

non

-rot

atin

g fr

ame

whic

h is

una

ccel

erat

edre

lati

ve t

o ea

rth’

s ce

ntre

of

grav

ity

(CG)

–N

on-r

otat

ing

w.r.

t.th

e ‘fi

xed

star

s’

XY

ZΩ

Earth

rota

tes

rela

tive

to

iner

tial f

ram

e

Iner

tial p

ositi

on v

ecto

r of e

arth

sate

llite

L2:1

0

Abs

olut

e sp

ace?

•

As

the

scal

e of

the

obs

erva

tion

s in

crea

se …

–Fr

ames

una

ccel

erat

edre

lati

ve t

o CG

of

sola

r sy

stem

–N

on-r

otat

ing

rela

tive

to

the

‘fix

ed s

tars

’–

Beyo

nd t

hat,

rel

ativ

isti

c ef

fect

s lik

ely

to b

e im

port

ant:

New

toni

an m

echa

nics

bre

ak d

own

•A

s th

e sc

ale

of t

he o

bser

vati

ons

incr

ease

…–

Fram

es u

nacc

eler

ated

rela

tive

to

CG o

f so

lar

syst

em–

Non

-rot

atin

g re

lati

ve

to t

he ‘f

ixed

sta

rs’

–Be

yond

tha

t, r

elat

ivis

tic

effe

cts

likel

y to

be

impo

rtan

t:N

ewto

nian

mec

hani

cs b

reak

dow

n

L2:1

1

Back

to

our

assign

men

t …

Back

to

our

assign

men

t …

R

O1 In

ertia

l re

fere

nce

fram

e

•Co

nsid

er, f

irst

, the

pat

hfo

llowe

d by

th

e pl

ane

•Ch

oose

, for

exa

mpl

e, it

s CG

as

a re

fere

nce

‘par

ticl

e’

•D

escr

ibe

the

inst

anta

neou

s po

siti

onof

the

pla

ne w

ith

a po

siti

on v

ecto

r

()t

=R

R

•In

erti

al p

osit

ion

vect

or:

Pat

h

•Th

e ti

p of

the

pos

itio

n ve

ctor

tra

ces

out

the

path

In w

ritt

en w

ork,

deno

te v

ecto

rs th

us:

(typ

eset

ting:

wig

gly

unde

rlin

e den

otes

bol

dfac

e)R

L2:1

2

Iner

tial v

eloc

ity

Iner

tial v

eloc

ity

0lim t

td dtδ

δ δ→

= ==R

V

RR

O1(

)t

tδ+

R(

)tR

Pat

h

•Ra

te o

f ch

ange

of

posi

tion

wit

h ti

me

Iner

tial

refe

renc

e fra

me

•Ve

loci

ty v

ecto

r is

ta

ngen

tto

pat

h:V

=V

t

•M

agni

tude

of

velo

city

vec

tor

(V) c

alle

d th

e sp

eed

•ti

s a

unit

vec

tor,

tang

ent

to

the

path

δR ≈

Vδt

Vt

L2:1

3

Iner

tial a

cceler

ation

Iner

tial a

cceler

ation

O1

V(t)

Pat

h

•Ra

te o

f ch

ange

of

velo

city

wit

h ti

me

0lim t

td dtδ

δ δ→

= ==

=

VA

VV

R

Iner

tial

refe

renc

e fra

me

•N

ote

that

vel

ocit

y ve

ctor

may

cha

nge:

–in

mag

nitu

de

(str

etch

ing)

, and

–in

dir

ecti

on

(swi

ngin

g)

()

ttδ

+V

Α

V(t)

()

ttδ

+V

δV ≈

Aδt

Vec

tor a

dditi

on:

()

()

tt

tδ

δ+

=+

VV

V

L2:1

4

Hod

ogra

phHod

ogra

ph•

Plot

vel

ocit

y ve

ctor

s fr

om c

omm

on p

oint

O1V1

Pat

h

V2

V3

V4

R1

R2

R3

R4

•Ti

p of

vel

ocit

y ve

ctor

tr

aces

out

a h

odog

raph

•A

ccel

erat

ion

vect

or

is t

ange

ntto

hod

ogra

ph

V1

V2

V3

V4

A1

A2

A3

A4

Hod

ogra

ph

•Ti

p of

pos

itio

n ve

ctor

tr

aces

out

pat

h•

Velo

city

vec

tor

is

tang

ent

to p

ath

L2:1

5

Coor

dina

te S

yste

ms

Coor

dina

te S

yste

ms

•Po

siti

on, v

eloc

ity

and

acce

lera

tion

ve

ctor

s re

pres

ent

obse

rvat

ions

fro

m a

gi

ven

refe

renc

e fr

ame

•To

des

crib

eth

em w

e in

trod

uce

a co

ordi

nate

sys

tem

•M

any

alte

rnat

ive

coor

dina

te s

yste

ms

may

be

empl

oyed

to

desc

ribe

the

sam

e ve

ctor

•Th

e ve

ctor

s ex

ist,

inde

pend

entl

yof

th

eir

desc

ript

ion

in a

ny c

oord

inat

e sy

stem

RP

ath

O1

Ref

eren

cefra

me

VA

L2:1

6

Intr

insic

(pat

h) c

oord

inat

esIn

trinsic

(pat

h) c

oord

inat

es

•Sc

alar

sm

easu

res

leng

th o

f pa

th

from

som

e da

tum

poi

nt

R

Pat

h le

ngth

O1

Ref

eren

cefra

me

Dat

um

s

•Ch

ange

in p

osit

ion

is δ

R

dd

dsV

dtds

dt=

=⋅

=R

RV

t

δs

R +

δR

0lim s

ds

dsδ

δ δ→

==

RR

t

•In

tim

e in

crem

ent

δt, p

ath

leng

thin

crea

ses

by δ

s

•Ta

ngen

t un

it v

ecto

r:

•D

escr

ipti

on o

f ve

loci

ty v

ecto

r:

t

Spee

d: V

s=

Dire

ctio

n:

d ds=

Rt

L2:1

7

Intr

insic

coor

dina

tes

Intr

insic

coor

dina

tes

R

O1

Ref

eren

cefra

me

tns

b

δs

R +

δR

Osc

ulat

ing

plan

e

Cen

tre o

f cu

rvat

ure

Rad

ius

of

curv

atur

eρ

δs/ρ

δR V

•Ta

ngen

tun

it v

ecto

r:d ds

=R

t

•(P

rinc

ipal

) no

rmal

unit

vect

or:

d dsρ

=t

n

•ta

nd n

lie

in t

he

oscu

lati

ng p

lane

•b

is p

erpe

ndic

ular

to

oscu

lati

ng p

lane

•Bi

norm

alun

it v

ecto

r:=

×b

tn

tn

bδt

V/ρ

δs/ρ

1s

δδ

ρ≈

⋅⋅

tn

11

,,

11

dd

dsds

d ds

ρσ σ

ρ

==

−

=×

⇒=

−

tb

nn

nn

bt

bt

σ=

radi

us o

f tor

sion

tn

b

()

1s

δδ

σ≈

⋅⋅

−b

n

δs/σ

V/σ

L2:1

8

2 0

t n b

VA

VA A

ρ

=

Mat

rix

form

Compo

nent

s of

vec

tors

in

intr

insic

coor

dina

tes

•Ve

loci

ty v

ecto

rd

dds

dtds

dt=

=⋅

RR

V

sV

==

Vt

ti.e

.,

•A

ccel

erat

ion

dd

VV

dtdt

dds

VV

dsdt

==

+

=+V

tA

t tt

2V

Vρ

=+

At

ni.e

.,

Com

pone

nts

0 0

t n bVs

V V

= = =

Tang

enti

al c

ompo

nent

: N

orm

al c

ompo

nent

: Bi

norm

al c

ompo

nent

:

'sw

ingi

ng'

com

pone

nt

0 0

t n bVs

V V

=

Mat

rix

form

'stre

tchi

ng'

com

pone

nt V

s=

2V ρ

nVt

A

L2:1

9

Intr

insic

coor

dina

tes

for

plan

e mot

ion

•Fo

r pl

ane

mot

ion,

tor

sion

10

σ=

θsn s

δδθ

ρ=

t n+δ

nt+

δt

δss

=V

t1

dd

dsV

dtds

dtρ

==

⋅t

tn

d dtθ

=tn

i.e.,

Vθ

ρ=

()

1t

δθδ

≈⋅

⋅−

nt

d dtθ

=−

nt

i.e.,

swin

ging

co

mpo

nent

on

ly

ρ

+

n

t

δn

δt

δθ

L2:2

0

Cart

esian

coor

dina

tes:

x, y

, zCa

rtes

ian

coor

dina

tes:

x, y

, z

•Po

siti

on:

R

O1

x

z

y

i

j

k

xy

z=

++

Ri

jk

xR

y z

=

•Ve

loci

ty:

xy

z

xy

zV

VV

=+

+=

++

Vi

jk

ij

k

x y z

xV

Vy

Vz

V

==

•A

ccel

erat

ion:

xy

z

xy

zA

AA

=+

+=

++

Ai

jk

ij

k

x yz

x y z

xA

Ay

Az

A

==

e.g.

, x

= E

AS

Ty

= N

OR

THz

= AL

TITU

DE

L2:2

1Cy

lindr

ical c

oord

inat

es:

r, θ,

zCy

lindr

ical c

oord

inat

es:

r, θ,

z•

Posi

tion

:

x

z

y

e r

rz

rz

=+

Re

e

0r

Rz

=

•

Velo

city

:r

rz

rr

z=

++

Ve

ee

r z

rV

Vr

Vz

Vθθ

==

•A

ccel

erat

ion:

zR

O1

e θe z

r θr

r

θ

θ

θ θ= =

−

ee

ee

rz

rr

zθ

θ=

++

Ve

ee

i.e.,

()

rz

rz

rr

rz

rr

zθ

θ

θθ

θ

=+

++

++

+

Ae

ee

ee

e

()

20

rz

r

rr

rz

rr

θ

θ

θθ

θθ

=+

++

+−

+

Ae

ee

ee

i.e.,

θ

tθδ

e r1r

tθ

δθδ

≈⋅

⋅e

ee θ

x

y

2

2r z

rr

AA

rr

Az

Aθ

θθ

θ

−

=+

=

L2:2

2Sp

herica

l co

ordina

tes:

R, θ

, φSp

herica

l co

ordina

tes:

R, θ

, φ•

Posi

tion

:

x

z

y

RR

=R

e

0 0R

R

=

•Ve

loci

ty:

RR

RR

=+

Ve

e

cos

RR

VV

RV

RVθ φ

θφ

φ

==

R

O1

e θe R

θ

0co

s

RR

RR

dR

dtR

θφ

θφ

θφ

φθφ

∂∂

∂=

++

∂∂

∂

=+

+

ee

ee

ee

R φ

(

)

()2

22

2

22

cos

cos

2si

n

1si

nco

s

RR

RR

Ad

AR

RA

Rdt

Ad

RR

Rdt

θ φ

φθ

φφ

θθφ

φ

φθ

φφ

−−

=−

=

+

•A

ccel

erat

ion:

e φ

cosφ

L2:2

3

Art

icles

from

Mer

iam

& Kr

aige

(5th

edn

)re

leva

nt t

o th

is lec

ture

•2/

1, 2

/2–

shou

ld b

e fa

mili

ar f

rom

hig

h sc

hool

phy

sics

•2/

3 Pl

ane

curv

iline

ar m

otio

n–

conc

epts

of

vect

oria

l pos

itio

n, v

eloc

ity

and

acce

lera

tion

; ho

dogr

aph

•2/

4 Re

ctan

gula

r co

ordi

nate

s (x

-y)

–sh

ould

be

fam

iliar

fro

m h

igh

scho

ol p

hysi

cs•

2/5

Nor

mal

and

tan

gent

ial c

oord

inat

es–

intr

oduc

tory

intr

insi

c co

ordi

nate

s, f

or p

lane

mot

ion

only

•2/

6 Po

lar

coor

dina

tes

(r-θ

)–

plan

e m

otio

n si

mpl

ific

atio

n of

cyl

indr

ical

coo

rdin

ates

•2/

7 Sp

ace

curv

iline

ar m

otio

n–

rect

angu

lar,

cyl

indr

ical

and

sph

eric

al c

oord

inat

es