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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