Physics 111 Lecture 05 - web.njit.edujanow/Physics 111 Spring 2012/Lectures... · Physics 111...
27
Copyright R. Janow – Spring 2012 Physics 111 Lecture 05 Force and Motion II – Friction, Circular Motion SJ 8th Ed.: Ch. 5.8, 6.1 – 6.3 • Dynamics Summary • Friction Basics – Static Friction – Kinetic Friction • Properties of Friction • Sample Problems • Uniform Circular Motion - Centripetal Force – Free Body Diagrams – Sample Problems • Non-Uniform Circular Motion • Accelerated Frames of Reference • Drag Forces and Terminal Speed 5.8 Friction Forces 6.1 Extending the Uniform Circular Motion Model 6.2 Non-uniform Circular Motion 6.3 Motion in Accelerated Frames
Transcript of Physics 111 Lecture 05 - web.njit.edujanow/Physics 111 Spring 2012/Lectures... · Physics 111...
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Physic
s 1
11 L
ectu
re 0
5Force a
nd M
otion I
I –
Fric
tion, Cir
cula
r M
otion
SJ 8
th E
d.: C
h. 5.8
, 6.1
–6.3
•D
ynam
ics S
um
mary
•Fric
tion B
asic
s–
Sta
tic F
ric
tion
–Kin
etic F
ric
tion
•Properties o
f Fric
tion
•Sam
ple
Proble
ms
•U
niform
Cir
cula
r M
otion -
Centr
ipeta
l Force
–Free B
ody D
iagram
s–
Sam
ple
Proble
ms
•N
on-U
niform
Cir
cula
r M
otion
•Accele
rate
d F
ram
es o
f Refe
rence
•D
rag F
orces a
nd T
erm
inal Speed
5.8
Fri
cti
on
Fo
rces
6.1
Exte
nd
ing
th
e U
nif
orm
Cir
cu
lar
Mo
tio
n M
od
el
6.2
No
n-u
nif
orm
Cir
cu
lar
Mo
tio
n
6.3
Mo
tio
n i
n A
ccele
rate
d F
ram
es
Co
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Sum
mary –
Dynam
ics s
o far
Now
: Add fric
tion…
.
Mass:
in
ert
ia, re
sis
tan
ce t
o a
ccele
rati
on
, in
here
nt
pro
pert
y
Fo
rce:
pu
sh
es o
r p
ulls o
n b
od
ies, c
on
tact
or
thro
ug
h f
ield
, cau
ses a
bo
dy t
o a
ccele
rate
FB
D:
Fre
e B
od
y D
iag
ram
, vit
al to
pro
ble
m a
naly
sis
. In
clu
de A
LL
fo
rces a
cti
ng
on
a b
od
y
Net
Fo
rce:
vecto
r su
m o
f all f
orc
es a
cti
ng
ON
a b
od
y (
su
perp
osit
ion
).
∑ ∑∑∑≡ ≡≡≡
ii
ne
tF
Fr
v
Fir
st
Law
: A
bo
dy’s
velo
cit
y i
s c
on
sta
nt
if t
he n
et
exte
rnal fo
rce a
cti
ng
on
it
iszero
(eq
uilib
riu
m)
Seco
nd
Law
:
am
F n
et
rr
= ===A
ccele
rati
on r
esu
ltin
g fr
om F
net
Th
ird
Law
: I
f b
od
y A
exert
s a
fo
rce o
n b
od
y B
, th
en
bo
dy B
exert
s a
fo
rce e
qu
al in
mag
nit
ud
e a
nd
op
po
sit
e in
dir
ecti
on
on
bo
dy A
.
For
each
Cart
esi
an
com
pone
nt
New
ton
’s3 L
aw
sh
old
in
in
ert
ial
(un
-accele
rate
d)
refe
ren
ce f
ram
es
Un
its:
SYSTEM
FO
RCE
M
ASS
A
CCELERATIO
NSI
Newto
n (N
)
Kg
m/s
2
CGS
Dyne
gm
cm
/s2
Bri
tish
Pou
nd (
lb)
slug
ft/s
2
Co
ncep
ts
Co
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Fric
tion B
asic
sA c
onta
ct f
orce
betw
een
surf
ace
sdue
to
their
rou
ghne
ss
Fri
ctio
n fo
rce f
alw
ays
oppo
ses
the m
otio
nM
otio
n m
ay b
e:
•Act
ual m
otio
n � ���
kine
tic
(slidin
g) f
rict
ion
f k•
Im
pend
ing
mot
ion � ���
stati
cfr
icti
onf s
v’
fv f
Mod
el fo
r so
lid s
urfa
ces
in c
onta
ct:
•S
urf
ace r
ou
gh
ness m
easu
red
by “
fric
tio
n c
oe
ffic
ien
ts µ µµµ
kan
d µ µµµ
s
•F
ricti
on
fo
rce
pro
po
rtio
nal
to p
ressu
re b
etw
een
su
rface a
nd
µ µµµ•
Fri
cti
on
fo
rce
in
dep
en
den
t o
f sp
eed
v•
Oth
er
dis
sip
ati
ve
fo
rces (
e.g
. in
liq
uid
s)
de
pen
d o
n s
peed
an
dvis
co
sit
y
N
f
ss
µ µµµ≤ ≤≤≤
WNf s
F
ST
AT
IC F
RIC
TIO
Nf s
self
-vari
es t
ocan
cel
F,
up
to
a “
bre
akaw
ay”
lim
it f k<
fs,m
ax
µ µµµk
< µ µµµ
s
N
f
kk
µ µµµ= ===
KIN
ET
IC F
RIC
TIO
N
WNf k
0≠ ≠≠≠
v F
Co
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Fric
tion force a
s a
function o
f applied force F
WNf
F
0
W
N
F
iy
= ===− −−−
= ===∑ ∑∑∑
m
a
f
F
F
ix
= ===− −−−
= ===∑ ∑∑∑
Slo
wly
in
cre
ase F
an
d o
bserv
e f
ricti
on
fo
rce
0
a
= ===
N
f
F
ss
µ µµµ≤ ≤≤≤
= ===
Sta
tic f
ricti
on
:
N
f
F
0
a
sm
ax
,s
µ µµµ= ===
= ==== ===
Imp
en
din
g m
oti
on
:
µ µµµk
an
d µ µµµ
sd
ep
en
d o
n
su
rface c
on
dit
ion
s
an
d m
ate
rials
µ µµµk
<
µ µµµs o
therw
ise
bre
akaw
ay c
ou
ld n
ot
hap
pen
Kin
eti
c f
ricti
on
aft
er
“b
reakaw
ay”:
f
F
ma
x,
s≥ ≥≥≥
f k<
f s
,max
an
d
0
m
f
F
a
0
a
k> >>>
− −−−= ===
≠ ≠≠≠
N
f
F
s
kµ µµµ
= ===≥ ≥≥≥
Co
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Fric
tion force a
s a
function o
f applied force F
WNf
FS
low
ly in
cre
ase F
an
d o
bserv
e f
ricti
on
fo
rce
ay
= 0
so
N
=
W
BR
EA
KA
WA
Y
f s,m
ax
fs=
F
f k=
µ µµµkN
f s,m
ax
= µ µµµ
sN
F
N
f
ss
µ µµµ≤ ≤≤≤
f k<
fs,m
ax
s
kµ µµµ
< <<<µ µµµ
Co
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Som
e F
ric
tion C
oeff
icie
nts
sk
µ µµµ< <<<
µ µµµ
Co
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2012
Exam
ple:
Kin
eti
c Fri
ctio
n
Fg=
ms
ledg
f k=
µ µµµkN
The s
led a
nd loa
d a
re p
ulle
d a
t co
nsta
nt v
elo
city
Fin
d t
he t
ens
ion
T in
the c
ord
Sub
stit
ute (
1)
into
(2)
)sin
(T
F
)
co
s(
Tk
gk
φ φφφµ µµµ
− −−−µ µµµ
= ===φ φφφ
F
] )
sin
(
)co
s(
[ T
gk
kµ µµµ
= ===φ φφφ
µ µµµ+ +++
φ φφφ
)sin
(
)co
s(
g
m
T
k
k
φ φφφµ µµµ
+ +++φ φφφ
µ µµµ= ===
Eva
luate
:
T =
91 N
.
Ms
led
= 7
5 k
gµ µµµ
k=
0.1
0
φ φφφ=
42
o
FB
D
N
Fri
ctio
n fo
rce d
epe
nds
on N
–th
e n
orm
al
forc
e –
whic
h o
ften
doe
s N
OT =
the w
eig
ht
ma
)
sin
(T
FN
Fg
y= ===
= ===φ φφφ
+ +++− −−−
= ===∑ ∑∑∑
0
)sin
(T
FN
gφ φφφ
− −−−= ===
(1)
App
ly S
eco
nd L
aw
0= ===
= ===− −−−
φ φφφ= ===
∑ ∑∑∑x
kx
ma
f)
co
s(
TF
v xis
con
stant
N
)c
os
(T
k
µ µµµ= ===
φ φφφ(2
)
Co
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2012
Do y
ou p
ush o
r p
ull (
for s
mallest F)?
Ass
ume c
hild s
lides
at
cons
tant
velo
city
v,
and
ay
= 0
WNf k
F 30
o
030
= ==== ===
− −−−− −−−
= ===∑ ∑∑∑
yo
ym
a)
sin
(F
WN
F
W
)
sin
(F
WN
o> >>>
+ +++= ===
30
Larg
er
N,
Larg
er
fric
tion
WN’
f k’
F30
o
030
= ==== ===
+ +++− −−−
= ===∑ ∑∑∑
yo
ym
a)
sin
(F
W'
NF
W
)
sin
(F
W'
N o
< <<<− −−−
= ===30
Sm
aller
N,
smaller
fric
tion
030
= ===µ µµµ
− −−−= ===
∑ ∑∑∑N
)co
s(
FF
ko
x )
co
s(N
F
ok 3
0
µ µµµ= ===
How
muc
h f
orce
F is
requ
ired t
o ov
erc
ome f
rict
ion
Co
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2012
Exam
ple:
Sta
tic
Fri
ctio
n on
a R
am
pA c
oin
is j
ust
abou
t to
slide d
own
the b
ook
when
the
ang
le θ θθθ
= 1
5o .
Fin
d t
he s
tati
c fr
icti
on c
oeff
icie
nt
µ µµµs.
Mo
del
as a
blo
ck o
n a
ra
mp
Ch
oo
se x
-axis
alo
ng
th
e r
am
p’s
su
rface
N =
th
e n
orm
al
forc
e is
alo
ng
y-a
xis
Weig
ht
Fg
acts
str
aig
ht
do
wn
N
f
is f
sm
ax
,s
sµ µµµ
= ===g
m
F
co
in
of
w
eig
ht
g= ===
= ===
Th
e y
an
d x
acc
ele
rati
on
s =
0
mo
tio
n)
(o
pp
os
es
po
sit
ive
is
sn
eg
ati
ve
is
gx
f
F⇒ ⇒⇒⇒
FB
DN
0= ===
θ θθθ− −−−
= ===∑ ∑∑∑
)co
s(
FN
Fg
y
mo
tio
n)
(i
mp
en
din
gx
gs
x
ma
)
sin
(F
N
F
0= ===
= ===θ θθθ
− −−−µ µµµ
= ===∑ ∑∑∑
)co
s(
F
N g
θ θθθ= ===
(1)
)sin
(F
N
g
sθ θθθ
= ===µ µµµ
(2)
Div
ide (
2)
by (
1)
)co
s(
F
)sin
(F
N
N
ggs
θ θθθθ θθθ= ===
µ µµµ
.23
)(
nta
s
≈ ≈≈≈θ θθθ
= ===µ µµµ
Co
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2012
Accele
ration o
f Connecte
d O
bje
cts
with F
ric
tion
A b
lock
of
mass
m2
on a
rou
ghhor
izon
tal su
rface
is c
onne
cted t
o hang
ing
ball m
1by a
cor
d p
ass
ing
over
a m
ass
less
pulley.
For
ce F
is
app
lied t
o blo
ck m
2at
an
ang
le θ θθθ
.Dete
rmin
e t
he m
agn
itud
e o
f th
e a
ccele
rati
on a
.Ass
ume a
is
posi
tive
(ri
ghtw
ard
) and
the s
am
e f
or b
oth
m1
m2
m1T m
1g
aFB
Da
mg
mT
Fy
11
1= ===
− −−−= ===
∑ ∑∑∑
)g
a(m
T + +++
= ===1
(1)
mm
]m
m[ g
)]sin
()
[co
s(
F
a
21
kk
+ +++
µ µµµ+ +++
− −−−θ θθθ
µ µµµ+ +++
θ θθθ= ===
21
Fo
r F
= 0
, a
is
neg
ati
ve
Fo
r µ µµµ
k=
0
mm
gm
)c
os
(F
a
21
+ +++
− −−−θ θθθ
= ===1
If F
als
o =
0
mm
gm
a
21
+ +++
− −−−= ===
1
0 )
Fsin
(
g
mN
Fy
= ===θ θθθ
+ +++− −−−
= ===∑ ∑∑∑
22
m2g
m2
T
N
θ θθθf k
a
FB
DF
)F
sin
(
g
mN
θ θθθ− −−−
= ===2
(2) a
m f
T)
Fco
s(
F
2k
x= ===
− −−−− −−−
θ θθθ= ===
∑ ∑∑∑2
N
f
kk
µ µµµ= ===
f
T )
Fco
s(
am
k2
− −−−− −−−
θ θθθ= ===
(3)
Co
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ng
2012
Fric
tion force o
f a w
all
5 –
1:
A s
tud
en
t p
resses h
er
ph
ysic
s b
oo
k a
gain
st
a r
ou
gh
ve
rtic
alw
all
, w
ith
her
han
d e
xert
ing
a f
orc
e n
orm
al
to t
he w
all
. W
hat
is t
he d
irecti
on
of
the f
ricti
on
fo
rce e
xe
rted
by t
he w
all
on
th
e b
oo
k?
A.
No
rmal
to t
he w
all, o
pp
osit
e t
o t
he f
orc
e o
f h
er
han
d
B.
Do
wn
ward
C.
Up
ward
D.
Into
th
e w
all
E.
Th
e f
orc
e i
s z
ero
, sin
ce t
he b
oo
k i
s in
eq
uil
ibri
um
Co
pyri
gh
t R
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–S
pri
ng
2012
Exam
ple:
Blo
ck S
lidin
g alo
ng a
Wall
θ θθθ
F
vert
ical
wall
blo
ck
The b
lock
is
free t
o sl
ide u
p or
dow
n alo
ng t
he v
ert
ical
wall.
It’s
weig
ht
is 1
00 N
. T
he s
tati
c fr
icti
on
coeff
icie
nt is
0.5
0.
The a
ngle
θ θθθ= 3
0o
A)
What
is t
he m
inim
um f
orce
F t
hat
will pr
eve
nt t
he
blo
ck f
rom
slidin
g dow
n th
e w
all?
B)
What
min
imum
for
ce F
will st
art
the b
lock
mov
ing
up
the w
all?
Bo
th c
ases a
re i
mp
en
din
g m
oti
on
:N
f
is
fs
ma
x,
ss
µ µµµ= === )
(co
sF
F
)(
ins
F
Fy
xθ θθθ
= ===θ θθθ
= ===0
0
a
a
yx
= ==== ===
W
Nf s
xF
y
x
yF
Case
A:
f sis
up
0
W
f
FF
max
s,
yy
= ===− −−−
+ +++= ===
∑ ∑∑∑
0
N
F
Fx
x= ===
− −−−= ===
∑ ∑∑∑
N )
sin
(F
= ===θ θθθ
)]
sin
(
)[c
os(
W
F
sA
θ θθθµ µµµ
+ +++θ θθθ
= ===(sam
e f
or c
ase
B)
N
107.2
F A
= ===
W
N
f s
xF
y
x
yF
Case
B:
f sis
dow
n
0
W
f
FF
max
s,
yy
= ===− −−−
− −−−= ===
∑ ∑∑∑
)]sin
(
)[c
os(
W
F
sB
θ θθθµ µµµ
− −−−θ θθθ
= ===
N
493
F B
1= ===
If
µ µµµs� ���
0,
bot
h r
esu
lts
go t
o th
e s
am
e r
esu
lt:
F =
W/c
os
(θ θθθ)
Co
pyri
gh
t R
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an
ow
–S
pri
ng
2012
Uniform
Cir
cula
r M
otion
Loo
king
dow
n on
a p
uck
forc
ed t
o m
ove in
a c
ircl
eon
a t
able
Magn
itud
es
r, v
, & a
c co
nsta
ntVect
ors
r, v
, a
calw
ays
poin
t to
“P”
/T2
f
2
sp
ee
d
an
gu
lar
22 T
on
"a
cc
ele
rati
lc
en
trip
eta
"c
r
rv
a
π πππ= ===
π πππ= ===
≡ ≡≡≡ω ωωω
ω ωωω= ===
= ==== ===
The c
ent
ripe
tal acc
ele
rati
on m
ust
be
caus
ed b
y a
“ce
ntri
peta
l fo
rce”
and
otherw
ise f
it int
o th
e 2
ndLaw.
Cent
ripe
tal fo
rce c
hang
es
the d
irect
ion
but
no
t th
e m
agn
itud
e o
f a b
ody’s v
elo
city
; i.e….
v
F
0v
F)
c(
c
rr
vo
v⊥ ⊥⊥⊥
= ===
App
lica
tion
of
Seco
nd L
aw t
o pr
oble
ms:
][
ma
r
mv
F
forc
es
ra
dia
l
real
all
c
2 Tc
∑ ∑∑∑= ===
= ==== ===
The “
ma”
side o
f 2
ndLaw
The a
ctua
l ph
ysi
cal fo
rces
rr
car
Tvr
P
m
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Application to 2
ndLaw
Proble
ms, continued
rop
e)
th
e
in (t
en
sio
n
2 T
T
r
mv
= ===
Exam
ple:
Puc
k on
the e
nd o
f a r
ope (
hor
izon
tal)
Exam
ple:
Pass
eng
er
in a
car
turn
ing
left
r
mv
2 T= ===
do
or
o
r
be
lt
se
at
o
f
forc
e
co
nta
ct
o
r
se
at,
th
e
wit
hfr
icti
on
Pass
eng
er
is in
a r
otati
ng (
non-
inert
ial) r
efe
renc
efr
am
e a
nd f
eels
a f
icti
tiou
s “c
ent
rifu
gal”
forc
e
Proble
m S
olv
ing M
eth
ods
•U
se s
am
e p
rocess a
s for o
ther 2
nd
Law
proble
ms inclu
din
g F
BD
’s•
Form
ula
te p
roble
m in inertial (non-r
ota
ting) r
efe
rence fram
e•
Treat radia
l dir
ection a
t som
e m
om
ent as a
n a
xis
•D
ynam
ical quantities r
ota
te � ���
show
snapshot FBD
s•
Som
etim
es p
roble
ms n
eed s
everal snapshots
•Set th
e s
um
of real radia
l fo
rces =
mv
2/r term
(previo
us s
lide)
][
ma
r
mv
F
forc
es
ra
dia
l
real
all
c
2 Tc
∑ ∑∑∑= ===
= ==== ===
If
rope
bre
aks
, T v
ani
shes
as
doe
s a
c.
Mass
m f
lies
off
tang
ent
to
circ
le
rr
car
Tvr
P
m
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Exam
ple:
Eart
h S
ate
llit
e in
Cir
cula
r O
rbit
EA
RT
Hrr
car
Tvr
m
Giv
en
:A
ltit
ud
e h
= 5
20 k
mO
rbit
al
sp
eed
v =
7.6
km
/s =
7600 m
/s
Mass m
= 1
000 k
g
Fin
d:
Cen
trip
eta
l accele
rati
on
ac
Fo
rce F
exert
ed
by E
art
h
Sol
utio
n:
rv
a
ma
F
F
2 Tc
cc
g= ===
= ==== ===
Tre
at
Eart
h a
s a p
oint
mass
at
it’s c
ent
er
(via
Shell
Theor
em
)Cent
er
of o
rbit
is
at
Eart
h’s c
ent
er
km
r
alt
itu
de,
h
r
h r
e
e6
37
0= ===
= ===+ +++
= ===
m
10
x
6.8
9
km
6890
r
6= ===
= ===
Cent
ripe
tal acc
ele
rati
on:
m
/s
8.4
x.7
600
rv
a
2
22 T
c= ===
= ==== ===
61
08
96
Gra
vita
tion
al For
ce:
N 8400
m/s
8.4
x
kg
1000
m
a
F
2c
g= ===
= ==== ===
IND
EP
EN
DE
NT
OF
MA
SS
OB
JE
CT
S I
NS
IDE
TH
E S
AT
EL
LIT
E F
EE
L “
WE
IGH
TL
ES
S”
SIN
CE
TH
EY
AL
L H
AV
E T
HE
SA
ME
AC
CE
LE
RA
TIO
N
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Exam
ple:
Con
ical Pe
ndul
um
Th
e b
all s
win
gs a
rou
nd
in
a c
ircu
lar
path
L a
nd
θ θθθare
as
su
med
kn
ow
n (
init
ial
co
nd
itio
ns)
Fin
d t
he s
peed
v
Sketc
h a
nd
FB
D s
ho
w a
sn
ap
sh
ot
of
mo
tio
n
)L
sin
(
r
p
ath
cir
cu
lar
o
f
rad
ius
= ===θ θθθ
= ===
Vert
icall
y,
ball i
s i
n e
qu
ilib
riu
m
Ho
rizo
nta
lly,
ball
is in
un
ifo
rm c
ircu
lar
mo
tio
n
0
ma
mg
)co
s(
TF
yy
= ==== ===
− −−−θ θθθ
= ===∑ ∑∑∑
m
g)
co
s(
T = ===
θ θθθ
r
mv
m
a
)sin
(T
F2
cr
= ==== ===
θ θθθ= ===
∑ ∑∑∑
(1)
(2)
Div
ide (
2)
by (
1):
rgv
)
tan
(2
= ===θ θθθ
)g
rtan
(
v
θ θθθ= ===
))t
an
(g
Lsin
(
v
θ θθθθ θθθ
= ===
v is
ind
epe
ndent
of
mass
If
you
sta
rt w
ith v
= 0
, th
en
θ θθθ= 0
(ze
ro p
ath
radiu
s)v
mus
t be inf
init
e f
or θ θθθ
to a
ppro
ach
90
o
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Exam
ple:
The A
mus
em
ent
Park
Rot
or
Rid
ers
sta
nd a
roun
d t
he w
alls
of a
larg
e c
ylind
er
while it
spin
s.
Then
the f
loor
dro
ps o
ut.
What
min
imum
tang
ent
ial sp
eed v
min
mus
t th
e
rider
have
in
order
to n
ot f
all t
hro
ugh?
f s Fg
FN
R
y
x
Vert
icall
y,
rid
er
mu
st
be k
ep
t in
eq
uilib
riu
mb
y s
tati
c f
ricti
on
bala
ncin
g t
he w
eig
ht.
T
he
sm
all
est
valu
e o
f v c
orr
esp
on
ds t
o i
mp
en
din
g
mo
tio
n
Th
e n
orm
al
forc
e F
N is
th
e c
en
trip
eta
l fo
rce
R
mv
m
a
F
F2
cN
x= ===
= ==== ===
∑ ∑∑∑
F
F
Ns
ma
xs
,µ µµµ
= ===
0= ===
= ===− −−−
µ µµµ= ===
− −−−= ===
∑ ∑∑∑y
Ns
gm
ax
,s
ym
a
mg
F
F
fF
R
mv
mg
F
2 min
sN
sµ µµµ
= ==== ===
µ µµµ
g
R
v
s
2 min
µ µµµ= ===
gR
v
sm
inµ µµµ
= ===
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Exam
ple:
Bank
ed R
oadway w
/o F
rict
ion
Nx
Ny
Nr
Wr
x
y
mg
W
)
co
s(
N
N
)sin
(N
Ny
x= ===
θ θθθ= ===
θ θθθ= ===
Requ
ire e
quilib
rium
in
the y
dir
ect
ion
0= ===
= ===− −−−
= ===∑ ∑∑∑
yy
ym
a
W
NF
m
g
W
)c
os
(N
= ==== ===
θ θθθ(1
)
r
mv
F
)s
in(
N F
2
cx
= ==== ===
θ θθθ= ===
∑ ∑∑∑
Cent
ripe
tal acc
ele
rati
on a
long
radia
l (x
) dir
ect
ion:
r
mv
)sin
(N
2
= ===θ θθθ
(2)
Div
ide (
2)
by (
1):
g
r
v
)(
nat
2
= ===θ θθθ
Bank
ang
le is
indepe
ndent
of m
ass
(lu
ckily)
)ta
n(
r g
v
θ θθθ
= ===
What
speed s
hou
ld
I h
ave
on
this
icy
, bank
ed c
urve
?
46.3
o= ===
θ θθθ
Ro
ad
s a
re “
ban
ked
”at
cu
rves s
o t
hat
cars
do
no
t h
ave t
o
rely
on
fri
cti
on
to
sta
y o
n t
he r
oad
as t
he
y g
o a
rou
nd
cu
rves
. A
ss
um
e:
•cir
cu
lar
cu
rve r
ad
ius, r
= 7
0 m
–h
ori
zo
nta
l p
lan
e
•d
esig
n s
peed
v =
60 m
i/h
r =
26.8
m/s
•n
o f
ricti
on
at
all � ���
µ µµµs
= 0
Fin
d t
he b
an
k a
ng
le θ
.θ
.θ
.θ
.
For
larg
er
v, c
ar
slid
es
upFor
sm
aller
v, c
ar
slid
es
dow
n
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Exam
ple:
Weig
ht
on a
Ferr
is W
heel, T
ang
ent
ial Acc
ele
rati
on
The a
ppare
nt w
eig
hts
of
pass
eng
ers
at
the t
op a
nd b
otto
m o
f a
Ferr
is w
heel are
dif
fere
nt.
Mod
el th
e F
err
is w
heel as
a
vert
ical dis
k (r
adiu
s 7.2
m)
rota
ting
at
cons
tant
speed,
com
pleti
ng 1
revo
luti
on in
28 s
. Ass
ume t
he s
eats
rem
ain
hor
izon
tal as
the W
heel ro
tate
s.
Fin
d t
he a
ppare
nt w
eig
ht
at
the t
op (
Nu)
and
bot
tom
(N
b).
All
fo
rces
are
ra
dia
l at
top
an
d b
ott
om
To
p:
Rvm
N
mg
F2
uto
p,
y= ===
− −−−= ===
∑ ∑∑∑ ]
Rv g[
m
N 2
u− −−−
= ===N
N
u5
20
= ===
sm
aller
than
mg
off
fl
ies
p
assen
ger
an
du
weig
ht
the
o
rF
0
N
gR
v≤ ≤≤≤
≥ ≥≥≥
Bo
tto
m:
Rvm
mg
NF
2
bb
ot
,y
= ===− −−−
= ===∑ ∑∑∑
]Rv
g[ m
N
2
b+ +++
= ===N
N
b5
60
= ===
larg
er
than
mg
Fo
r θ θθθ,
N a
nd
mg
ha
ve r
ad
ial
an
d t
an
gen
tia
l co
mp
on
en
ts
)co
s(
N
Nr
θ θθθ= ===
θ θθθ )
sin
(N
N
tθ θθθ
= ===θ θθθ
)g
co
s(
gr
θ θθθ= ===
)g
sin
(
g
tθ θθθ
= ===
rr
2
mg
N
Rvm
− −−−= ===
] )R
co
s(
v
g[
m
N
2
θ θθθ+ +++
= ===θ θθθ
Red
uces t
oA
bo
ve f
or
θ θθθ� ���
0 o
r θ θθθ� ���
180
RA
DIA
L
tt
tN
mg
ma
− −−−
= === )
tan
(Rv
m
a
m 2
tθ θθθ
− −−−= ===
TA
NG
EN
TIA
L
Sp
eed
v w
ou
ld v
ary
(n
ot
in U
CM
) if
ma
tn
ot
can
celled
by s
ym
metr
y,
inert
ia o
f w
ho
le w
heel, b
rake…
bo
tto
m
θ θθθmg
mg
mg
Nθ θθθ
Nu
Nb
ar
ar
ar
R
v
v
top
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Non-U
niform
Cir
cula
r M
otion
Th
e a
ccele
rati
on
an
d f
orc
e c
an
have t
an
gen
tial as w
ell a
s r
ad
ial
(cen
trip
eta
l) co
mp
on
en
ts a
t p
oin
ts a
lon
g a
cir
cu
lar
path
.
gta
nra
dF
F
F∑ ∑∑∑
∑ ∑∑∑∑ ∑∑∑
+ +++= ===
rr
r
c
on
accele
rati
lcen
trip
eta
p
rod
uces
rad
a
F
rr
∑ ∑∑∑
t
on
accele
rati
ta
ng
en
tial
p
rod
uces
gta
na
F
r
r
∑ ∑∑∑
v
F
0v
F)
rad
(ra
d
rr
vo
v⊥ ⊥⊥⊥
= ===
We f
orc
e t
he v
elo
cit
y t
o r
em
ain
ta
ng
en
t to
th
e p
ath
, b
ut
the s
peed
ch
an
ges d
ue t
o a
t
Th
e n
et
rad
ial fo
rce c
an
no
t ch
an
ge
the s
peed
as lo
ng
as t
he v
elo
cit
y is
tan
gen
t to
th
e p
ath
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Helical Loop-t
he-Loop
Tre
at
the r
ide a
s a v
ert
ical ci
rcle
.Spe
ed v
ari
es
alo
ng t
he p
ath
, as
tang
ent
ial fo
rces
exis
t.
Nor
mal fo
rce is
alw
ays
radia
lly
inward
, equ
ivale
nt t
o te
nsio
nSim
ilar:
Ball s
pun
in a
vert
ical ci
rcle
on
a c
ord u
nder
tens
ion
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Exam
ple:
Loo
p-th
e loo
p: m
otio
n aro
und a
vert
ical ci
rcul
ar
track
bo
tto
m
θ θθθmg
mg
mg
Nrθ θθθ
Nu
Nb
ar
ar
ar
R
vu
vb
vt
top
NO
T U
nifo
rm C
ircu
lar
Mot
ion:
vva
ries
wit
h p
osit
ion.
All f
orce
s are
radia
l at
top
and
bot
tom
. E
lsewhere
(e
.g.,
at
θ)
θ)
θ)
θ)
weig
ht
mg
has
tang
ent
ial co
mpo
nent
. The n
orm
al fo
rce o
f th
e t
rack
is
radia
lly
inward
; no
ta
ngent
ial co
mpo
nent
as
fric
tion
is
abse
nt.
Bo
tto
m:
Rvm
mg
NF
2 bb
bo
t,
y= ===
− −−−= ===
∑ ∑∑∑ ]
gRv
[ m
g
N
2 bb
1+ +++
= ===la
rger
than
Nu
off
fa
lls
m
mass
an
du
forc
e
no
rmal
th
eu
o
rF
0
N
gR
v< <<<
< <<<
To
p:
Rvm
N
mg
F
2 uu
top
,y
= ===+ +++
= ===∑ ∑∑∑
]g
Rv[
mg
N
2 uu
1− −−−
= ===sm
aller
than
Nb
Fo
r p
oin
t θ θθθ, w
eig
ht
mg
has r
ad
ial an
d
tan
gen
tial co
mp
on
en
ts:
)m
gco
s(
mg
rθ θθθ
= === )
mg
sin
(
m
gt
θ θθθ= ===
tt
mg
ma
− −−−
= === )
gs
in(
a
tθ θθθ
− −−−= ===
TA
NG
EN
TIA
LC
CW
= p
osit
ive
so
atis
CW
here
Red
uces t
o t
op
&
bo
tto
m c
ases a
bo
ve
for
θ θθθ� ���
180 o
r θ θθθ� ���
0
RA
DIA
L r
mg
NRv
m− −−−
= ===θ θθθ
2
] )c
os
(g
Rv [ m
gN
θ θθθ+ +++
= ===θ θθθ
2
Wh
at
sh
ou
ld v
bb
e a
t b
ott
om
to
sta
y o
n t
rack a
t to
p?
: U
se
en
erg
y…
.W
hat
ch
an
ges
if
there
is f
ric
tio
n?
2
212
212
bb
mv
mg
R
m
v+ +++
= ===
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Motion in A
ccele
rate
d F
ram
es o
f Refe
rence
A f
icti
tiou
s fo
rce
due
to
inert
ia a
ppears
when
the f
ram
e
of r
efe
renc
e is
acc
ele
rati
ng (
non-
inert
ial).
•O
bje
cts
app
ear
to r
esp
ond t
o a f
orce
…but
…•
There
is
no o
bje
ct e
xert
ing
the f
orce
.•
Real fo
rces
are
alw
ays
inte
ract
ions
bet
wee
nob
ject
s.
Lin
earl
y a
ccele
rati
ng
syste
ms i
mp
ly f
icti
tio
us lin
ear
forc
e:
•R
ocket
in s
pa
ce f
irin
g m
oto
rs. P
eo
ple
an
d o
bje
cts
in
sid
e
see
m t
o f
eel
art
ific
ial “g
ravit
y”.
•T
rain
or
car
ac
cele
rati
ng
alo
ng
a l
inear
pa
th.
Passen
gers
an
d l
oo
se o
bje
cts
are
pu
sh
ed
to
ward
th
e b
ac
k
Ro
tati
ng
syste
ms (
UC
M o
r w
ith
an
gu
lar
accele
rati
on
) im
ply
“cen
trif
ug
al
forc
e”
at
co
nsta
nt
rad
ius,
du
e t
o c
en
trip
eta
l accele
rati
on
:•
Passen
gers
in
a c
ar
turn
ing
fe
el “cen
trif
ug
al
forc
e:.
•
Fic
titi
ou
s o
utw
ard
rad
ial
forc
e w
hen
ro
tati
ng
in
sid
e a
dru
m r
ide, sp
ace
sta
tio
n,…
Als
o i
n r
ota
tin
g s
yste
ms:
“C
ori
oli
sF
orc
e”
ap
pears
to
defl
ect
ob
jects
ch
an
gin
g r
ad
ius f
rom
th
e r
ota
tio
n c
en
ter:
•W
hir
lpo
ols
, c
yclo
nic
sto
rms,
ocean
cu
rren
ts f
low
CC
W in
No
rth
ern
Hem
isp
here
•B
all
th
row
n r
ad
iall
yin
sp
inn
ing
me
rry-g
o-r
ou
nd
ap
pears
to
defl
ec
t
Inertial fr
am
es a
re p
riv
ileged in that N
ew
ton’s
Law
s a
re o
beyed w
ithout in
venting fic
titious forces
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Ele
vato
rs a
s n
on-i
nertial syste
ms
5-2
:
A p
ers
on
weig
hin
g 0
.70 k
Nri
des i
n a
n e
leva
tor
that
has
an
up
ward
accele
rati
on
of
1.5
m/s
2.
Wh
at
is t
he m
ag
nit
ud
e o
f th
e f
orc
e o
f th
e e
levato
r fl
oo
r o
n t
he p
ers
on
?
A.
0.1
1 k
N
B.
0.8
1 k
NC
.0.7
0 k
ND
.0.5
9 k
NE
.0.6
4 k
N
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Exam
ple:
Fic
titi
ous
For
ces
in a
Lin
earl
y A
ccele
rati
ng T
rain
Tra
in is
acc
ele
rati
ng t
o th
e r
ight
alo
ng x
-axis
. Pe
ndul
um h
ang
s fr
eely
.
Bot
h ine
rtia
l and
non
-in
ert
ial ob
serv
ers
agr
ee:
•co
rd a
ngle
θ θθθ•
equ
ilib
rium
for
the y
-dir
ect
ion
•dis
agr
ee a
long
x
mg
- )T
co
s(
Fy
θ θθθ= ===
= ===∑ ∑∑∑
0
Inert
ial
fra
me o
bserv
er:
m a
cce
lera
tes
du
e t
o h
ori
zo
nta
l co
mp
on
en
t o
f T
m
g
)T
co
s(
= ===
θ θθθ
m
a
)T
sin
(
Fx
= ===θ θθθ
= ===∑ ∑∑∑
(no
fict
itio
us f
orce
s)
No
n-i
nert
ial
fra
me o
bserv
er:
m is i
n
eq
uil
ibri
um
alo
ng
x:
zero
acce
lera
tio
n
fic
tx
F - )
Tsin
(
0
F
θ θθθ= ===
= ===∑ ∑∑∑ (m
ust
intr
oduc
e f
icti
tiou
s fo
rce
to e
xpl
ain
the e
quilib
rium
)
Use
pend
ulum
to
measu
re a
: )
tan
(
g
a
θ θθθ= ===
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Exam
ple:
“Cent
rifu
gal”
forc
e in
a c
ar
maki
ng a
left
tur
n
A c
ar
is m
aki
ng a
left
tur
n on
a c
ircu
lar
sect
ion
of leve
l ro
ad a
t co
nsta
nt s
peed.
Obse
rvers
in
inert
ial and
rot
ati
ng f
ram
es
agr
ee t
hat
a
pass
eng
er
may s
lide t
o th
e r
ight
side o
f th
e c
ar.
rea
lce
ntr
ipe
tal
forc
e
Inert
ial
fra
me o
f th
e E
art
h:
the c
ar
seat
exert
s a
real
cen
trip
eta
l fo
rce o
n p
assen
ger
tow
ard
th
e c
en
ter
of
rota
tio
n,
du
e t
o f
ric
tio
n.
m
a
F
F
cs
ea
tra
d= ===
= ===∑ ∑∑∑ If t
he f
ricti
on
fo
rce is n
ot
larg
e e
no
ug
h, p
asse
ng
er
ten
ds
to
mo
ve
tan
gen
t to
th
e p
ath
du
e t
o N
ew
ton
’s
Fir
st
Law
fic
titi
ou
sce
ntr
ifu
ga
lfo
rce
No
n-i
nert
ial
fra
me (
accele
rati
ng
wit
h t
he c
ar)
: p
assen
ger
is i
n e
qu
ilib
riu
m, w
ith
fic
titi
ou
s “
cen
trif
ug
al
forc
e”
in
rad
ial
ou
tward
dir
ec
tio
n,
can
ce
led
by s
ea
t fr
icti
on
p
ush
ing
left
ward
.
Rv
m
F
F
FF
F2
c
fic
tfi
ct
se
at
rad
− −−−= ===
− −−−= ===
− −−−= ===
= ===∑ ∑∑∑
0
Inertial fr
am
es a
re p
riv
ileged in that N
ew
ton’s
Law
s a
re o
beyed w
ithout in
venting fic
titious forces
Co
pyri
gh
t R
. J
an
ow
–S
pri
ng
2012
Exam
ple:
“Cor
iolis
For
ce”
in a
rot
ati
ng s
yst
em
An
obje
ct m
ovin
g in
a r
otati
ng c
oord
inate
syst
em
may c
hang
e its
radia
l dis
tanc
e f
rom
the r
otati
on a
xis
.
•In
an
inert
ial sy
stem
it
follow
s N
ewto
n’s
Fir
st L
aw.
•In
the n
on-in
ert
ial (r
otati
ng)
syst
em
, ob
ject
resp
onds
to a
fic
titi
ous
“Cor
iolis
For
ce”,
resu
ltin
g in
cur
ved p
ath
s.
Exam
ple (
CW
rot
ati
on)
Exam
ple:
Air
or
wate
r m
ovin
g a
lon
g E
art
h’s
su
rface (
No
rth
of
Eq
uato
r) t
o:
•N
ort
h f
eels
We
stw
ard
Co
rio
lis
forc
e
•S
ou
th f
eels
Eastw
ard
Co
rio
lis
forc
eA
s a
resu
lt:
•W
ate
r sp
irals
CC
W a
rou
nd
a d
rain
•H
urr
ican
es
, to
rnad
oes,
Mo
nso
on
,e
tc s
pir
al
CC
W
Co
rio
lis
eff
ec
ts r
evers
e t
o C
W S
ou
tho
f th
e E
qu
ato
r.
