1.053/
2.003 Dynami
csandConr
to
lI
Fal
l2007
Probl
em Set9
Out: Tue
s
day
,Nove
mbe
r27th,2007
Due
: Monday
,De
ce
mbe
r3rd,2007
Eq
ui
l
i
bra
i,
l
i
nearz
iat
i
on,andst
abi
l
i
t
y
1.
Probl
em 1
Adi
s
c ofmas
sm andr
adi
usr i
ss
andwi
che
dbe
twe
e
na hor
i
z
ontals
ur
f
aceanda
s
l
e
nde
rr
odofmas
sm whi
chcans
l
i
dei
nandout ofa hor
i
z
ontalgr
oovewi
thout
f
r
i
cti
onass
howni
nFi
gur
e1. Thedi
s
c rol
l
swi
t
houts
l
i
p
page
wi
thr
e
s
pe
ct toboth
thehor
i
z
ontalbottom aswe
l
lasther
od. Ther
odi
sconne
cte
dtoa s
pr
i
ngofa
s
pr
i
ngcons
tant k ass
howni
nthef
i
gur
e
,andtheothe
re
ndofthes
pr
i
ngi
s
attache
dtothei
ne
r
ti
alf
r
ame
. As
s
umes
omel
e
ngthL0 oftheun-s
tr
e
tche
ds
pr
i
ng
(al
thoughthat wi
l
ltur
nout tobei
r
r
e
l
e
vant tothepr
obl
e
m). Ther
odi
spul
l
e
dout
ofthegr
ooves
uchthat thes
pr
i
ngi
ss
tr
e
tche
dbys
omel
e
ngthandthe
nr
e
l
e
as
e
d.
a. De
r
i
ve the e
quati
on ofmoti
on ofthe s
y
s
te
m us
i
ng the Lagr
angi
an
appr
oach.
b. Fi
ndthee
qui
l
i
br
i
a ofthes
y
s
te
m.
c. Li
ne
ar
i
z
e the e
quati
on ofmoti
on about the e
qui
l
i
br
i
um poi
ntsand
e
xami
nethes
tabi
l
i
tyofthee
qui
l
i
br
i
a.
Fi
gure1
Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J
Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
2.
Probl
em 2
Adi
s
c ofmas
sm andr
adi
usr r
ol
l
swi
thout s
l
i
ppi
ngonani
ncl
i
newhi
chi
sat
angl
e� wi
thr
e
s
pe
ct tothehor
i
z
ontalass
howni
nFi
gur
e2 be
l
ow. As
pr
i
ngofa
s
pr
i
ngcons
tant k i
sconne
cte
dtoa f
i
xe
ds
uppor
t anda pi
vot at thece
nte
rofthe
l
ythedi
s
ci
she
l
ds
uch
di
s
c ass
howni
nthef
i
gur
e
. Notethat gr
avi
tyacts
.I
ni
ti
al
that thes
pr
i
ngi
sun-s
tr
e
tche
d. Thedi
s
ci
sthe
ns
udde
nl
yr
e
l
e
as
e
d. I
ts
tar
tsr
ol
l
i
ng
wi
thout s
l
i
ppi
ngonthei
ncl
i
ne
.
a. De
r
i
vethee
quati
onofmoti
onofthes
y
s
te
m.
b. Fi
ndthee
qui
l
i
br
i
a ofthes
y
s
te
m.
c. Li
ne
ar
i
z
e the e
quati
on ofmoti
on about the e
qui
l
i
br
i
um poi
ntsand
e
xami
nethes
tabi
l
i
tyofthee
qui
l
i
br
i
a.
Fi
gure2
Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J
Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
3.
Probl
em 3
Cons
i
de
rthes
y
s
te
ms
howni
nthef
ol
l
owi
ngf
i
gur
easi
nPr
obl
e
m 2 ofExam 2.
You mayus
ethee
quati
onofmoti
onf
r
om thes
ol
uti
onspos
te
donl
i
ne
.
a. Fi
ndthee
qui
l
i
br
i
a ofthes
y
s
te
m.
b. Li
ne
ar
i
z
ethee
quati
onofmoti
onabout thee
qui
l
i
br
i
um q=0 ande
xami
ne
thes
tabi
l
i
tyofthi
se
qui
l
i
br
i
um (ony).
l
g
m, l
k
C
k
M, R
Fi
gure3
Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J
Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
4.
Probl
em 4
I
nthel
i
nkages
howni
nFi
gur
e4be
l
ow,thedi
s
kBands
l
i
de
rCe
achhavea mas
s
m. Thel
i
nkse
achhavea l
e
ngthlandar
eofne
gl
i
gi
bl
emas
s
. Thes
pr
i
nghasa
s
ti
f
f
ne
s
skanduns
tr
e
tche
dl
e
ngth2l
. Thes
l
i
de
ri
spus
he
ddownbya s
mal
l
di
s
tances
uchthat thes
pr
i
ngi
scompr
e
s
s
e
d,andthes
y
s
te
mi
sr
e
l
e
as
e
d. Notethat
gr
avi
tyacts
. Al
s
onoti
cethat poi
nt Ai
spi
nne
ds
oi
t doe
snot move
.
a. Fi
ndthee
quati
onofmoti
on.
b. Fi
ndthee
qui
l
i
br
i
a ofthes
y
s
te
m.
c. Li
ne
ar
i
z
e the e
quati
on ofmoti
on about the e
qui
l
i
br
i
um poi
ntsand
e
xami
nethes
tabi
l
i
tyofthee
qui
l
i
br
i
a.
g
Fi
gure4
Cite as: Sanjay Sarma, Nicholas Makris, Yahya Modarres-Sadeghi, and Peter So, course materials for 2.003J/1.053J
Dynamics and Control I, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].