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PROBLEM 19.80 (Continued)
Conservation of energy.
222 22 2
11 2 2 0
11 1 1 1
:00
22 3 2 2 2
AB m m AB m
l
TV T V mr ml kr mg
θ
θθ
⎛⎞
+=+ + +=+ +
⎜⎟
⎝⎠
For simple harmonic motion,
22222 2
0
2
2
11
23 2
mnm
AB n m AB m
AB
l
mt m l kr m g
kr m gl
θωθ
ωθ θ
=
⎛⎞⎛⎞
+=+
⎜⎟⎜⎟
⎝⎠⎝⎠
+
S
O
O
LUTION
P
R
A
s
coll
con
s
B c
equ
i
dis
p
vib
r
R
OBLEM 1
9
s
lender 10-kg
ars of negli
g
s
tant
1.5
k
k=
an slide freel
i
librium whe
n
p
lacement an
d
r
ations.
9
.81
bar AB of
l
g
ible weight.
k
N/m
and ca
n
y on a vertic
n
bar AB is v
e
d
released,
d
(
1
2
Vkl
θ
=
11
212
T
m
⎡
=
⎢
⎢
⎣
l
ength
0.
6
l
=
Collar A is
n
slide on a h
o
al rod. Kno
w
e
rtical and tha
t
d
etermine the
)
2
2
4
l
mg
θ
θ
⎛
+⎜
⎜
⎝
2
2
2
l
m
lm
⎛⎞
+⎜⎟
⎝⎠
6
m
is conn
e
attached to
o
rizontal rod,
w
ing that the
s
t
collar A is
g
period of t
h
2
⎞
⎟
⎟
⎠
2
θ
⎤
⎥
⎥
⎦
e
cted to two
a spring of
while collar
s
ystem is in
g
iven a small
h
e resulting
PROBLEM 19.82
A slender 5-kg bar AB of length l = 0.6 m is connected to two collars,
each of mass 2.5 kg. Collar A is attached to a spring of constant
k = 1.5 kN/m and can slide on a horizontal rod, while collar B can
slide freely on a vertical rod. Knowing that the system is in
equilibrium when bar AB is vertical and that collar A is given a small
displacement and released, determine the period of the resulting
vibrations.
SOLUTION
Let m be the mass of rod and C
m be mass of each collar
The
n
22 2 2
24 2
C
kl mgl l
Vmg
θ
θθ
⎛⎞
=+ +
⎜⎟
⎜⎟
⎝⎠
()
22
2
11
23 2
C
l
Tm ml
θ
θ
⎛⎞
=+
⎜⎟
⎜⎟
⎝⎠
2
mgl m gl
n
S
O
P
o
s
O
LUTION
s
ition
1
(
P
R
A
n
k
=
at
p
e
1
2
1(
2
1
(
)12
G
GAB
TI
I
m
l
=
=
R
OBLEM
1
n
800-g rod
A
12 N/m
=
is
a
C. Knowing
e
riod of small
o
2
2
1
)2
1(0.8)(
0
12
G
AB m
A
m
l
θ
+
=
1
9.83
A
B is bolted
t
a
ttached to th
e
that the dis
k
o
scillations o
f
2
2
2
0
.6) 0.024
k
A
Bm
lr
θ
⎛⎞
−
⎜⎟
⎝⎠
=
t
o a 1.2-kg d
i
e
center of th
e
k
rolls witho
u
f
the system.
disk
2
1()
2
k
gm
G
m
I
θ
+
⋅
i
sk. A spring
e
disk at A an
d
u
t sliding, d
e
22
disk
1
2
m
mr
θ
+
of constant
d
to the wall
e
termine the
2
m
θ
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 19.83 (Continued)
2
2
2
11 2 2
222
22 2
2
2
2
1[0.750 2.354]
2
1(3.104) N m
2
11
(0.1385) 0 0 (3.104)
22
(3.104 N m)
(0.1385 kg m )
22.41 s
m
m
mnm
mn m
n
V
TV T V
θ
θ
θωθ
θω θ
ω
−
=+
=
⋅
+=+
=
+=+
⋅
=⋅
=
22
22.41
n
n
π
π
τω
== 1.327 s
n
τ
=
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 19.84
Three identical 3.6-kg uniform slender bars are connected by pins as
shown and can move in a vertical plane. Knowing that bar BC is given
a small displacement and released, determine the period of vibration of
the system.
SOLUTION
22
12
0, 2
222
l
VVmgl mg
θ
θ
⎛⎞
== +⎜⎟
⎜⎟
⎝⎠
2
mgl
θ
=
2
22 2
21
11
0, 2
223
ml
TTml
θ
θ
⎡
⎤
== +
⎢
⎥
⎣
⎦
22
5
6
ml
θ
=
2
222 22
56
:,
65
nnn
ml g
mgl l
θωθ ωω
== =
()
()
22
1.586 s
9.81 m/s
6
0.75 m
5
n
π
π
τω
== =
SO
P
o
s
P
R
A
1
rot
a
ro
d
LUTION
s
ition
1
R
OBLEM 1
9
1
4-oz sphere
A
a
te in a verti
c
d
.
9
.85
A
and a 10-o
z
c
al plane abo
u
z
sphere C ar
e
u
t an axis at
B
e
attached to
t
B
. Determine
t
he ends of a
the period o
f
20-oz rod
AC
f
small oscilla
t
C
which can
t
ions of the
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 19.85 (Continued)
P
osition 2
2
2
2
2
2
2
2
2
2
2
0
58 1
(1cos) (1cos) (1cos)
12 12 8
1cos 2sin 22
14 5 10 8 20 1 (lb ft)
16 12 16 12 16 8 2
[ 0.3646 0.4167 0.1563] 2
0.2084
2
A
mC mAC m
mm
m
m
m
m
T
VW W W
V
V
V
θ
θθ
θθ
θ
θ
θ
θ
=
=− − + − + −
−= ≈
⎡⎤
⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞
=− + + ⋅
⎢⎥
⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟
⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠
⎣⎦
=− + +
=
Conservation of energy. 22
11 2 2
1 0.2084
: (0.01775) 0 0
22
mm
TV T V
θ
θ
+=+ +=+
Simple harmonic motion.
20.2084 11.738
0.01775
22
11.738
mnm
n
n
n
θωθ
ω
ππ
τω
=
==
==
1.834 s
n
τ
=
SO
P
o
s
LUTION
s
ition
1
PR
O
A 1
0
whi
c
Kn
o
sma
1
1
2
2
T
=
O
BLEM 1
9
0
-lb uniform
r
c
h is welded
t
o
wing that th
e
ll oscillations
A
B
WW
=
=
2
disk
2
()
2
Am
I
θ
+
9
.86
r
od CD is we
l
t
o the center
s
e
disks roll w
i
of the syste
m
disk
W
disk
2
1(
2
Wr
g
θ
⎛⎞
⎜⎟
⎝⎠
l
ded at C to
a
s
of two 20-l
b
i
thout sliding
,
m
.
2
)
m
θ
a
shaft of neg
l
b
uniform dis
k
,
determine t
h
l
igible mass
k
s A and B.
h
e period of
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 19.86 (Continued)
Small angles:
2
2
2
2
2
2
1cos 2sin 22
1
22
1(10) (1.5)
2
1(15)
2
mm
m
m
CD
m
m
VWl
θ
θ
θ
θ
θ
θ
−= ≈
=
=
=
Conservation of energy and simple harmonic motion.
11 2 2
22 2
2
11
(70) 0 0 (15)
22
15
70
mnm
nm m
n
TV T V
g
g
θωθ
ω
θθ
ω
+=+
=
+=+
=
Period of oscillations. 270
2(15)(32.2)
n
n
π
τπ
ω
== 2.39 s
n
τ
=
