S
O
O
LUTION
PROB
A 6-kg
u
and is a
t
is attac
h
Knowin
g
position
(b) the
m
LEM 19.3
9
u
niform cyli
n
t
tached by a
p
h
ed to two s
p
g
that the ba
r
and released,
m
agnitude of t
h
9
n
der can roll
w
p
in at point C
p
rings, each
r
is moved 1
2
determine (a
)
h
e maximum
v
w
ithout slidin
g
to the 4-kg h
o
of constant
k
2
mm to the
)
the period o
f
v
elocity of b
a
g
on a horizo
n
o
rizontal bar
A
k
= 5 kN/m
right of the
f
vibration of
a
r AB.
n
tal surface
A
B. The bar
as shown.
equilibrium
the system,
2
2
6
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 19.40
A 6-kg uniform cylinder is assumed to roll without sliding on a horizontal
surface and is attached by a pin at point C to the 4-kg horizontal bar AB.
The bar is attached to two springs, each of constant k = 3.5 kN/m as
shown. Knowing that the coefficient of static friction between the
cylinder and the surface is 0.5, determine the maximum amplitude of the
motion of point C which is compatible with the assumption of rolling.
SOLUTION
From Problem 19.39
(
)
223500N/m
2538.46; 23.2048 rad/s
34 kg 9 kg
2
nn
k
M
m
ωω
== = =
+
+
()
(
)
()
(
)
22
6kg 9.81m/s 4kg 9.81m/s 98.1NNW Mgmg== + = + =
22
11
22
x
Fr Mr Mr r
θ
⎛⎞
==
⎜⎟
⎝⎠
(
)
12
26kg
max , ; Amplitude
98.1 N
ω
===
mmn
Fxx AA
N
()
()
2
16kg
2
0.5 23.2048 rad/s
98.1 N
s
A
μ
==
or 0.03036 mA=
30.4 mmA=
SO
Eq
u
LUTION
u
ation of moti
o
o
n.
M
Σ
PR
O
A 15-
l
b
elt i
s
rod at
down
maxi
m
e
f
()
BB
M
M=Σ
L
O
BLEM 19.
4
l
b slender ro
d
s
attached to
t
rest in the p
o
and release
d
m
um velocity
o
f
f
:cos
2
L
mg
θ
4
1
d
AB is rivete
d
t
he rim of th
e
o
sition shown
.
d
, determine
o
f end A.
AB
kxr I
θα
−=
d
to a 12-lb u
e
disk and to
a
.
If end A of
t
(a) the peri
o
2
L
L
m
α
α
⎛⎞⎛
+⎜⎟⎜
⎝⎠⎝
niform disk
a
a
spring whi
c
t
he rod is mo
v
o
d of vibrati
o
disk
2
L
I
α
⎞+
⎟
⎠
a
s shown. A
c
h holds the
v
ed 0.75 in.
o
n, (b) the
(1)
e
a
a
a
n
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 19.41 (Continued)
2
2
2
2
disk disk
2
2
1
12
1(0.46584)(3.0)
12
0.34938 lb s ft
1
2
1(0.37267)(0.83333)
2
0.1294 lb s ft
AB
ImL
Imr
=
=
=⋅⋅
=
=
=⋅⋅
22
1
0.34935 (0.46584)(3.0) 0.1294 (360)(0.83333) 0
4
θθ
⎡⎤
+++=
⎢⎥
⎣⎦
1.5269 250 0
θθ
+=
or 163.73 0
θθ
+
=
(a) Natural frequency and period. 22
163.73 (rad/s)
n
ω
=
12.796 rad/s
22
12.796
n
n
ω
π
π
τω
=
== 0.491 s
τ
=
(b) Maximum velocity. (12.796)(0.75)
mnm
vx
ω
== 9.60 in./s
m
v=
Su
b
Copyrig
h
b
stituting Eq.
(
ht
© McGra
w
(
2) into Eq. (1
w
-Hill Educ
a
P
R
A
3
b
e
l
cy
l
m
o
of
cy
l
C
) and noting t
h
0
0
0
0
3
2
x
rmx I r
mrx
x
+
⎛
+
⎜
⎝
a
tion. Permis
s
R
OBLEM
1
3
0-lb unifor
m
l
t is attached
l
inder at rest
i
o
ved 2 in. do
w
vibration, (b
l
inder.
st
h
at
0
,
x
r
θ
=
0
0
0
40
1
2
40
80
3
rkx
Im
rkx
kx
m
+=
=
+=
⎞=
⎟
⎠
s
ion require
d
1
9.42
m
cylinder can
to the rim o
f
i
n the positio
n
w
n the incline
) the maxim
u
0
,
x
r
θ
=
2
m
r
d
for reprodu
roll without s
l
f
the cylinde
r
n
shown. If th
e
and released,
u
m accelerat
i
ction or disp
l
iding on a 15
°
r
, and a sprin
g
e
center of th
e
determine (a
)
i
on of the c
e
lay.
°
–
incline. A
g
holds the
e
cylinder is
)
the period
e
nter of the
PROBLEM 19.42 (Continued)
Natural frequency.
2
1
(30 lb)
(32.2 ft/s )
8(8)(3012 lb/ft)
32.1 s
3(3)
n
k
m
ω
−
×
== =
π
π
0max
SO
(a)
t
t
t
t
h
l
l
k
k
i
i
i
i
d
e
t
e
e
e
c
c
u
u
m
m
LUTION
Small vert
i
Let the pla
t
downward
horizontal
i
cal displace
m
t
e be displace
d
a distance x
a
springs exert
n
PROBL
E
A square pl
a
that each s
p
frequency
o
displaceme
n
about G an
d
m
ent.
d
downward
a
a
nd the four v
e
n
egligible ch
a
E
M 19.43
a
te of mass m
p
ring can ac
t
o
f the resultin
g
n
t and releas
e
d
released.
a
distance x fr
o
e
rtical springs
a
nge.
is held by eig
h
t
in either te
g
vibration (
a
e
d, (b) if the
p
o
m the equili
b
exert additio
n
:Fma
Σ
=
h
t springs, ea
c
nsion or co
m
a
) if the plate
p
late is
r
otat
e
b
rium positio
n
n
al forces kx
f
4kx mx−=
c
h of constant
k
m
pression, de
t
is given a s
m
e
d through a
s
n
. Each corner
f
or each sprin
g
k
. Knowing
t
ermine the
m
all vertical
s
mall angle
moves
g
. The
:
:
+
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 19.43 (Continued)
2
:4 4( /2)
G
MI FlI kl
α
θθ
==−
2
2
1
6
12 0
m
12
m
n
ml
k
k
θ
θθ
ω
=
+=
=
Frequency: 112
22
nk
fm
ω
ππ
== 0.551 k
fm
=
SO
Eq
u
LUTION
u
ation of moti
o
M
Σ
PRO
B
Two s
m
and wei
the angl
o
n.
ef
f
()
CC
M
M=Σ
B
LEM 19.4
4
m
all weights
w
ght W. Deno
t
e
β
for whic
h
f
:sin(wr
β
−
4
w
are attached
t
ing by
0
τ
the
h
the period o
f
2
1
2
t
D
ar r
W
I
r
g
αθ
αθ
=
=
=
=
)sin(
wr
θ
β
−
−
at A and B t
o
period of s
m
f
small oscilla
t
2
θ
2
)wr
a
g
β
θ
+=
o
the rim of a
m
all oscillatio
n
t
ions is
0
2.
τ
t
a
I
α
+
uniform dis
k
n
s when
0
β
=
k
of radius r
0
,
determine
2
τ
=
π
Copyri
g
g
h
t
© McGra
w
PRO
B
Two 4
0
10
0
r=
w
-Hill Educ
a
B
LEM 19.4
5
0
-g weights a
r
0
mm. Deter
m
a
tion. Permi
s
5
r
e attached at
A
m
ine the frequ
e
2
n
π
s
sion require
d
A
and B to th
e
e
ncy of small
o
2
π
d
for reprod
u
e
rim of a 1.5
–
o
scillations
w
u
ction or dis
p
–
kg uniform d
i
w
hen
60 .
β
=°
n
p
lay.
i
sk of radius
u
i