PROBLEM 19.74
A connecting rod is supported by a knife edge at Point A; the period of its small
oscillations is observed to be 1.03 s. Knowing that the distance ra is 6 in. determine the
centroidal radius of gyration of the connecting rod.
S
O
P
A
d
f
O
LUTION
P
ROBLE
M
A
uniform ro
d
istance c ab
o
f
or which the
M
19.75
d
AB
can ro
t
o
ve the mass
c
frequency of
t
t
ate in a vert
i
c
enter G of th
e
t
he motion wi
i
cal plane ab
o
e
rod. For sm
a
ll be maximu
m
o
ut a horizon
t
a
ll oscillation
s
m
.
t
al axis at C
s
, determine t
h
located at a
h
e value of c
S
O
O
LUTION
PROB
L
A homog
e
about a
oscillatio
n
small osc
i
L
EM 19.76
e
neous wire
o
frictionless
p
n
s when
β
=
i
llations is
2
τ
o
f length 2l is
b
p
in at B. D
e
0,
determine
0
.
b
ent as show
n
e
noting by
τ
the angle
β
n
and allowed
0
τ
the perio
d
for which th
e
to oscillate
d
of small
e
period of
t
u
t
a
i
t
i
a
≈
=
y
2
c
(
NPROBLEM 19.76 (Continued)
PROBLEM 19.77
A uniform disk of radius r and mass m can roll without slipping on a cylindrical
surface and is attached to bar ABC of length L and negligible mass. The bar is
attached to a spring of constant k and can rotate freely in the vertical plane about
Point B. Knowing that end A is given a small displacement and released,
determine the frequency of the resulting oscillations in terms of m, L, k, and g.
23 3
n
π
S
O
O
LUTION
PROB
L
Two uni
welded t
o
each spr
i
and rele
a
L
EM 19.78
form rods, e
a
o
gether to for
m
i
ng is k = 0.6
a
sed, determi
n
a
ch of weigh
t
m
the assemb
lb/in. and th
a
n
e the frequen
c
t
W = 1.2 lb
ly shown. Kn
o
a
t end A is g
i
c
y of the resu
l
and length l
o
wing that th
e
i
ven a small
d
l
ting motion.
= 8 in., are
e
constant of
d
isplacement
a
p
r
n
f
0
0
PROBLEM 19.78 (Continued)
P
osition 2 2
2
2
2
2
2
0
1
(1 cos ) 2
222
(1.2)(0.66667) 1 1 0.66667
(2) (7.2)
22 2 2
0.6
mm
mm
m
T
Wl l
Vk
θθ
θθ
θ
=
⎛⎞⎛ ⎞
=− − + ⎜⎟⎜ ⎟
⎝⎠⎝ ⎠
⎛⎞⎛⎞⎛ ⎞
≈− +
⎜⎟⎜⎟⎜ ⎟
⎝⎠⎝⎠⎝ ⎠
≈
Conservation of energy. 11 2 2
TV T V
+
=+
32 2
3.4506 10 0 0 0.6
13.186
mm
mm
θ
θ
θ
θ
−
×+=+
=
Simple harmonic motion. mnm
θ
ωθ
=
13.186 rad/s
n
ω
=
Frequency. 2
n
n
f
ω
π
= 2.10 Hz
n
f=
S
O
t
O
LUTION
P
A
is
m
p
e
c
y
ROBLEM
1
15-lb unifor
m
attached to
a
m
oved 0.4 in.
e
riod of vibra
y
linder.
1
9.79
m
cylinder ca
n
a
spring AB a
s
down the i
n
tion, (b) the
m
n
roll withou
t
s
shown. If th
e
n
cline and re
l
m
aximum ve
l
t
sliding on a
n
e
center of th
l
eased, deter
m
l
ocity of the
c
n
incline and
e cylinder is
m
ine (a) the
c
enter of the
)
)
PROBLEM 19.79 (Continued)
Substituting Eq. (2) into Eq. (1)
22 2 2
22 2 2 2
2
2
()
mmm
mnmm m nm
mmn
kr I mv
vr r
kr I mr
kr
θθ
θ
ωθ θ ωθ
θθω
=+
===
=+
1
S
O
O
LUTION
PR
O
A 3-
k
const
posit
i
relea
s
O
BLEM 19.
k
g slender ro
d
ant 280 N/m
i
on shown. If
s
ed, determin
e
80
d
AB is bolte
is attached
end B of the
e
the period o
f
d to a 5-kg
u
to the disk
a
rod is given
f
vibration of
t
u
niform disk.
a
nd is unstre
t
a small displ
a
t
he system.
A spring of
t
ched in the
a
cement and
(
(
PROBLEM 19.80 (Continued)
Conservation of energy.
11 1 1 1
l
⎛⎞
58.55
n
n
n
S
O
O
LUTION
θ
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
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
l
θ
⎛
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
⎞
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.
I
m
2
(
0
0
1
(
0
P
R
A
n
k
=
at
p
e
R
OBLEM
1
n
800-g rod
A
12 N/m
=
is
a
C. Knowing
e
riod of small
o
1
9.83
A
B is bolted
t
a
ttached to th
e
that the dis
k
o
scillations o
f
t
o a 1.2-kg d
i
e
center of th
e
k
rolls witho
u
f
the system.
i
sk. A spring
e
disk at A an
d
u
t sliding, d
e
of constant
d
to the wall
e
termine the
(
I
0
(
k
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 )
m
m
mnm
mn m
n
V
TV T V
θ
θ
θωθ
θω θ
ω
−
=+
=
⋅
+=+
=
+=+
⋅
=⋅
22.41
n
n
n
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.
()
()
9.81 m/s
6
0.75 m
5
n
SO
P
R
A
1
rot
a
ro
d
LUTION
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
PROBLEM 19.85 (Continued)
P
osition 2
2
2
2
2
2
0
58 1
(1cos) (1cos) (1cos)
12 12 8
1cos 2sin 22
A
mC m AC m
mm
m
T
VW W W
θ
θθ
θθ
θ
=
=− − + − + −
−= ≈
11.738
n
n
n
SO
2
(
5
2
2
(
5
2
LUTION
PR
O
A 1
0
whi
c
Kn
o
sma
O
BLEM 1
9
0
-lb uniform
r
c
h is welded
t
o
wing that th
e
ll oscillations
WW
=
=
9
.86
r
od CD is we
l
t
o the center
s
e
disks roll w
i
of the syste
m
W
l
ded at C to
a
s
of two 20-l
b
i
thout sliding
,
m
.
a
shaft of neg
l
b
uniform dis
k
,
determine t
h
l
igible mass
k
s A and B.
h
e period of
PROBLEM 19.86 (Continued)
θ
2
2
mm
θ
θ
n
n
ω
n