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S
O
O
LUTION
PROB
L
A perio
d
suspend
e
1.25-in.-
d
radius o
f
(
MM
Σ=Σ
L
EM 19.67
d
of 6.00 s is
e
d from a wir
e
d
iameter steel
f
gyration of t
h
eff
):
M
K
θ
−
=
2
n
K
I
θ
θ
ω
τ
+=
=
=
observed fo
r
e
as shown.
K
sphere is sus
p
h
e rotor. (Spe
c
I
θ
=
0
2
K
I
I
K
π
=
=
=
r
the angular
o
K
nowing that
p
ended in the
c
ific weight o
f
o
scillations o
f
a period of 3
same fashion
,
f
steel = 490 l
b
f
a 4-oz gyro
.80 s is obtai
n
,
determine th
b
/ft
3
.)
scope rotor
n
ed when a
e centroidal
(1)
PROBLEM 19.67 (Continued)
From Eq. (2):
26
2
6
4 (9.7719 10 )
(3.80)
26.716 10 lb ft/rad
K
π
−
−
×
=
=×⋅
41
W
⎛⎞⎛ ⎞
the
Let
Th
e
Eq
u
Copyrig
h
vertical direc
t
F
b
e the hori
z
e
two forces
F
u
ation of moti
o
ht
© McGra
w
t
ion. Then,
si
n
z
ontal compo
n
F
form a coupl
e
o
n:
w
-Hill Educ
a
PRO
B
The c
e
suspe
n
is rota
t
releas
e
deter
m
n
(10 ft ) /
ϕ
θ
=
0:
2
c
F
T
Σ
=
=
n
ent of T.
s
FT
=
e
of moment
M
=
y
M
Σ=
a
tion. Permis
s
B
LEM 19.
6
e
ntroidal radi
u
n
ding the airpl
t
ed through
a
e
d. Knowing
m
ine the centr
o
5
6
(12 ft ) .
θ
=
2cos
c
os 2
T
W
WW
ϕ
ϕ
−
≈
5
s
in 26
W
ϕθ
≈⋅
(20 ft)F
=
−=
:20
(20
)
1
2
y
I
α
θ
⎛
=
−⎜
⎝
+
s
ion require
d
6
8
u
s of gyratio
n
ane by two 1
2
a
small angle
a
that the obs
o
idal radius o
f
0
W
=
5
12 W
θ
θ
=
5
(20) 12
W
θ
⎛⎞
−⎜⎟
⎝⎠
2
5
12
)
(5) 0
2
y
WW
k
g
g
k
θ
θ
⎛
⎞=
⎟
⎝
⎠
=
d
for reprodu
n
y
k of an a
i
2
-ft-long cabl
e
a
bout the ver
t
erved period
f
gyration .
y
k
W
θ
2
y
k
θ
ction or disp
i
rplane is det
e
e
s as shown.
T
t
ical through
of oscillatio
n
lay.
e
rmined by
T
he airplane
G and then
n
is 3.3 s,
PROBLEM 19.68 (Continued)
Natural frequency:
2
222
(20)(5) (20)(5)(32.2) 268.33
12 12
n
yyy
g
kkk
ω
== =
y
S
O
O
LUTION
P
T
w
p
i
a
r
a
t
is
i
n
m
P
ROBLEM
1
w
o blocks, e
a
i
n-connected
t
r
e negligible,
a
t
tached to a s
p
at rest when
n
itial velocit
y
m
aximum disp
l
(
2
1
2
Tmb
=+
2
1
2
Vkc
θ
=
2
2
2
n
kc
ω
=
1
9.69
a
ch of mass 1
t
o bar BC as s
a
nd the block
s
p
ring of const
a
it is struck h
o
y
of 250 m
m
l
acement of b
l
)
22
c
θ
+
2
θ
2
2
.5 kg, are att
a
hown. The m
a
s
can slide wi
t
a
nt k = 720 N/
o
rizontally w
i
m
/s determin
e
l
ock D during
a
ched to link
s
a
sses of the li
n
t
hout friction.
m. Knowing
t
i
th a mallet a
n
e
the magnit
u
the resulting
m
s
which are
n
ks and bar
Block D is
t
hat block A
n
d given an
u
de of the
m
otion.
PROBLEM 19.70
Two small spheres, A and C, each of mass m, are attached to rod AB,
which is supported by a pin and bracket at B and by a spring CD of
constant k. Knowing that the mass of the rod is negligible and that the
system is in equilibrium when the rod is horizontal, determine the
frequency of the small oscillations of the system.
SOLUTION
2
22 2
11
22 2
B
l
TI mlm
θ
θ
⎡
⎤
⎛⎞
==+
⎢
⎥
⎜⎟
⎝⎠
⎢
⎥
⎣
⎦
25
n
π
S
O
Da
t
P
o
s
P
R
A
w
e
os
c
O
LUTION
t
um at 1:
s
ition 1
R
OBLEM 1
14-oz sphere
e
ight which c
a
c
illations of t
h
9.71
A and a 10-
o
a
n rotate in a
h
e rod.
1
1
C
T
V
h
=
=
=
o
z sphere C
a
vertical plan
e
0
(1 cos
CC AA
m
Wh Wh
BC
θ
−
−
a
re attached
t
e
about an ax
i
)
t
o the ends o
f
i
s at B. Deter
m
f
a rod AC o
f
m
ine the peri
o
f
negligible
o
d of small
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 19.71 (Continued)
22
2
22
2
2
22
11
() ()
22
1815
(0.019410) (0.027174)
212212
0.013344 2
0.013344 2
CCm AAm
m
m
nm
Tmv mv
θ
θ
ωθ
=+
⎡⎤
⎛⎞ ⎛⎞
=+
⎢⎥
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
⎢⎥
⎣⎦
=
=
Conservation of energy.
222
11 2 2
: 0 0.05208 0.013344 0
22
mnm
TV T V
θωθ
+=+ + = +
20.05208 3.902
0.013344
1.9755 rad/s
n
n
ω
ω
==
=
Period of oscillations. 2
n
n
π
τ
ω
= 3.18 s
n
τ
=
S
O
Da
t
P
o
s
P
o
s
O
LUTION
t
um at :
s
ition
1
s
ition
2
PRO
B
Deter
m
move
s
1
1
0
T
V
W
=
=
1
m
v
R
=
2
0
V
=
B
LEM 19.
7
m
ine the peri
s
without fric
t
0
(1 cos
)
m
W
R
θ
−
2
2
1
2
m
R
T
m
θ
=
0
7
2
od of small
o
t
ion inside a
c
)
22
1
2
m
m
vmR
θ
=
o
scillations o
f
c
ylindrical s
u
2
m
θ
f
a small par
t
u
rface of radi
u
t
icle which
u
s R.
S
O
D
a
P
o
s
P
o
s
O
LUTION
a
tum at
1
:
s
ition
1
s
ition
2
PROBL
E
The inner
r
of its sm
a
moment o
f
1
1
2
T
I
=
0
T
=
E
M 19.73
r
im of an 85-
l
a
ll oscillation
s
f
inertia of the
2
01
0
m
I
V
θ
=
Vmgh
=
l
b flywheel is
s
is found t
o
flywheel.
placed on a
k
o
be 1.26 s.
D
k
nife edge, a
n
D
etermine th
n
d the period
e centroidal
