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PROBLEM 18.89 (Continued)
Let the reference frame Dxyz be rotating with angular velocity
1
.=Ωω
( )
( )
( )
11
0 sin cos
G G G xx yy
Gxyz
II
ω βω β ω ω
= +× =+ + × +HH Η i j ij
Ω
22
11
sin ,
D
Fd mgL
β
= −Mk k
where
22
d Lr= +
( )
/
eff
D G GD G
m=+×M Hr a
22
1sin cos
GmL
ω ββ
= +Hk
Equating
( )
eff
DD
=MM
and taking the z component,
2 2 2 2 22
11
11
sin sin sin cos sin cos
24
C
L
F d mgL mr mr m L
r
β β ββω ω ββ
−= − +
2 22
1
11
sin sin cos cos
24
m rL r L
ωβ β β β
= −−
Solving for
,
C
F
22 2
1
sin 1 1
cos cos sin
42
C
m
F gL L r rL
d
βωβ β β
= − +−
Additional data:
2
1
20.062112 lb s /ft, 4 rad/s
32.2
m
ω
==⋅=
22
20 4 1.69967 ft
12 12
d
= +=
PROBLEM 18.89 (Continued)
( ) ( ) ( )
22
2
0.061112 sin 30 20 20 1 4 1 20 4
32.2 4 cos30 cos30 sin 30
1.69967 12 12 4 20 2 12 12
C
F
°
= − °+ °− °
0.311 lb=
11
4
tan 30 tan 18.7
20
r
L
αβ α
−−
= − = °− = °
0.311 lb
18.7°
PROBLEM 18.90
The slender rod AB is attached by a clevis to arm BCD which
rotates with a constant angular velocity
ω
about the centerline
of its vertical portion CD. Determine the magnitude of the
angular velocity
.
ω
SOLUTION
Let
0.3 m, and 0.1 m.AB L BC b= = = =
Choose
,,xyz
axes as shown.
2
1
0, 12
x yz
I I I mL≈==
PROBLEM 18.90 (Continued)
2
cos30 sin 30
22
BG
LL
mg mr
ω
Σ = °= ° +
M k kH
22
3 13
4 4 48
mgL mLr mL
ω
= +
kk
2
3 13
4 4 48
grL
ω
= +
( ) ( ) ( )
2
31 3
9.81 0.22990 0.3
4 4 48
ω
= +
2
62.194
ω
=
7.89 rad/s
ω
=
PROBLEM 18.91
The slender rod
AB is attached by a clevis to arm BCD which rotates with a constant
angular velocity
ω
about the centerline of its vertical portion CD. Determine the
magnitude of the angular velocity
.
ω
SOLUTION
Let
0.3 m, and 0.1 m.AB L BC b= = = =
Choose
,,xyz
axes as shown.
2
1
0, 12
x yz
I I I mL≈==
PROBLEM 18.91 (Continued)
2
cos30 sin 30
22
BG
LL
mg mr
ω
Σ =− °=− ° +
M k kH
22
3 13
4 4 48
mgL mLr mL
ω
− =−+
kk
2
3 13
4 4 48
grL
ω
= +
( ) ( ) ( )
2
31 3
9.81 0.029904 0.3
4 4 48
ω
= +
2
232.11
ω
=
15.24 rad/s
ω
=
PROBLEM 18.92
The essential structure of a certain type of aircraft turn indicator is shown.
Springs AC and BD are initially stretched and exert equal vertical forces at A
and B when the airplane is traveling in a straight path. Each spring has a
constant of 600 N/m and the uniform disk has a mass of 250 g and spins at
the rate of 12,000 rpm. Determine the angle through which the yoke will
rotate when the pilot executes a horizontal turn of 800-m radius to the right at
a speed of 720 km/h. Indicate whether point A will move up or down.
x
y
G Gxyz
2
2
3
() 0 ( )
1
2
1(0.25 kg)(0.05 m) (1256.6 rad/s)( 0.250 rad/s)
2
(98.172 10 N m)
G G Gxyz G y x x y y
xxy xy
G
II
I mr
ω ωω
ωω ωω
−
= +× =+ × +
=−=−
=−−
=+ ×⋅
HH ΩH j i j
kk
k
Hk
PROBLEM 18.92 (Continued)
Free Body and kinetic diagrams
PROBLEM 18.93
The 10-oz disk shown spins at the rate
1750
ω
=
rpm, while
axle AB rotates as shown with an angular velocity
ω
2 of
6 rad/s. Determine the dynamic reactions at A and B.
PROBLEM 18.94
The 10-oz disk shown spins at the rate
1750 rpm,
ω
=
while axle AB rotates as shown with an angular velocity
ω
2. Determine the maximum allowable magnitude of
2
ω
if the dynamic reactions at A and B are not to exceed
0.25 lb each.
PROBLEM 18.95
Two disks, each of mass 5 kg and radius 300 mm, spin as shown at
the rate
11200 rpm
ω
=
about a rod AB of negligible mass which
rotates about the horizontal z axis at the rate
2
60 rpm.
ω
=
(a) Determine the dynamic reactions at points C and D. (b) Solve part
a assuming that the direction of spin of disk A is reversed.
22
DC=
PROBLEM 18.95 (Continued)
2
221
0 21
AB
mr
cC mr C c
ωω
ωω
Σ= = + = =M iH H i
(a) Data:
5 kg, 0.3 m, 0.6 m,mr c= = =
( ) ( )
12
2 1200 2 60
40 rad/s 2 rad/s
60 60
ππ
ω πω π
= = = =
( )( ) ( )( )
2
5 0.3 40 2 592 N
0.6
C
ππ
= =
( )
592 N= −Cj
( )
592 N=Dj
(b) For disk A:
12 1 2
,
A Ay z
II
ωω ω ω
=−− =− −jk H j kω
2
21
1, 0
2
A AB
mr
ωω
=− +=H i HH
0= =CD
PROBLEM 18.96
Two disks, each of mass 5 kg and radius 300 mm, spin as shown at
the rate
11200 rpm
ω
=
about a rod AB of negligible mass which
rotates about the horizontal z axis at the rate
2
.
ω
Determine the
maximum allowable value of
2
ω
if the magnitudes of the dynamic
reactions at points C and D are not to exceed 350 N each.
22
DC=
PROBLEM 18.96 (Continued)
2
221
0 21
AB
mr
cC mr C c
ωω
ωω
Σ= = + = =M iH H i
Solving for
2
,
ω
22
1
cC
mr
ωω
=
Data:
5 kg , 0.3 m, 0.6 mmr c= = =
( )
1
2 1200 40 rad/s, 350 N
60 CD
π
ωπ
= = = =
( )( )
( )( ) ( )
22
0.6 350 3.7136 rad/s
5 0.3 40
ωπ
= =
235.5 rpm
ω
=
PROBLEM 18.97
A stationary horizontal plate is attached to the ceiling by means of a
fixed vertical tube. A wheel of radius a and mass m is mounted on a light
axle AC which is attached by means of a clevis at A to a rod AB fitted
inside the vertical tube. The rod AB is made to rotate with a constant
angular velocity
Ω
causing the wheel to roll on the lower face of the
stationary plate. Determine the minimum angular velocity
Ω
for which
contact is maintained between the wheel and the plate. Consider the
particular cases (a) when the mass of the wheel is concentrated in the
rim, (b) when the wheel is equivalent to a thin disk of radius a.
PROBLEM 18.97 (Continued)
Equating moments about A:
2
()
G
RW
R
×− =
i jH
PROBLEM 18.98
PROBLEM 18.97 A stationary horizontal plate is attached to the ceiling
by means of a fixed vertical tube. A wheel of radius a and mass m is
mounted on a light axle AC which is attached by means of a clevis at A to
a rod AB fitted inside the vertical tube. The rod AB is made to rotate with
a constant angular velocity
Ω
causing the wheel to roll on the lower face
of the stationary plate. Determine the minimum angular velocity
Ω
for
which contact is maintained between the wheel and the plate. Consider
the particular cases (a) when the mass of the wheel is concentrated in the
rim, (b) when the wheel is equivalent to a thin disk of radius a.
g
and
2
2 32
32
83
ft 15.53 10 lb ft s
12
20
15.53 10 (25) 48.52
4
xx
G
I mk g
−
−
= = = × ⋅⋅
=−× =
H kk
PROBLEM 18.98 (Continued)
Taking moments about A:
()
20 ( 8 ) 48.52
12
5( 8) 48.52
3
3(48.52) 8 21.1 lb
G
R DW
D
D
×− − =
×− − =−
− +=−
i j jH
i jj k
kk
PROBLEM 18.99
A thin disk of mass
4 kgm=
rotates with an angular velocity
2
ω
with respect to arm ABC, which itself rotates with an angular
velocity
1
ω
about the y axis. Knowing that
ω
1
5 rad/s=
and
2
15 rad/s
ω
=
and that both are constant, determine the force-couple
system representing the dynamic reaction at the support at A.
PROBLEM 18.99 (Continued)
(4)( 11.25 ) (45 N)
C
m=−=−a ii
A
