PROBLEM 18.77 (Continued)
(b)
Reactions at and for the case 0.AB
ω
=
2
PROBLEM 18.78
For the sheet–metal component of Problem 18.77, determine
(a) the angular velocity of the component 0.6 s after the
couple M0 has been applied to it, (b) the magnitude of the
dynamic reactions at A and B at that time.
SOLUTION
The sheet metal component rotates about the fixed z axis with angular acceleration
2
(12 rad/s ) .=αk
(a) Angular velocity at
0.6 s.t=
0
(b) Dynamic reactions.
zz
Calculation of the required moment and products of inertia.
22
Let
r
be the mass per unit area.
2
35.556 kg/m
m
A
r
= =
The component is comprised of 3 parts: triangle , triangle , and rectangle as shown. Let the lengths of
75 mm be labeled b.
el
xz
el
yz
PROBLEM 18.78 (Continued)
Coordinates of the element mass center:
el
,xb= −
el
1,
2
yy=
el
zz=
el el el
()
xz xz
dI dI x z dm
= +
PROBLEM 18.78 (Continued)
Coordinates of the element mass center:
1,
PROBLEM 18.79
The blade of an oscillating fan and the rotor of its motor have a total
2(0.6 rad/s) .
PROBLEM 18.79 (Continued)
Data:
0.300 kg
0.075 m
0.125 m
x
m
k
b
=
=
=
32
(0.300)(75 10 ) (188.5)(0.6) 1.527 N
0.125
y
B
−
×
=−=−
PROBLEM 18.80
The blade of a portable saw and the rotor of its motor have a
total weight of 2.5 lb and a combined radius of gyration of
1.5 in. Knowing that the blade rotates as shown at the rate
ω
1
1500 rpm,=
determine the magnitude and direction of the
couple M that a worker must exert on the handle of the saw to
rotate it with a constant angular velocity
2
(2.4 rad/s) .= −ωj
SOLUTION
1
11
(2 )(1500)
60
π
ω
=
=k
ω
ω
Angular velocity:
21
ωω
= +jkω
Angular momentum of rotor:
21Gy z
II
ωω
= +H jk
Let the reference frame Gxyz be rotating with angular velocity
2
.
ω
=j
Ω
2 21
12
()
0( )
G G Gxyz G
yz
z
II
I
ω ωω
ωω
= +×
=+× +
=
HH H
j jk
i
Ω
Couple exerted on the saw:
12
2
12
G
z
z
I
mk
ωω
ωω
=
=
=
MH
i
i
Data:
2
2.5 lb
2.5
32.2
0.07764 lb s /ft
W
m
=
=
= ⋅
2
1.5 in.
0.125 ft
(0.07764)(0.125) (157.08)( 2.4)
z
k=
=
= −Mi
(0.457 lb ft)=−⋅Mi
PROBLEM 18.81
The flywheel of an automobile engine, which is rigidly attached to the crankshaft, is equivalent to a 400-mm–
diameter, 15–mm–thick steel plate. Determine the magnitude of the couple exerted by the flywheel on the
horizontal crankshaft as the automobile travels around an unbanked curve of 200-m radius at a speed of
90 km/h, with the flywheel rotating at 2700 rpm. Assume the automobile to have (a) a rear–wheel drive with
the engine mounted longitudinally, (b) a front–wheel drive with the engine mounted transversely. (Density of
steel
3
7860 kg/m .=
)
SOLUTION
Let the x axis be a horizontal axis directed along the engine mounting, i.e., longitudinally for rear-wheel drive
and transversely for front–wheel drive.
Let the y axis be vertical.
22
x
Using Equation (1),
(0.29632)(282.74)(0.125) (10.47 N m)
′= = ⋅M kk
PROBLEM 18.82
Each wheel of an automobile has a mass of 22 kg, a diameter of 575 mm, and a radius of gyration of 225 mm.
The automobile travels around an unbanked curve of radius 150 m at a speed of 95 km/h. Knowing that the
transverse distance between the wheels is 1.5 m, determine the additional normal force exerted by the ground
on each outside wheel due to the motion of the car.
PROBLEM 18.83
The uniform thin 5-lb disk spins at a constant rate
2
ω
= 6 rad/s
about an axis held by a housing attached to a horizontal rod that
rotates at the constant rate
1
ω
=
3 rad/s. Determine the couple
which represents the dynamic reaction at the support A.
1,
x
y
2.
z
2
+
ωω
1
ikω=
Angular momentum:
++
O xx yy zz
II I
ωωω
=H i jk
PROBLEM 18.84
The essential structure of a certain type of aircraft turn indicator is shown.
Each spring has a constant of 500 N/m, and the 200-g uniform disk of 40-mm
radius spins at the rate of 10,000 rpm. The springs are stretched and exert
equal vertical forces on yoke AB when the airplane is traveling in a straight
path. Determine the angle through which the yoke will rotate when the pilot
executes a horizontal turn of 750-m radius to the right at a speed of 800 km/h.
Indicate whether Point A will move up or down.
PROBLEM 18.84 (Continued)
The spring forces
A
F
and
B
F
exerted on the yoke provide the couple
.
G
M
The force exerted by spring B is
0.1 0.049646
0.49646 N.
GG
F
F
=
=
Compression of spring B:
3
0.49646
500
0.99291 10 m
0.9929 mm
B
F
k
δ
−
=
=
= ×
=
Point B moves 0.9929 mm down. Point A moves 0.9929 mm up.
0.9929 0.9929
100
+
0.019858 rad=
1.138= °θ
Point moves .A up
PROBLEM 18.85
A slender rod is bent to form a square frame of side 6 in. The
frame is attached by a collar at A to a vertical shaft which
rotates with a constant angular velocity
.ω
Determine the value
of
ω
for which line AB forms an angle
48
β
= °
with the
horizontal x axis.
5 sin (5)(0.5)sin 48
a
°
PROBLEM 18.86
A uniform square plate of side a = 225 mm is hinged at points
A and B to a clevis which rotates with a constant angular
velocity
ω
about a vertical axis. Determine (a) the constant
angle
β
that the plate forms with the horizontal x axis when
12 rad/s,
ω
=
(b) the largest value of
ω
for which the plate
remains vertical
( )
90 .
β
= °
3
PROBLEM 18.86 (Continued)
( )( )
( )( )( )
22
3 9.81
3
22 0.225 12
g
a
(b)
Set
90 , sin 1
ββ
=°=
( )( )
( )( )
2
3 9.81
365.4
2 2 0.225
g
a
ω
= = =
8.09 rad/s
ω
=
PROBLEM 18.87
A uniform square plate of side a = 300 mm is hinged at points
A and B to a clevis which rotates with a constant angular
velocity
ω
about a vertical axis. Determine (a) the value of
ω
for which the plate forms a constant angle
60
β
= °
with the
horizontal x axis, (b) the largest value of
ω
for which the plate
remains vertical
( )
90 .
β
= °
3
PROBLEM 18.87 (Continued)
( )( )
( )( ) ( )
2
2
3 9.81
356.638 rad/s
2 sin 2 0.3 sin 60
g
a
°
7.53 rad/s
ω
=
(b)
Set
90 , sin 1
ββ
=°=
( )( )
( )( ) ( )
2
2
3 9.81
349.05 rad/s
2 2 0.3
g
a
ω
= = =
7.00 rad/s
ω
=
PROBLEM 18.88
The 2–lb gear A is constrained to roll on the fixed gear B, but is free to
rotate about axle AD. Axle AD, of length 20 in. and negligible weight, is
connected by a clevis to the vertical shaft DE which rotates as shown
with a constant angular velocity
1.
ω
Assuming that gear A can be
approximated by a thin disk of radius 4 in., determine the largest
allowable value of
1
ω
if gear A is not to lose contact with gear B.
G xx yy
PROBLEM 18.88 (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
D
where
22
d Lr= +
( )
/
eff
D G GD G
m=+×M Hr a
22
11
24
C
r
2 22
1
11
sin sin cos cos
24
m rL r L
ωβ β β β
= −−
2
1
( )
2
122
22
20
32.2 12
11 20 1 4 1 4 20
cos cos sin cos30 cos30 sin 30
42 12 4 12 2 12 12
gL
L r rL
ω
βββ
= =
+− °+ °− °
23.427=
1
4.84 rad/s
ω
=
PROBLEM 18.89
Determine the force
F exerted by gear B on gear A of Prob. 18.88 when
shaft
DE rotates with the constant angular velocity
1
4=ω
rad/s. (Hint:
The force
F must be perpendicular to the line drawn from D to C.)
G xx yy