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PROBLEM 15.187
At the instant considered the radar antenna shown rotates about
the origin of coordinates with an angular velocity
xyz
ωωω
=++i jk
ω
. Knowing that
( )
15
Ay
v=
in./s,
( )
9
By
v=
in./s, and
( )
18
Bz
v=
in./s, determine (a) the angular
velocity of the antenna, (b) the velocity of point A.
12
PROBLEM 15.187 (Continued)
( ) .a Angular velocity
( ) ( ) ( )
0.600 rad/s 2.00 rad/s 0.750 rad/s = −+ij kω
From (1),
( )
10 20.0 in./s
Ay
x
v
ω
=−=
From (3),
( )
12 24.0 in./s
Ay
z
v
ω
=−=
( ) . b Velocity of point A
( ) ( ) ( )
20.0 in./s 15.00 in./s 24.0 in./s
A
v=++i jk
PROBLEM 15.188
The rotor of an electric motor rotates at the constant rate
1
1800 rpm.
ω
=
Determine the angular acceleration of the rotor as the
motor is rotated about the y axis with a constant angular velocity
2
ω
of 6 rpm counterclockwise when viewed from the positive y axis.
SOLUTION
1
1800 rpm
60 rad/s
ω
π
=
=
PROBLEM 15.189
The disk of a portable sander rotates at the
constant rate
1
4400 rpm
ω
=
as shown.
Determine the angular acceleration of the disk as
a worker rotates the sander about the z axis with
an angular velocity of 0.5 rad/s and an angular
acceleration of 2.5 rad/s2, both clockwise when
viewed from the positive z axis.
SOLUTION
22
(230 rad/s ) (2.5 rad/s )= −ikα
PROBLEM 15.190
A flight simulator is used to train pilots
on how to recognize spatial
disorientation. It has four degrees of
freedom, and can rotate around a
planetary axis as well as in yaw, pitch,
and roll. Knowing that the simulator is
rotating around the planetary axis with
a constant angular velocity of 20 rpm
counterclockwise as seen from above,
determine the angular acceleration of
the cab if (a) the cab has a constant
pitch angular velocity of +3 k rad/s,
(b) the cab has a constant roll angular
velocity of −4 i rad/s.
SOLUTION
AB
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 15.191
In the system shown, disk A is free to rotate about the horizontal rod
OA. Assuming that disk B is stationary
2
( 0),
ω
=
and that shaft OC
rotates with a constant angular velocity
1
,
ω
determine (a) the angular
velocity of disk A, (b) the angular acceleration of disk A.
PROBLEM 15.192
In the system shown, disk A is free to rotate about the horizontal rod
OA. Assuming that shaft OC and disk B rotate with constant angular
velocities
1
ω
and
2
,
ω
respectively, both counterclockwise, determine
(a) the angular velocity of disk A, (b) the angular acceleration of disk A.
dt r
11 2
A
r
PROBLEM 15.193
The L-shaped arm BCD rotates about the z axis with a constant
angular velocity
1
ω
of 5 rad/s. Knowing that the 150-mm-
radius disk rotates about BC with a constant angular velocity
2
ω
of 4 rad/s, determine (a) the velocity of Point A, (b) the
acceleration of Point A.
SOLUTION
.Total angular velocity
21
(4 rad/s) (5 rad/s)
ωω
= +
= +
jk
jk
ω
ω
.Angular acceleration
Frame Oxyz is rotating with angular velocity
1
.
ω
=Ωk
1 21
12
2
0( )
(5)(4)
20
(20.0 rad/s )
Oxyz
α
ω ωω
ωω
α
=
= +×
=+× +
= −
= −
= −
= −
Ω
k jk
i
i
i
i
ω
ωω
α
(a) Velocity of Point A.
(0.15 m) (0.12 m)
0 45
0.15 0.12 0
0.6 0.75 0.6
A
AA
= +
×
=
=−+ −
r ij
vr
i jk
i jk
=ω
(0.600 m/s) (0.750 m/s) (0.600 m/s)
A
=−+ −v i jk
(b) Acceleration of Point A.
20 0 0 0 4 5
0.15 0.6 0 0.6 0.75 0.6
2.4 6.15 3 2.4
AA A
=× +×
=−+
−−
=− − −+
aαr v
i jk i j k
k ij k
ω
6.15 3=−−ij
22
(6.15 m/s ) (3.00 m/s )
A
=−−a ij
PROBLEM 15.194
A gun barrel of length
4 mOP =
is mounted on a turret as
shown. To keep the gun aimed at a moving target, the azimuth
angle
β
is being increased at the rate
/ 30 /sd dt
β
= °
and the
elevation angle
γ
is being increased at the rate
/ 10 /s.
d dt
γ
= °
For the position
β
90= °
and
30 ,
γ
= °
determine (a) the
angular velocity of the barrel, (b) the angular acceleration of
the barrel, (c) the velocity and acceleration of Point P.
SOLUTION
30 rad/s rad/s
d
βπ π
P
PROBLEM 15.194 (Continued)
2
PP P
α
π ππ
=× +×
arv
i jk
ω
P=−−a i jk
PROBLEM 15.195
A 3-in.-radius disk spins at the constant rate
2
4
ω
=
rad/s
about an axis held by a housing attached to a horizontal rod
that rotates at the constant rate
15
ω
=
rad/s. For the position
shown, determine (a) the angular acceleration of the disk,
(b) the acceleration of Point P on the rim of the disk if
θ
= 0,
(c) the acceleration of Point P on the rim of the disk
if
90 .
θ
= °
SOLUTION
P
PROBLEM 15.195 (Continued)
(c)
θ
90 .= °
Acceleration at Point P.
(0.25 ft)
(5 4 ) 0.25
(1.25 ft/s) (1 ft/s)
20 0.25 (5 4 ) ( 1.25 )
0 0 6.25 4 0
P
PP
PP P
=
= ×
=+×
=−+
=× +×
=− × + + ×− +
=+− − +
rj
vr
ik j
ij
aαr v
j j i k ij
jj
ω
ω
2
P= −aj
PROBLEM 15.196
A 3-in.-radius disk spins at the constant rate
2
4
ω
=
rad/s
about an axis held by a housing attached to a horizontal rod
that rotates at the constant rate
15
ω
=
rad/s. Knowing that
30 ,
θ
= °
determine the acceleration of Point P on the rim of
the disk.
SOLUTION
P
PROBLEM 15.197
The cone shown rolls on the zx plane with its apex at the origin of
coordinates. Denoting by
ω
1 the constant angular velocity of the axis OB
of the cone about the y axis, determine (a) the rate of spin of the cone
about the axis OB, (b) the total angular velocity of the cone, (c) the
angular acceleration of the cone.
SOLUTION
tan
β
PROBLEM 15.198
At the instant shown, the robotic arm ABC is
being rotated simultaneously at the constant rate
10.15
ω
=
rad/s about the y axis, and at the
constant rate
20.25
ω
=
rad/s about the z axis.
Knowing that the length of arm ABC is 1 m,
determine (a) the angular acceleration of the arm,
(b) the velocity of Point C, (c) the acceleration of
Point C.
SOLUTION
C
PROBLEM 15.199
In the planetary gear system shown, gears A and B are
rigidly connected to each other and rotate as a unit about
the inclined shaft. Gears C and D rotate with constant
angular velocities of 30 rad/s and 20 rad/s, respectively
(both counterclockwise when viewed from the right).
Choosing the x axis to the right, the y axis upward, and the
z axis pointing out of the plane of the figure, determine
(a) the common angular velocity of gears A and B, (b) the
angular velocity of shaft FH, which is rigidly attached to
the inclined shaft.
SOLUTION
xy
PROBLEM 15.199 (Continued)
FH
FH
PROBLEM 15.200
In Problem 15.199, determine (a) the common angular acceleration of gears A and B, (b) the acceleration of
the tooth of gear A which is in contact with gear C at Point I.
SOLUTION
See the solution to part (a) of Problem 15.199 for the calculation of the common angular velocity of unit AB.
1
PROBLEM 15.201
Several rods are brazed together to form the robotic guide arm shown
which is attached to a ball-and-socket joint at O. Rod OA slides in a
straight inclined slot while rod OB slides in a slot parallel to the z axis.
Knowing that at the instant shown
(9 in./s) ,
B
=vk
determine (a) the
angular velocity of the guide arm, (b) the velocity of Point A, (c) the
velocity of Point C.
PROBLEM 15.201 (Continued)
C
