PROBLEM 15.227 (Continued)
D=+−a ij k
PROBLEM 15.228
Manufactured items are spray–painted as
they pass through the automated work station
shown. Knowing that the bent pipe ACE
rotates at the constant rate
10.4 rad/s
ω
=
and
that at Point D the paint moves through the
150 mm/s,u=
determine, for the position
shown, (a) the velocity of the paint at D,
(b) the acceleration of the paint at D.
SOLUTION
Use a frame of reference CE rotating about the x–axis with angular velocity
D=−+a i jk
PROBLEM 15.229
Solve Problem 15.227, assuming that at the instant shown the angular
velocity
ω
1 of the plate is 10 rad/s and is decreasing at the rate of 25 rad/s2,
while the relative speed u of Point D of strap CD is 1.5 m/s and is decreasing
at the rate of 3 m/s2.
PROBLEM 15.227 The circular plate shown rotates about its vertical
diameter at the constant rate
110 rad/s.
ω
=
Knowing that in the position
shown the disk lies in the XY plane and Point D of strap CD moves upward at
a constant relative speed u
1.5 m/s,=
determine (a) the velocity of D, (b) the
acceleration of D.
SOLUTION
PROBLEM 15.229 (Continued)
D
PROBLEM 15.230
Solve Prob. 15.225, assuming that at the instant shown the
angular velocity
1
ω
of the rod is 5 rad/s and is increasing at
the rate of 10
2
rad/s ,
while the relative speed u of the collar C
is 39 in./s and is decreasing at the rate of 260
2
in./s .
C
PROBLEM 15.231
Using the method of Section 15.7A, solve Problem 15.192.
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 A, (b) the
angular acceleration of disk A.
2 12 xx
PROBLEM 15.231 (Continued)
PROBLEM 15.232
Using the method of Section 15.7A, solve Problem 15.196.
PROBLEM 15.196 A 3–in.–radius disk spins at the constant rate
ω
2
4=
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
Let frame
Oxyz
rotate with angular velocity
PROBLEM 15.233
Using the method of Section 15.7A, solve
Problem 15.198.
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
2
0.25
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
Geometry: Dimensions in meters.
PROBLEM 15.233 (Continued)
PROBLEM 15.234
The 400-mm bar AB is made to rotate at the constant rate
2
ω
=
/d dt
θ
= 8 rad/s
with respect to the frame CD which itself rotates at the constant rate
1
ω
= 12 rad/s about the Y axis. Knowing that
θ
=
60°
at the instant shown,
determine the velocity and acceleration of point A.
/
AF
PROBLEM 15.234 (Continued)
Acceleration of point A:
//
2
A A AF AF
′
=+ +×aaa vΩ
( )
( ) ( )
22 2
20.8 m/s 6.4 3 m/s 19.2 3 m/s
A
=−− +a ij k
( ) ( ) ( )
2 22
20.8 m/s 11.09 m/s 33.3 m/s
A=−− +a i jk
PROBLEM 15.235
The 400-mm bar AB is made to rotate at the rate
2
ω
=
/d dt
θ
with respect
to the frame CD which itself rotates at the rate
1
ω
about the Y axis. At the
instant shown
1
ω
= 12 rad/s,
1/d dt
ω
=
−
16
2
rad/s ,
2
ω
= 8 rad/s,
2
/d dt
ω
= 10
2
rad/s ,
and
θ
=
60 .°
Determine the velocity and
acceleration of point A at this instant.
A=−+ −v i jk
PROBLEM 15.235 (Continued)
Coriolis acceleration:
( )( )
( ) ( )
2
/
2 2 12 0.8 3 0.8 19.2 3 m/s
AF
× = ×− + =v j ij kΩ
Acceleration of point A:
//
2
A A AF AF
′
=+ +×aaa vΩ
( ) ( ) ( )
20.8 3 6.4 3 1.0 19.2 3 1.2
A
=−+− −+ +a ij k
( ) ( ) ( )
2 22
22.5 m/s 10.09 m/s 34.9 m/s
A=−− +a i jk
PROBLEM 15.236
The arm AB of length 16 ft is used to provide an elevated
platform for construction workers. In the position shown, arm
AB is being raised at the constant rate
/ 0.25d dt
θ
=
rad/s;
simultaneously, the unit is being rotated about the Y axis at the
constant rate
ω
1 = 0.15 rad/s. Knowing that
θ
= 20°, determine
the velocity and acceleration of Point B.
SOLUTION
Frame of reference. Let moving frame Axyz rotate about the Y axis with angular velocity
PROBLEM 15.236 (Continued)
Motion of Point B relative to the frame.
2
/ 2/
(0.25 rad/s)
BF BA
d
dt
θ
= −
= −
= ×
k
k
vr
ω
ω
B=−−a i jk
PROBLEM 15.237
The remote manipulator system (RMS) shown is used
to deploy payloads from the cargo bay of space
shuttles. At the instant shown, the whole RMS is
rotating at the constant rate
10.03 rad/s
ω
=
about the
axis AB. At the same time, portion BCD rotates as a
rigid body at the constant rate
2/ 0.04 rad/sd dt
ωβ
= =
about an axis through B
parallel to the X axis. Knowing that
30 ,
β
= °
determine (a) the angular acceleration of BCD, (b) the
velocity of D, (c) the acceleration of D.
SOLUTION
At the instant given, Points A, B, C, and D lie in a plane which is parallel to the YZ plane. The plane ABCD is
rotating with angular velocity.
2
(0.03 ) ( 0.168875 )
(0.0050662 m/s )
DD
′′
= × = ×−
=
aΩv j i
k
PROBLEM 15.237 (Continued)
Motion of D relative to the frame: This motion is a rotation about B with angular velocity.
2
/frame 2 /
(0.04 rad/s)
(0.04 ) (5.75 5.6292 )
D DB
=
= ×
= ×−
i
vr
ij k
ω
ω
D
PROBLEM 15.238
A disk of radius 120 mm rotates at the constant rate
2
5
ω
=
rad/s
with respect to the arm AB, which itself rotates at the constant
rate
13
ω
=
rad/s. For the position shown, determine the velocity
and acceleration of Point C.
SOLUTION
C=− −+a ii
C= −ai
PROBLEM 15.239
The crane shown rotates at the constant rate
1
0.25
ω
=
rad/s;
simultaneously, the telescoping boom is being lowered at the
constant rate
20.40
ω
=
rad/s. Knowing that at the instant
shown the length of the boom is 20 ft and is increasing at the
constant rate
1.5 ft/s,u=
determine the velocity and
acceleration of Point B.
SOLUTION
/
(1.5 ft/s)sin 30 (1.5 ft/s) cos30
0
BF
= °+ °
=
jk
a