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Problem 17.166 In Problem 17.165, bar AB rotates
with a constant angular velocity of 6 rad/s in the
counterclockwise direction. Determine the acceleration
of point D.
y
D
C
8 in
12 in
angular acceleration of point Cis
Problem 17.167 Point Cis moving to the right at
20 in./s. What is the velocity of the midpoint Gof bar
BC?
A
B
4 in
3 in
y
G
Solution:
Problem 17.168 In Problem 17.167, point Cis mov-
ing to the right with a constant velocity of 20 in/s. What
is the acceleration of the midpoint Gof bar BC?
B
4 in
3 in
y
G
Solution: See the solution of Problem 17.167.
aB=aA+αAB ×rB/A −ω2
AB rB/A
452
Problem 17.169 In Problem 17.167, if the velocity of
point Cis vC=1.0i(in./s), what are the angular veloc-
ity vectors of arms AB and BC?
B
4 in
3 in
y
G
Solution: Use the solution to Problem 17.167: The velocity of the
point Bis determined from the known velocity of point Cand the
The angular velocity of bar AB is determined from the velocity of B.
ij k
1.0−7ωBC =−4ωAB ,−10ωBC =4ωAB .
Problem 17.170 Points Band Care in the x–y
plane. The angular velocity vectors of arms AB and
BC are ωAB =−0.5k(rad/s) and ωBC =−2.0k(rad/s).
Determine the velocity of point C.
760 mm
y
z
B
C
Solution: The vector
rB/A =760(icos 15◦−jsin 15◦)=734.1i−196.7j(mm).
rC/B =900(icos 50◦+jsin 50◦)=578.5i+689.4j(mm).
The velocity of point Bis
Problem 17.171 In Problem 17.170, if the velocity
vector of point Cis vC=1.0i(m/s), what are the
angular velocity vectors of arms AB and BC?
760 mm
y
C
Solution: Use the solution to Problem 17.170.
The vector
rB/A =760(icos 15◦−jsin 15◦)=734.1i−196.7j(mm).
578.5 687.40
1000i=196.7ωAB i+734.1ωAB j−687.4ωBCi+578.5ωBCj(mm/s).
Problem 17.172 The angular velocity vectors of arms
AB and BC are ωAB =−0.5k(rad/s) and ωBC =
2.0k(rad/s), and their angular accelerations are αAB =
1.0k(rad/s2), and αBC =1.0k(rad/s2). What is the
acceleration of point C?
A15⬚
760 mm
900 mm
y
x
z
B
50⬚
C
Solution: Use the solution to Problem 17.170.
454
Problem 17.173 The velocity of point Cis vC=
1.0i(m/s) and aC=0. What are the angular velocity
and angular acceleration vectors of arm BC?
A15⬚
760 mm
900 mm
y
x
z
B
50⬚
C
578.5 689.40
1000i=196.7ωAB i+734.1ωAB j−689.4ωBCi
734.1−196.70
−(ω2
AB )(734.1i−196.7j)(mm/s)2.
aB=196.7αAB i+734.1αAB j−639.0i+171.3j(mm/s2)
Solve: αAB =2.24 rad/s2,αBC =−1.465 (rad/s2)
Problem 17.174 The crank AB has a constant
clockwise angular velocity of 200 rpm. What are the
velocity and acceleration of the piston P?
6 in2 in
2 in A
B
P
Solution:
456
Problem 17.175 Bar AB has a counterclockwise
angular velocity of 10 rad/s and a clockwise angular
acceleration of 20 rad/s2. Determine the angular
acceleration of bar BC and the acceleration of point C.
y
6 in
8 in
10 rad/s
AB
Solution: Choose a coordinate system with the origin at the left
The velocity of point Bis
vB=ωBC ×rB/IC =
ij k
00ωBC
=−6ωBCj=80j,
80
6−40
The acceleration of point Cis constrained to be parallel to the xaxis.
Separate components:
Problem 17.176 The angular velocity of arm AC is
1 rad/s counterclockwise. What is the angular velocity
of the scoop?
D
C
1 m
Problem 17.177 The angular velocity of arm AC in
Problem 17.176 is 2 rad/s counterclockwise, and its
angular acceleration is 4 rad/s2clockwise. What is the
angular acceleration of the scoop?
D
C
1 m
Problem 17.178 If you want to program the robot so
that, at the instant shown, the velocity of point Dis
vD=0.2i+0.8j(m/s) and the angular velocity of arm
CD is 0.3 rad/s counterclockwise, what are the neces-
sary angular velocities of arms AB and BC?
x
D
C
y
B
30°
20°
A
250 mm
300 mm
250 mm
Solution: The position vectors are:
460
Problem 17.179 The ring gear is stationary, and the
sun gear rotates at 120 rpm (revolutions per minute) in
the counterclockwise direction. Determine the angular
velocity of the planet gears and the magnitude of the
velocity of their centerpoints.
20 in
Sun gear
34 in
Planet gears (3)
Ring gear
7 in
from which ωS=4πk(rad/s). At the point of contact between the sun
The velocity of the centerpoint of the top most planet gear is
S
Problem 17.180 Arm AB is rotating at 10 rad/s in the
clockwise direction. Determine the angular velocity of
the arm BC and the velocity at which the arm slides
relative to the sleeve at C.
30⬚
A
B
C
2 m
1.8 m
1.56 0.90
The unit vector from Bto Cis
462
Problem 17.181 In Problem 17.180, arm AB is rotat-
ing with an angular velocity of 10 rad/s and an angular
acceleration of 20 rad/s2, both in the clockwise direction.
Determine the angular acceleration of arm BC.
30⬚
A
B
C
2 m
1.8 m
Solution: Use the solution to 17.180. The vector
Collecting terms,
Problem 17.182 Arm AB is rotating with a constant
counterclockwise angular velocity of 10 rad/s. Deter-
mine the vertical velocity and acceleration of the rack R
of the rack-and-pinion gear.
B
C
12 in
D
6 in
10 in
Solution: The vectors:
464
Problem 17.183 The rack Rof the rack-and-pinion
gear is moving upward with a constant velocity of
10 ft/s. What are the angular velocity and angular
acceleration of bar BC ?
B
C
12 in
6 in
10 in
Problem 17.184 Bar AB has a constant counterclock-
wise angular velocity of 2 rad/s. The 1-kg collar C slides
on the smooth horizontal bar. At the instant shown, what
is the tension in the cable BC?
A
B
C
1 m 2 m
2 m
Solution:
ijk
466
Problem 17.185 An athlete exercises his arm by rais-
ing the 8-kg mass m. The shoulder joint A is stationary.
The distance AB is 300 mm, the distance BC is 400 mm,
and the distance from Cto the pulley is 340 mm. The
angular velocities ωAB =1.5 rad/s and ωBC =2 rad/s
are constant. What is the tension in the cable?
A
B
m
C
ω
AB
ω
BC 30°
60°
Solution:
y
e
Problem 17.186 The hydraulic actuator BC of the
crane is extending (increasing in length) at a constant
rate of 0.2 m/s. When the angle β=35o, what is the
angular velocity of the crane’s boom AD ?
3m 2m
2 m
AB
C
D
b
Solution: Using AD we have
Problem 17.187 The coordinate system shown is fixed
relative to the ship B. The ship uses its radar to measure
the position of a stationary buoy A and determines it to
be 400i+200j(m). The ship also measures the velocity
of the buoy relative to its body-fixed coordinate sys-
tem and determines it to be 2i−8j(m/s). What are the
ship’s velocity and angular velocity relative to the earth?
(Assume that the ship’s velocity is in the direction of the
yaxis).
y
x
A
B
Solution:
468
