Problem 17.42 The upper grip and jaw of the pli-
ers ABC is stationary. The lower grip DEF is rotating
at 0.2 rad/s in the clockwise direction. At the instant
shown, what is the angular velocity of the lower jaw
CFG?G
70 mm
B
A
D
C
30 mm
30 mm
EF
30 mm
Stationary
Solution:
vE=vB+ωBE ×rE/B =0+ωBEk×(0.07i−0.03j)m
Problem 17.43 The horizontal member ADE support-
ing the scoop is stationary. If the link BD is rotating in
the clockwise direction at 1 rad/s, what is the angular
velocity of the scoop? 1 ft 6 in
2 ft 6 in
1 ft
5 ft
2 ft
C
B
DE
A
Scoop
Problem 17.44 The diameter of the disk is 1 m, and
the length of the bar AB is 1 m. The disk is rolling, and
point Bslides on the plane surface. Determine the angu-
lar velocity of the bar AB and the velocity of point B.
4 rad/s
A
B
Solution: Choose a coordinate system with the origin at O, the
354
Problem 17.45 A motor rotates the circular disk
mounted at A, moving the saw back and forth. (The saw
is supported by a horizontal slot so that point Cmoves
horizontally). The radius AB is 4 in, and the link BC is
14 in long. In the position shown, θ=45◦and the link
BC is horizontal. If the angular velocity of the disk is
one revolution per second counterclockwise, what is the
velocity of the saw?
x
y
B
A
C
θ
The angular velocity of Bis
vB=vA+ωAB ×rB/A,
vB=0+2π(2√2)
ijk
001
110
=4π√2(−i+j)(in/s).
The radius vector from Bto Cis rC/B =(4 cos 45◦−14)i.
The velocity of point Cis
vC=vB+ωBC ×rC/B =vB
ijk
00ωBC
2√2−14 0 0
=−
√24πi+((2√2−14)ωBC +1)j.
Problem 17.46 In Problem 17.45, if the angular
velocity of the disk is one revolution per second counter-
clockwise and θ=270◦, what is the velocity of the saw?
Problem 17.47 The disks roll on a plane surface.
The angular velocity of the left disk is 2 rad/s in the
clockwise direction. What is the angular velocity of the
right disk?
2 rad/s
3 ft
356
Problem 17.48 The disk rolls on the curved surface.
The bar rotates at 10 rad/s in the counterclockwise
direction. Determine the velocity of point A.
10 rad/s 40 mm
A
x
y
Solution: The radius vector from the left point of attachment of
vO=ωO×rO/P =
ijk
00ωO
−40 0 0
=−40ωOj.
Equate the two expressions for the velocity of the center of the disk
and solve: ωO=−30 rad/s. The radius vector from the center of the
disk to point Ais rA/O =40j(mm). The velocity of point Ais
vA=vO+ωO×rA/O =1200j−(30)(40)
ijk
001
010
=1200i+1200j(mm/s)
x
P
O
Problem 17.49 If ωAB =2 rad/s and ωBC =4 rad/s,
what is the velocity of point C, where the excavator’s
bucket is attached?
y
B
C
5 m
5.5 m
A
BC
AB
v
v
Problem 17.50 In Problem 17.49, if ωAB =2 rad/s,
what clockwise angular velocity ωBC will cause the
vertical component of the velocity of point Cto be zero?
What is the resulting velocity of point C?
y
B
C
5.5 m
BC
AB
v
v
Solution: Use the solution to Problem 17.49. The velocity of
2.3=2.61 rad/s clockwise.
Problem 17.51 The steering linkage of a car is shown.
Member DE rotates about fixed pin E. The right brake
disk is rigidly attached to member DE. The tie rod CD is
pinned at Cand D. At the instant shown, the Pitman arm
AB has a counterclockwise angular velocity of 1 rad/s.
What is the angular velocity of the right brake disk?
180 mm 220 mm
100 mm
460
mm
340
mm
70
mm
200
mm
Steering link
Brake disks
B
A
CD
E
358
Problem 17.52 An athlete exercises his arm by raising
the mass m. The shoulder joint Ais stationary. The
distance AB is 300 mm, and the distance BC is 400 mm.
At the instant shown, ωAB =1 rad/s and ωBC =2 rad/s.
How fast is the mass mrising?
A
B
m
C
ω
AB
ω
BC 30°
60°
Problem 17.53 The distance AB is 12 in., the distance
BC is 16 in., ωAB =0.6 rad/s, and the mass mis rising
at 24 in./s. What is the angular velocity ωBC?
Problem 17.54 Points Band Care in the x–yplane.
The angular velocity vectors of the arms AB and BC are
ωAB =−0.2k(rad/s), and ωBC =0.4k(rad/s). What is
the velocity of point C.
A
y
x
z
30°
40°
B
C
920 mm
760 mm
Solution: Locations of Points:
360
Problem 17.55 If the velocity at point Cof the
robotic arm shown in Problem 17.54 is vC=0.15i+
0.42j(m/s), what are the angular velocities of the arms
AB and BC?
Problem 17.56 The link AB of the robot’s arm is rotat-
ing at 2 rad/s in the counterclockwise direction, the link
BC is rotating at 3 rad/s in the clockwise direction, and
the link CD is rotating at 4 rad/s in the counterclockwise
direction. What is the velocity of point D?
y
20°
250 mm
300 mm
0.3 cos 30◦0.3 sin 30◦0
=−0.3i+0.520j(m/s).
0.25 cos 20◦−0.25 sin 20◦0
Problem 17.57 The person squeezes the grips of the
shears, causing the angular velocities shown. What is
the resulting angular velocity of the jaw BD?
0.12 rad/s
D
25 mm
18 mm
25 mm
Solution:
vD=vC+ωCD ×rD/C =0−(0.12 rad/s)k×(0.025i+0.018j)m
=(0.00216i−0.003j)m/s
vB=vD+ωBD ×rB/D
=(0.00216i−0.003j)m/s +ωBDk×(−0.05i−0.018j)m
Problem 17.58 Determine the velocity vWand the
angular velocity of the small pulley. 50 mm
0.6 m/s
Solution: Since the radius of the bottom pulley is not given, we
cannot use Eq (17.6) (or the equivalent). The strategy is to use the fact
r=0.2
0.05 =4 rad/s
362
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Problem 17.59 Determine the velocity of the block
and the angular velocity of the small pulley. 9 in/s
2 in
3 in
Solution: Denote the velocity of the block by vB. The strategy
Problem 17.60 The device shown is used in the semi-
conductor industry to polish silicon wafers. The wafers
are placed on the faces of the carriers. The outer and
inner rings are then rotated, causing the wafers to move
and rotate against an abrasive surface. If the outer ring
rotates in the clockwise direction at 7 rpm and the inner
ring rotates in the counterclockwise direction at 12 rpm,
what is the angular velocity of the carriers?
0.6 m
Inner ring
1.0 m
Carriers (3)
Outer ring
12 rpm
7 rpm
0.4
=−35.5 rpm.
Problem 17.61 In Problem 17.60, suppose that the
outer ring rotates in the clockwise direction at 5 rpm
and you want the centerpoints of the carriers to remain
stationary during the polishing process. What is the
necessary angular velocity of the inner ring? 0.6 m
Inner ring
1.0 m
Carriers (3)
Outer ring
12 rpm
7 rpm
364
Problem 17.62 The ring gear is fixed and the hub
and planet gears are bonded together. The connecting
rod rotates in the counterclockwise direction at 60 rpm.
Determine the angular velocity of the sun gear and the
magnitude of the velocity of point A.
A
240 mm
720 mm
340
mm
Planet gear
Connecting
rod
Hub gear
140
mm
Solution: Denote the centers of the sun, hub and planet gears by
Problem 17.63 The large gear is fixed. Bar AB has a
counterclockwise angular velocity of 2 rad/s. What are
the angular velocities of bars CD and DE?
4 in
2 rad/s
16 in
A
BCD
E
10 in
4 in
10 in
Solution: The strategy is to express vector velocity of point Din
366
Problem 17.64 If the bar has a clockwise angular
velocity of 10 rad/s and vA=20 m/s, what are the
coordinates of its instantaneous center of the bar, and
what is the value of vB?
y
x
AB
vv
−10 0
Problem 17.65 In Problem 17.64, if vA=24 m/s
and vB=36 m/s, what are the coordinates of the
instantaneous center of the bar, and what is its angular
velocity?
y
x
AB
vv
Problem 17.66 The velocity of point Oof the bat is
vO=−6i−14j(ft/s), and the bat rotates about the z
axis with a counterclockwise angular velocity of 4 rad/s.
What are the xand ycoordinates of the bat’s instanta-
neous center?
y
x
O
368
Problem 17.67 Points Aand Bof the 1-m bar slide on
the plane surfaces. The velocity of Bis vB=2i(m/s).
(a) What are the coordinates of the instantaneous cen-
ter of the bar?
(b) Use the instantaneous center to determine the
velocity at A.
x
y
A
B
G
70°
Problem 17.68 The bar is in two-dimensional motion
in the x–yplane. The velocity of point Ais vA=
8i(ft/s), and Bis moving in the direction parallel to the
bar. Determine the velocity of B(a) by using Eq. (17.6)
and (b) by using the instantaneous center of the bar.
x
y
B
A
4 ft
30°
Solution:
The velocity of point Ais
370
Problem 17.69 Point Aof the bar is moving at 8 m/s
in the direction of the unit vector 0.966i−0.259j, and
point Bis moving in the direction of the unit vector
0.766i+0.643j.
(a) What are the coordinates of the bar’s instanta-
neous center?
(b) What is the bar’s angular velocity?
x
y
B
A
2 m
30⬚
Problem 17.70 Bar AB rotates with a counterclock-
wise angular velocity of 10 rad/s. At the instant shown,
what are the angular velocities of bars BC and CD? (See
Active Example 17.4.)
A
10 rad/s
D
BC
2 m 1 m
2 m
Solution: The location of the instantaneous center for BC is shown,
Problem 17.71 Use instantaneous centers to determine
the horizontal velocity of B.
A
1 rad/s
Solution: The instantaneous center of OA lies at O,bydefinition,
since Ois the point of zero velocity, and the velocity at point Ais
372