298
CHAPTER 16
Planar Kinematics of Rigid Bodies
Problems
4 m/s
16.19 At a certain instant, the velocity of end A of the bar AB is 4 m/s in the
direction shown. Knowing that the magnitude of the velocity of end B is 3 m/s,
determine the angular velocity of bar AB.
60°
A
B
1m
16.20 The wheel rolls without slipping. In the position shown, the vertical component of the velocity of point B is 4 m/s directed upward. For this position,
calculate the angular velocity of the wheel and the velocity of its center C.
Fig. P16.19
16.21 The disk rolls without slipping with the constant angular velocity ω.
For the position shown, find the angular velocity of link AB and the velocity of
slider A.
2m
B C
0.4
0.18 m
ω
R
B
C
Fig. P16.20
R
45°
A
Fig. P16.21
16.22 The pinion gear meshes with the two racks. If the racks are moving with
the velocities shown, determine the angular velocity of the gear and the velocity
of its center C.
0.6 m/s
90 mm
B
C
R
θ
150 mm
O
vO
0.8 m/s
B
ω0
rB
rA
A
Fig. P16.24, P16.25
Fig. P16.22
Fig. P16.23
16.23 The wheel rolls without slipping to the right with constant angular velocity. The velocity of the center of the wheel is v O . Determine the speed of point B
on the rim as a function of its angular position θ .
16.24 The arm joining the two friction wheels rotates with the constant angular
velocity ω0 . Assuming that wheel A is stationary and that there is no slipping
between the wheels, determine the angular velocity of wheel B.
16.19–16.39
299
Problems
16.25 Solve Prob. 16.24 if wheel A is rotating clockwise with the angular
velocity ω A = 2ω0 .
16.26 Gear A of the planetary gear train is rotating clockwise at ω A = 8 rad/s.
Calculate the angular velocities of gear B and the arm AB. Note that the outermost
gear C is stationary.
ωA
24
A
30
24
A
B
θ
L
C
Dimensions in mm
B
Fig. P16.26
Fig. P16.27
16.27 The bar AB is rotating counterclockwise with the constant angular
speed ω0 . (a) Find the velocities of ends A and B as functions of θ. (b) Differentiate the results of part (a) to determine the accelerations of A and B in terms
of θ .
D
16.28 End A of bar AD is pushed to the right with the constant velocity
v A = 0.6 m/s. Determine the angular velocity of AD as a function of θ .
B
16.29 The angular speed of link AB in the position shown is 2.8 rad/s clockwise.
1m
Compute the angular speeds of links BC and CD in this position.
vA
60
θ
A
C
D
Fig. P16.28
Dimensions in mm
D 0.12 m
0.16 m
60
0.30 m
6 rad/s
30
A
B
2.8 rad/s
Fig. P16.29
B
0.12 m
A
E
Fig. P16.30
16.30 The link AB of the mechanism rotates with the constant angular speed of
6 rad/s counterclockwise. Calculate the angular velocities of links BD and DE in
the position shown.
300
CHAPTER 16
Planar Kinematics of Rigid Bodies
375
16.31 When the mechanism is in the position shown, the velocity of slider D
150
D
is v D = 1.25 m/s. Determine the angular velocities of bars AB and BD at this
instant.
vD
B
Dimensions in mm
16.32 When the linkage is in the position shown, bar AB is rotating
225
counterclockwise at 16 rad/s. Determine the velocity of the sliding collar C in
this position.
B
A
125
4m
0.2
Fig. P16.31
30°
6m
0.3
16 rad/s
A
30°
20°
0.5 m
A
C
0.40 m
B
0.5 m
C
Fig. P16.32
Fig. P16.33
16.33 At the instant shown, end A of the bar ABC has a downward velocity of
2 m/s. Find the angular velocity of the bar and the speed of end C at this instant.
0.25 m
B
D
16.34 Bar AB is rotating counterclockwise with the constant angular velocity
ω0 = 30 rad/s. Find the angular velocities of bars BD and DE in the position
shown.
30°
0.2
A
8m
4m
0.1
ω0
16.35 The wheel is rolling without slipping. Its center has a constant velocity
of 0.6 m/s to the left. Compute the angular velocity of bar BD and the velocity of
end D when θ = 0.
E
B
Fig. P16.34
m
0.1
8m
6
0.
0.2
m
B
θ
0.6 m/s A
D
D
Fig. P16.35
E
A
θ
0.36 m
Fig. P16.36
16.36 Crank AB rotates with a constant counterclockwise angular velocity of
16 rad/s. Calculate the angular velocity of bar BE when θ = 60◦ .
16.6
16.37 The hydraulic cylinder raises pin B at the constant rate of 30 mm/s.
Determine the speed of end D of the bar AD at the instant shown.
B
80
mm
D
m
0m
16
30°
80 mm
A
Hydraulic
cylinder
Fig. P16.37
16.38 In the position shown, the speeds of corners A and B of the right triangular plate are v A = 3 m/s and v B = 2.4 m/s, directed as shown. Find (a) the
angle α; and (b) the speed of corner D.
0.60 m
B
420
vB
α
0.50 m
mm
A
mm
0m
B
640
20°
D
35°
0.3
D
ω0
E
A
vA
Fig. P16.38
Fig. P16.39
16.39 Bar DE is rotating counterclockwise with the constant angular velocity
ω0 = 5 rad/s. Find the angular velocities of bars AB and BD in the position shown.
16.6
Instant Center for Velocities
The instant center for velocities of a body undergoing plane motion is defined to
be the point that has zero velocity at the instant under consideration.* This point
may be either in a body or outside the body (in the “body extended”). It is often
convenient to use the instant center of the body in computing the velocities of
points in the body.
* Three
“centers” are sometimes used in the kinematic analysis of plane motion: the instant center of
rotation for virtual motion (see Art. 10.6), the instant center for velocities, and the instant center for
accelerations. Each of these points is called simply the instant center when it is clear from the context
which center is being used. The discussion of instant center for velocities presented here parallels the
discussion of instant center of rotation for virtual motion in Art. 10.6.
Instant Center for Velocities
301