16–78.
If the ring gear A rotates clockwise with an angular velocity
of vA
=30 rad>s
, while link BC rotates clockwise with an
angular velocity of vBC
=15 rad>s
, determine the angular
velocity of gear D.
SOLUTION
Rotation About A Fixed Axis. The magnitudes of the velocity of Point E on the rim
and center C of gear D are
vE=
v
ArA=30(0.3) =9 m>s
vC=
250 mm
C
vBC 15 rad/s
300 mm
D
A
B
30 rad/
s
vA
711
16–79.
The mechanism shown is used in a riveting machine.It
consists of a driving piston A, three links, and a riveter which
is attached to the slider block D. Determine the velocity of
Dat the instant shown, when the piston at Ais traveling at
vA=20 m>s.
150 mm 300 mm
v=20m/s
200 mm
A
A
C
D
B
45°
45°
60°30°
45°
SOLUTION
Kinematic Diagram: Since link BC is rotating about fixed point B, then vCis always
Velocity Equation: Here,
gniylppA. dna
Equating iand jcomponents gives
Solving Eqs. [1] and [2] yields
Equating iand jcomponents gives
{–0.2598i+0.150j}mrC>A={–0.3 cos 30°i+0.3 sin 30°j}m =m>s
{–14.14i+14.14j}vA={–20 cos 45°i+20 sin 45°j}m>s=
Ans:
712
*16–80.
The mechanism is used on a machine for the manufacturing
of a wire product. Because of the rotational motion of link
AB and the sliding of block F, the segmental gear lever DE
undergoes general plane motion. If AB is rotating at
v
AB
=
5 rad
>
s, determine the velocity of point E at the
instant shown.
SOLUTION
vB
=
vAB
r
AB
=5(50) =250 mm>s 45°
v
Solving,
vC=353.6 mm>s;
v
BC =1.25 rad>s
v
p
=v
C
+v
p
>
C
v
Solving,
vD=353.6 mm>s
; v
v
vE=312 mm>s
C
A
45
45
B
50 mm
200 mm
20 mm
20 mm
50 mm
45
E
F
D
713
*16–80. Continued
Also;
v
B
=v
AB
*r
B
>
A
v
v
v
v
v
714
16–81.
In each case show graphically how to locate the
instantaneous center of zero velocity of link AB. Assume
the geometry is known.
SOLUTION
a)
AA
A
B
BB
C
(a)
(c)
(b)
v
v
v
715
16–82.
Determine the angular velocity of link AB at the instant
shown if block Cis moving upward at 12 in.>s.
SOLUTION
4
sin 45° =rIC–B
sin 30° =rIC –C
sin 105°
A
B
5in. 45
30
4in.
C
V
AB
716
16–83.
SOLUTION
Kinematic Diagram: Since linke AB is rotating about fixed point A,
Instantaneous Center: The instantaneous center of zero velocity of link BC at the
instant shown is located at the intersection point of extended lines drawn
perpendicular from vBand vC. Using law of sines, we have
The shaper mechanism is designed to give a slow cutting
stroke and a quick return to a blade attached to the slider
at C. Determine the angular velocity of the link CB at the
instant shown, if the link AB is rotating at 4 rad>s.
C
AB =4rad/s
ω
300 mm
45°
125 mm
B
Ans:
717
*16–84.
The conveyor belt is moving to the right at v
=
8 ft
>
s, and at
the same instant the cylinder is rolling counterclockwise at
v
=2 rad>s
without slipping. Determine the velocities of
the cylinder’s center C and point B at this instant.
SOLUTION
v
v
1 ft
C
B
A
Ans:
718
16–85.
The conveyor belt is moving to the right at
v
=
12 ft
>
s, and
at the same instant the cylinder is rolling counterclockwise
at
v
=
6 rad
>
s while its center has a velocity of 4 ft
>
s to the
left. Determine the velocities of points A and B on the disk
at this instant. Does the cylinder slip on the conveyor?
SOLUTION
v
v
1 ft
C
B
A
Ans:
719
16–86.
As the cord unravels from the wheel’s inner hub, the wheel
is rotating at at the instant shown. Determine
the velocities of points Aand B.
v=2 rad>s
SOLUTION
rB>IC =5+2=7 in.
rA>IC =222+52=229 in.
5in.
2in.
B
=2rad/s
ω
O
Ans:
720
16–87.
If rod CD is rotating with an angular velocity
v
CD
=
4 rad
>
s, determine the angular velocities of rods AB
and CB at the instant shown.
SOLUTION
Rotation About A Fixed Axis. For links AB and CD, the magnitudes of the
velocities of C and D are
General Plane Motion. With the results of vC and vB, the IC for link BC can be
located as shown in Fig. c. From the geometry of this figure,
B
30
C
D
A
vCD
0.4 m
1 m
0.5 m
4 rad/s
Ans:
721
*16–88.
SOLUTION
Kinematic Diagram: Since link AB is rotating about fixed point A,then
is always directed perpendicular to link AB and its magnitude is
.At the instant shown. vBis directed with an
angle with the horizontal. Also,block Cis moving horizontally due to the
constraint of the guide.
Instantaneous Center: The instantaneous center of zero velocity of bar BC at the
instant shown is located at the intersection point of extended lines drawn
perpendicular from and . Using law of sine, we have
vC
vB
45°
yB=vAB rAB =6(0.2) =1.20 m>s
vB
If bar AB has an angular velocity , determine
the velocity of the slider block Cat the instant shown.
vAB
=6 rad>s
30°
500 mm
200 mm
AB
=6rad/s
=45°
A
B
C
v
Ans:
722
16–89.
Show that if the rim of the wheel and its hub maintain
contact with the three tracks as the wheel rolls, it is necessary
that slipping occurs at the hub A if no slipping occurs at B.
Under these conditions, what is the speed at A if the wheel
has angular velocity
V
?
SOLUTION
B
A
v
r2
r1
Ans:
723
16–90.
Due to slipping, points Aand Bon the rim of the disk have
the velocities shown. Determine the velocities of the center
point Cand point Dat this instant.
SOLUTION
1.6 –x
C
A
B
F
D
E
v
B
10 ft
/
s
v
A
5ft/s
0.8 ft
30
45
Ans:
724
16–91.
SOLUTION
1.6 –x
5=x
10
Due to slipping, points Aand Bon the rim of the disk have
the velocities shown. Determine the velocities of the center
point Cand point Eat this instant.
C
A
B
F
D
E
v
B
10 ft
/
s
v
A
5ft/s
0.8 ft
30
45
Ans:
725
*16–92.
Member AB is rotating at vAB
=6 rad>s.
Determine the
velocity of point D and the angular velocity of members
BPD and CD.
SOLUTION
Rotation About A Fixed Axis. For links AB and CD, the magnitudes of the
velocities of B and D are
200 mm
B
A
D
C
P
250 mm
200 mm
200 mm200 mm
vAB 6 rad/s
6060
Ans:
726
16–93.
Member AB is rotating at vAB
=6 rad>s.
Determine the
velocity of point P, and the angular velocity of member BPD.
SOLUTION
Rotation About A Fixed Axis. For links AB and CD, the magnitudes of the
velocities of B and D are
And their direction are indicated in Fig. a and b
General Plane Motion. With the results of
vB
and
vD
, the IC for member BPD can
be located as shown in Fig. c. From the geometry of this figure
200 mm
B
A
D
C
P
250 mm
200 mm
200 mm200 mm
vAB 6 rad/s
6060
Ans:
727
16–94.
The cylinder B rolls on the fixed cylinder A without slipping.
If connected bar CD is rotating with an angular velocity
v
CD
=
5 rad
>
s, determine the angular velocity of cylinder
B. Point C is a fixed point.
SOLUTION
B
CA
D
0.1 m 0.3 m
vCD 5 rad/s
Ans:
728
16–95.
As the car travels forward at 80 ft
/
s on a wet road, due to
slipping, the rear wheels have an angular velocity
Determine the speeds of points A, B, and C
caused by the motion.
v=100 rad>s.
SOLUTION
80 ft/s
100 rad/s
1.4 ft A
C
B
729
*16–96.
The pinion gear A rolls on the fixed gear rack B with an
angular velocity v
=8 rad>s.
Determine the velocity of the
gear rack C.
SOLUTION
General Plane Motion. The location of IC for the gear is at the bottom of the gear
150 mm
A
B
C
v