690
*16–60.
The slider block C moves at 8 m
>
s down the inclined groove.
Determine the angular velocities of links AB and BC, at the
instant shown.
SOLUTION
Rotation About Fixed Axis. For link AB, refer to Fig. a.
General Plane Motion. For link BC, refer to Fig. b. Applying the relative velocity
equation,
Equating i and j components,
2 m
2 m
A
C
B
45
Ans:
691
16–61.
Determine the angular velocity of links AB and BC at the
instant
u=30°
. Also, sketch the position of link BC when
u=55°
, 45°, and 30° to show its general plane motion.
SOLUTION
Rotation About Fixed Axis. For link AB, refer to Fig. a.
General Plane Motion. For link BC, refer to Fig. b. Applying the relative velocity
equation,
Equating i and j components,
The general plane motion of link BC is described by its orientation when
u=30°
,
45°
and
55°
shown in Fig. c.
A
B
C
1 ft
3 ft
vC 6 ft/s
u
16–62.
SOLUTION
Kinematic Diagram:Since link BC is rotating about fixed point C. then vBis
The planetary gear Ais pinned at B. Link BC rotates
clockwise with an angular velocity of 8 rad/s, while the outer
gear rack rotates counterclockwise with an angular velocity
of 2 rad/s. Determine the angular velocity of gear A.
15 in.
C
BC =8rad/s
ω
20 in.
D
A
B
=2rad/s
ω
Ans:
693
16–63.
If the angular velocity of link AB is
determine the velocity of the block at Cand the angular
velocity of the connecting link CB at the instant
and f=30°.
u=45°
vAB =3 rad
>
s,
SOLUTION
Also,
vC=vB+v*rC>B
B
vC
;
R
=C6
30°cS+DvCB (3)
45°bT
vC=vB+vC>B
3ft
2ft
AB =3rad/s
ω
=45°
θ
=30°
φ
C
B
A
Ans:
694
*16–64.
SOLUTION
C
B
v0.3 ft
A
The pinion gear A rolls on the fixed gear rack B with an
angular velocity v = 4 rad>s. Determine the velocity of
the gear rack C.
695
16–65.
SOLUTION
Also:
vC=vB+v*rC>B
vC=vB+vC>B
The pinion gear rolls on the gear racks. If Bis moving to the
right
at and Cis moving to the left at , determine
the
angular velocity of the pinion gear and the velocity of its
center
A.
4ft>s8ft>s
C
B
V0.3 ft
A
Ans:
696
16–66.
SOLUTION
General Plane Motion:Applying the relative velocity equation to points Band C
and referring to the kinematic diagram of the gear shown in Fig. a,
Equating the icomponents yields
For points Oand C,
Determine the angular velocity of the gear and the velocity
of its center Oat the instant shown.
3ft/s
4ft/s
A
O0.75 ft
1.50 ft
45
Ans:
697
16–67.
Determine the velocity of point on the rim of the gear at
the instant shown.
A
SOLUTION
General Plane Motion:Applying the relative velocity equation to points Band C
and referring to the kinematic diagram of the gear shown in Fig. a,
Equating the icomponents yields
For points Aand C,
Equating the iand jcomponents yields
Thus, the magnitude of vAis
vA=vC+v*rA>C
3ft/s
4ft/s
A
O0.75 ft
1.50 ft
45
Ans:
698
*16–68.
Knowing that angular velocity of link AB is
v
AB
=
4 rad
>
s, determine the velocity of the collar at C
and the angular velocity of link CB at the instant shown.
Link CB is horizontal at this instant.
500 mm
45
CB
350 mm
vAB 4 rad/s
SOLUTION
vB=vAB
rAB
=4(0.5) =2 m>s
Ans:
699
C
vAB 60 rad/s
A
300 mm
600 mm
B
f
u
16–69.
Rod AB is rotating with an angular velocity of
v
AB
=
60 rad
>
s. Determine the velocity of the slider C at
the instant
u
=
60° and f
=
45°. Also, sketch the position
of bar BC when
u
=
30°, 60° and 90° to show its general
plane motion.
SOLUTION
Rotation About Fixed Axis. For link AB, refer to Fig. a.
General Plane Motion. For link BC, refer to Fig. b. Applying the relative velocity
equation,
Equating i components,
Then, equating j components,
The general plane motion of link BC is described by its orientation when
u=30°
,
60°
and
90°
shown in Fig. c.
Ans:
700
16–70.
The angular velocity of link AB is
v
AB
=
5 rad
>
s.
Determine the velocity of block C and the angular velocity
of link BC at the instant
u
=
45° and f
=
30°. Also, sketch
the position of link CB when
u
=
45°, 60°, and 75° to show
its general plane motion.
SOLUTION
Rotation About A Fixed Axis. For link AB, refer to Fig. a.
General Plane Motion. For link BC, refer to Fig. b. Applying the relative velocity
equation,
v
C
= v
B
+ V
BC
*r
C
>
B
Equating j components,
Then, equating i components,
f
A
B
2 m
3 m
vAB
5 rad/
s
u
701
16–70. Continued
Ans:
702
16–71.
The similar links AB and CD rotate about the fixedpins
at A and C. If AB has an angular velocity
v
AB
=
8 rad
>
s, determine the angular velocity of BDP and
the velocity of point P.
300 mm
B
A
D
C
v
>
700 mm
300 mm
300 mm300 mm
6060
vAB 8 rad/s
SOLUTION
v
D
=v
B
+v*r
D
>
B
703
*16–72.
If the slider block A is moving downward at
vA
=4 m>s,
determine the velocities of blocks B and C at
the instant shown.
SOLUTION
v
B
=v
A
+v
B
>
A
Solving,
v
AB =9.091 rad>s
v
v
v
4
5
3
250 mm
400 mm
300 mm
300 mm
E
B
C
D
A
30
vA 4 m/s
704
*16–72. Continued
vB=
v
AB(0.33)
Ans:
705
16–73.
If the slider block A is moving downward at vA
=
4 m
>
s,
determine the velocity of point E at the instant shown.
SOLUTION
See solution to Prob. 16–87.
Also:
See solution to Prob. 16–87.
4
5
3
250 mm
400 mm
300 mm
300 mm
E
B
C
D
A
30
vA 4 m/s
16–74.
The epicyclic gear train consists of the sun gear Awhich is
in mesh with the planet gear B.This gear has an inner hub C
which is fixed to Band in mesh with the fixed ring gear R.If
the connecting link DE pinned to Band Cis rotating at
about the pin at determine the angular
velocities of the planet and sun gears.
E,vDE =18 rad>s
SOLUTION
The velocity of the contact point Pwith the ring is zero.
Let be the contact point between Aand B.
P¿
BA
200 mm
100 mm
R
C
E
600 mm
D
707
16–75.
If link AB is rotating at
v
AB
=
3 rad
>
s, determine the
angular velocity of link CD at the instant shown.
SOLUTION
v
B
=v
AB
*r
B
>
A
v
=v
>
v
A
B
8 in.
30
6 in.
vAB
*16–76.
If link CD is rotating at
v
CD
=
5 rad
>
s, determine the
angular velocity of link AB at the instant shown.
SOLUTION
v
B
=v
AB
*r
B
>
A
v
=v
>
v
A
B
8 in.
30
6 in.
vAB
Ans:
709
16–77.
The planetary gear system is used in an automatic
transmission for an automobile. By locking or releasing
certain gears, it has the advantage of operating the car at
different speeds. Consider the case where the ring gear Ris
held fixed, , and the sun gear Sis rotating at
. Determine the angular velocity of each of the
planet gears Pand shaft A.
vS=5 rad>s
vR=0
SOLUTION
vA=5(80) =400 mm>s;
R
S
P
A
vS
vR
80 mm
40 mm