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PROBLEM 15.67
Robert’s linkage is named after Richard Robert (1789–1864) and can be used
to draw a close approximation to a straight line by locating a pen at Point F.
The distance AB is the same as BF, DF and DE. Knowing that the angular
velocity of plate BDF is 2 rad/s counterclockwise when
θ
= 90°, determine
(a) the angular velocities of bars AB and DE, (b) the velocity of Point F.
When point F may be assumed to coincide with point E, with
negligible error in the velocity analysis.
DE
DE
90 ,
θ
= °
PROBLEM 15.67 (Continued)
(12)( 1.4641) 17.569
D
=−− =
v ii
PROBLEM 15.68
In the position shown, bar DE has a constant angular
velocity of 10 rad/s clockwise. Knowing that
500 mm,h=
determine (a) the angular velocity of bar FBD, (b) the
velocity of Point F.
SOLUTION
F
F
PROBLEM 15.69
In the position shown, bar DE has a constant angular
velocity of 10 rad/s clockwise. Determine (a) the distance h
for which the velocity of Point F is vertical, (b) the
corresponding velocity of Point F.
PROBLEM 15.69 (Continued)
F
F
PROBLEM 15.70
Both 6-in.-radius wheels roll without slipping on
the horizontal surface. Knowing that the distance
AD is 5 in., the distance BE is 4 in. and D has a
velocity of 6 in./s to the right, determine the
velocity of Point E.
AB E
Velocity of Point E.
/
2.4605 6
E E EQ
ω
=×=− ×v kr k j
E
PROBLEM 15.71
The 80-mm-radius wheel shown rolls to the left with a velocity
of 900 mm/s. Knowing that the distance AD is 50 mm,
determine the velocity of the collar and the angular velocity of
rod AB when (a)
0,
β
=
(b)
90 .
β
= °
SOLUTION
PROBLEM 15.71 (Continued)
250
AB
AB
PROBLEM 15.72*
For the gearing shown, derive an expression for the angular velocity
C
ω
of
gear C and show that
C
ω
is independent of the radius of gear B. Assume that
Point A is fixed and denote the angular velocities of rod ABC and gear A by
ABC
ω
and
A
ω
respectively.
A B ABC
PROBLEM 15.72* (Continued)
B
PROBLEM 15.73
A juggling club is thrown vertically into the air. The center of
gravity G of the 20 in. club is located 12 in. from the knob.
Knowing that at the instant shown G has a velocity of 4 ft/s
upwards and the club has an angular velocity of 30 rad/s
counterclockwise, determine (a) the speeds of Point A and B,
(b) the location of the instantaneous center of rotation.
SOLUTION
B
Let
/CG x=ri
bet the position of the instantaneous center C relative to G.
PROBLEM 15.74
At the instant shown during deceleration, the velocity of an
automobile is 40 ft/s to the right. Knowing that the velocity of the
contact point A of the wheel with the ground is 5 ft/s to the right,
determine (a) the instantaneous center of rotation of the wheel,
(b) the velocity of point B, (c) the velocity of point D.
SOLUTION
40 5 35 rad/s
1
OA
AO
vv
l
ω
−−
= = =
(
a)
51
ft
35 7
A
CA
v
l
ω
= = =
1.714 in.=
lies 1.714 in. below .CA
(
b)
1 15
2 ft
77
CB
l=+=
15
35 75 ft/s
7
B CB
vl
ω
= =⋅=
75.0 ft/s
B=v
(
c)
18
1 ft, 1 ft
77
OD CO
ll= =+=
2
2
8
1 1.5186 ft
7
CD
l
=+=
( )( )
1.5186 35 53.2 ft/s
D CD
vl
ω
= = =
7
tan , 41.2
8
OD
CO
l
l
θθ
= = = °
53.2 ft/s
D
=v
41.2°
PROBLEM 15.75
A helicopter moves horizontally in the x direction at a speed of
120 mi/h. Knowing that the main blades rotate clockwise with an
angular velocity of 180 rpm, determine the instantaneous axis of
rotation of the main blades.
SOLUTION
PROBLEM 15.76
A 60-mm-radius drum is rigidly attached to a 100-mm-radius drum as
shown. One of the drums rolls without sliding on the surface shown, and a
cord is wound around the other drum. Knowing that end E of the cord is
pulled to the left with a velocity of 120 mm/s, determine (a) the angular
velocity of the drums, (b) the velocity of the center of the drums, (c) the
length of cord wound or unwound per second.
SOLUTION
Since the drum rolls without sliding, its instantaneous center lies at
D.
120 mm/s
EB
= =vv
//
,
A AD B BD
vv vr
ωω
= =
(
a)
/
120 3 rad/s
100 60
B
BD
v
v
ω
= = =
−
3.00 rad/s=ω
(
b)
(100)(3) 300 mm/s
A
v= =
300 mm/s
A
=v
Since
A
v
is greater than
,
B
v
cord is being wound.
300 120 180 mm/s
AB
vv−= − =
(
c) Cord wound per second
=
180.0 mm
PROBLEM 15.77
A 60-mm-radius drum is rigidly attached to a 100-mm-radius drum as shown.
One of the drums rolls without sliding on the surface shown, and a cord is
wound around the other drum. Knowing that end E of the cord is pulled to
the left with a velocity of 120 mm/s, determine (a) the angular velocity of the
drums, (b) the velocity of the center of the drums, (c) the length of cord
wound or unwound per second.
SOLUTION
Since the drum rolls without sliding, its instantaneous center lies at
B.
120 mm/s
ED
= =vv
//
,
A AB D DB
vr vr
ωω
= =
(
a)
/
120 3 rad/s
100 60
D
DB
v
r
ω
= = =
−
3.00 rad/s=ω
(
b)
(60)(3.00) 180 mm/s
A
v= =
180 mm/s
A
=v
Since
A
v
is to the right and
D
v
is to the left, cord is being unwound.
180 120 300 mm/s
AE
vv−= + =
(
c) Cord unwound per second
=
300 mm
PROBLEM 15.78
The spool of tape shown and its frame assembly are pulled upward at a
speed
750
A
v=
mm/s. Knowing that the 80-mm-radius spool has an angular
velocity of 15 rad/s clockwise and that at the instant shown the total
thickness of the tape on the spool is 20 mm, determine (a) the instantaneous
center of rotation of the spool, (b) the velocities of Points B and D.
SOLUTION
D
1.950 m/s
D
=v
PROBLEM 15.79
The spool of tape shown and its frame assembly are pulled upward at a
speed
100 mm/s.
A
v=
Knowing that end B of the tape is pulled downward
with a velocity of 300 mm/s and that at the instant shown the total thickness
of the tape on the spool is 20 mm, determine (a) the instantaneous center of
rotation of the spool, (b) the velocity of Point D of the spool.
( ) ( ) ( ) (80 25) mm 25 mm
DO OC OC
++
D=v
PROBLEM 15.80
The arm ABC rotates with an angular velocity of 4 rad/s
counterclockwise. Knowing that the angular velocity of the
intermediate gear B is 8 rad/s counterclockwise, determine
(a) the instantaneous centers of rotation of gears A and C,
(b) the angular velocities of gears A and C.
PROBLEM 15.80 (Continued)
Gear C:
2
1.2 0.8
0.200 0.2
C
C
vv
ω
−−
= =
2 rad/s
C
ω
=
1.2 0.6 m
2
C
C
C
v
ω
= = =
PROBLEM 15.81
The double gear rolls on the stationary left rack R. Knowing that the rack on the
right has a constant velocity of 2 ft/s, determine (a) the angular velocity of the
gear, (b) the velocities of Points A and D.
10 in.
B
CB
l
ω
= = =
