Thus,
and
aB=aA=Æ
#
*rB>A+Æ*(Æ*rB>A)+2Æ*(vB>A)xyz +(aB>A)xyz
20–43. Continued
Ans:
1098
*20–44.
At the instant shown, the rod AB is rotating about the
z axis with an angular velocity
v
1 = 4 rad
>
s and an
angular acceleration
v
#
1 = 3 rad
>
s2. At this same instant,
the circular rod has an angular motion relative to the
rod as shown. If the collar C is moving down around
the circular rod with a speed of 3 in.
>
s, which is
increasing at 8 in.
>
s2, both measured relative to the rod,
determine the collar’s velocity and acceleration at this
instant.
SOLUTION
Ω=54k6 rad>s
Ω
C
>
B
=52i6 rad>s
Ω
#
C
>
B=
5
8i
6
rad
>
s
2
>
>
z
x
y
5 in.
A
B
v1 4 rad/s
fi
v1 3 rad/s2
v2 2 rad/s
fi
v2 8 rad/s2
Ans:
1099
20–45.
SOLUTION
Relative to XYZ, let xyz have
Relative to xyz, let coincident ‚ ,‚ have
((rP/O)xyz changes direction relative to XYZ.)
(rP>O)xyz
z¿y¿x¿
The particle Pslides around the circular hoop with a
constant angular velocity of while the hoop
rotates about the xaxis at a constant rate of If
at the instant shown the hoop is in the x–y plane and the
angle determine the velocity and acceleration of
the particle at this instant.
u=45°,
v=4 rad>s.
u
#
=6 rad>s,
200 mm
z
P
y
O
θ
=4rad/s
V
Ans:
1100
20–46.
At the instant shown, the industrial manipulator is
rotating about the z axis at
v
1 = 5 rad
>
s, and about joint
B at
v
2 = 2 rad
>
s. Determine the velocity and
acceleration of the grip A at this instant, when f = 30°,
u
= 45°, and r = 1.6 m.
SOLUTION
Ω=55k6 rad>s
Ω
#
=0
rB=1.2 sin 30°j+1.2 cos 30°k=50.6j+1.0392k6 m
Ω
xyz
=52i6 rad>s
Ω
#
xyz =0
v
2
v
1
u
f
1.2 m
z
A
r
x
B
1101
20–47.
At the instant shown, the industrial manipulator is rotating
about the z axis at
v
1 = 5 rad
>
s, and v
#
1=
2 rad
>
s
2
; and
about joint B at
v
2 = 2 rad
>
s and v
#
2=
3 rad
>
s
2
. Determine
the velocity and acceleration of the grip A at this instant,
when f = 30°,
u
= 45°, and r = 1.6 m.
SOLUTION
Ω=55k6 rad>s
Ω
#
=
{2k} rad
>
s
2
v
2
v
1
u
f
1.2 m
z
A
r
x
B
Ans:
1102
Ans:
P=5
*20–48.
At the given instant, the rod is turning about the z axis
with a constant angular velocity
v
1 = 3 rad
>
s. At this
same instant, the disk is spinning at
v
2 = 6 rad
>
s when
v
#
2 = 4 rad
>
s2, both measured relative to the rod.
Determine the velocity and acceleration of point P on
the disk at this instant.
y
z
v2 6 rad/s
v2 4 rad/s2
O
1.5 m
0.5 m
0.5 m
x
2 m
v1 3 rad/s
P
fi
SOLUTION
Motion of point A. Point A is located at the center of the disk. At the instant
consider, the fixed XYZ frame and rotating
x′ y′ z′
frame are set coincident with
origin at Point O. The
x′ y′ z′
frame is set rotate with constant angular velocity
𝛀
Motion of P with Respect to A. The xyz and
x″ y″ z″
frame are set coincident with
origin at A. Here
x″ y″ z″
frame is set to rotate at
𝛀 ″ =
V
x″ y″ z″
of which
Motion of P. Here,
𝛀=
V
1=5–3
k6 rad>s
and
𝛀
#
=0
.
1103
y
5 ft
x
P
B
C
A
u
15 ft
3 ft
2 ft
4 ft
v1 0.8 rad/s
v1 1.30 rad/s2
v2 3 rad/s
v2 2 rad/s2
z
vO 2 ft/s
O
SOLUTION
𝛀=50.8k6 rad>s
𝛀
#
=5
1.3k
6
rad
>
s
2
𝛀
P
>
A
={–3j} rad>s
>
20–49.
At the instant shown, the backhoe is traveling forward
at a constant speed vO = 2 ft
>
s, and the boom ABC is
rotating about the z axis with an angular velocity
v
1 = 0.8 rad
>
s and an angular acceleration
v
#
1=
1.30 rad
>
s
2
. At this same instant the boom is
rotating with
v
2 = 3 rad
>
s when v
#
2=
2 rad
>
s
2
, both
measured relative to the frame. Determine the velocity
and acceleration of point P on the bucket at this instant.
1104
20–50.
SOLUTION
vP=vO+Æ*rP/O+(rP/O)xyz
Æ
#
=0
Æ=v1={6k} rad>s
At t
h
e
i
nstant s
h
own, t
h
e arm OA of t
h
e conveyor
b
e
l
t
i
s
rotating about the zaxis with a constant angular velocity
while at the same instant the arm is rotating
upward at a constant rate If the conveyor is
running at a constant rate determine the velocity
and acceleration of the package Pat the instant shown.
Neglect the size of the package.
r
#=5ft>s,
v2=4 rad>s.
v1=6 rad>s,
z
A
O
r6ft
u45
P
v
1
6 rad/s
=
=
=
Ans:
1105
20–51.
At t
h
e
i
nstant s
h
own, t
h
e arm OA of t
h
e conveyor
b
e
l
t
i
s
rotating about the zaxis with a constant angular velocity
while at the same instant the arm is rotating
upward at a constant rate If the conveyor is
running at a rate which is increasing at
determine the velocity and acceleration of the package Pat
the instant shown. Neglect the size of the package.
r
$=8ft>s2,r
#=5ft>s,
v2=4 rad>s.
v1=6 rad>s,
SOLUTION
(aP/O)xyz =8cos 45j+8 sin 45°k–96.18j–39.60k
Æ=v1={6k} rad>s
z
A
O
r6ft
u45
P
v
1
6rad/s
=
=
=
Ans:
1106
30
y
A
x
O
z
v1 fi 0.25 rad/s
v2 fi 0.4 rad/s
40 ft
*20–52.
The crane is rotating about the z axis with a constant rate
v
1 = 0.25 rad
>
s, while the boom OA is rotating downward
with a constant rate
v
2 = 0.4 rad
>
s. Compute the velocity
and acceleration of point A located at the top of the boom
at the instant shown.
SOLUTION
𝛀={0.25k} rad>s
𝛀
#
=0
r
(v
A
>
O
)
xyz
=(r
#
A
>
O
)
xyz
+𝛀
A
>
O
*r
A
>
O
v
Ans:
1107
20–53.
Solve Prob. 20–52 if the angular motions are increasing
at v
#
1=
0.4 rad
>
s
2
and v
#
2=
0.8 rad
>
s
2
at the instant
shown.
SOLUTION
𝛀=50.25k6 rad>s
(aA
>
O)xyz =
3
(r
$
A
>
O)xyz +𝛀A
>
O*(r
#
A
>
O)xyz
4
+𝛀
#
A
>
O*rA
>
O+𝛀A
>
O*r
#
A
>
O
30
y
A
x
O
z
v1 fi 0.25 rad/s
v2 fi 0.4 rad/s
40 ft
Ans:
1108
20–54.
At the instant shown, the arm AB is rotating about the fixed
bearing with an angular velocity
v
1 = 2 rad
>
s and angular
acceleration v
#
1=
6 rad
>
s
2
. At the same instant, rod BD is
rotating relative to rod AB at
v
2 = 7 rad
>
s, which is
increasing at v
#
2=
1 rad
>
s
2
. Also, the collar C is moving
along rod BD with a velocity
r
#
=2 ft>s
and a deceleration
r
$
=–
0.5 ft
>
s
2
, both measured relative to the rod.
Determine the velocity and acceleration of the collar at
this instant.
SOLUTION
𝛀=52k6 rad>s
𝛀
#
=5
6k
6
rad
>
s
2
aB=r
$
B=
3
(r
$
B)xyz +𝛀*(r
#
B)xyz
4
+𝛀
#
*rB+𝛀*r
#
B
>
(aC
>
B)xyz =
3
(r
$
C
>
B)xyz +𝛀C
>
B*(r
#
C
>
B)xyz
4
+𝛀
#
C
>
B*rC
>
B+𝛀C
>
B*r
#
C
>
B
u 30
A
C
B
v2 7 rad/s
v2 1 rad/s2
v1 2 rad/s
v1 6 rad/s2
r 1 ft
D
y
x2 ft
1.5 ft
z