PROBLEM 15.239 (Continued)
/boom
/boom
(1.5 ft/s)sin 30 (1.5 ft/s) cos30
0
B
B
= °+ °
=
jk
a
PROBLEM 15.239 (Continued)
2 /boom
22
/ /boom
2 (2)(0.40 ) (1.5sin 30 1.5cos30 )
(1.03923 ft/s ) (0.6 ft/s )
4 3 4 1.5sin 30 1.5cos30
B
BF B B
′′
× = × °+ °
=−+
= +
=− + + °+ °
Ωv i j k
jk
v vv
jk j k
PROBLEM 15.240
The vertical plate shown is welded to arm EFG, and the entire unit rotates at
the constant rate
1
1.6
ω
=
rad/s about the Y axis. At the same time, a
continuous link belt moves around the perimeter of the plate at a constant
speed
4.5 in./s.u=
For the position shown, determine the acceleration of the
link of the belt located (a) at Point A, (b) at Point B.
SOLUTION
Let the moving frame of reference be the unit, less the pulleys and belt. It rotates about the Y axis with
constant angular velocity
1(1.6 rad/s) .
ω
= =jjΩ
The relative motion is that of the pulleys and belt with speed
90 mm/s.u=
(a) Acceleration at Point A.
2
/
(5 in.) (19 in.)
1.6 ( 5 19 )
(8 in./s)
1.6 8
(12.8 in./s )
(4.5 in./s)
A
AA
AA
AF u
′
′′
=−+
= ×
= ×− +
=
= ×
= ×
=
= =
r ij
vr
j ij
k
av
jk
i
vk k
Ω
Ω
( )
2
/
2
2
/
2
//
4.5
3
6.75 in./s
2 (2)(1.6 ) (4.5 )
(14.4 in./s )
2
12.8 6.75 14.4
AF
AF
A A AF AF
u
r
′
= −
= −
= −
×= ×
=
=+ +×
=−+
aj
j
j
v jk
i
aaa v
i ji
Ω
Ω
22
(27.2 in./s ) (6.75 in./s )
A
= −a ij
PROBLEM 15.240 (Continued)
PROBLEM 15.241
The vertical plate shown is welded to arm EFG, and the entire unit rotates at
the constant rate
11.6
ω
=
rad/s about the Y axis. At the same time, a
continuous link belt moves around the perimeter of the plate at a constant
speed
4.5 in./s.u=
For the position shown, determine the acceleration of
the link of the belt located (a) at Point C, (b) at Point D.
SOLUTION
Let the moving frame of reference be the unit, less the pulleys and belt. It rotates about the Y axis with
constant angular velocity
1(1.6 rad/s) .
ω
= =jjΩ
The relative motion is that of the pulleys and belt with speed
90 mm/s.u=
PROBLEM 15.241 (Continued)
PROBLEM 15.242
A disk of 180-mm radius rotates at the constant rate
ω
2
12=
rad/s with respect to arm CD, which itself rotates at
the constant rate
18
ω
=
rad/s about the Y axis. Determine at
the instant shown the velocity and acceleration of Point A on
the rim of the disk.
SOLUTION
Geometry.
/
/
(0.15 m) (0.18 m) (0.36 m)
(0.18 m)
AD
AC
=+−
=
r i jk
rj
Let frame Dxyz, which coincides with the fixed frame DXYZ at the instant shown, be rotating about the y axis
with constant angular velocity
1(8rad/s) .
ω
= =Ωj j
Then the motion relative to the frame consists of a rotation
of the disk AB about the bent axle CD with constant angular velocity
22
(12 rad/s) .
ω
= =kkω
A=−−v ik
PROBLEM 15.242 (Continued)
PROBLEM 15.243
A disk of 180-mm radius rotates at the constant rate
ω
2
12=
rad/s with respect to arm CD, which itself rotates at
the constant rate
1
8
ω
=
rad/s about the Y axis. Determine at
the instant shown the velocity and acceleration of Point B on
the rim of the disk.
SOLUTION
.Geometry
/
/
(0.15 m) (0.18 m) (0.36 m)
(0.18 m)
BD
BC
=−−
= −
r i jk
rj
Let frame Dxyz, which coincides with the fixed frame DXYZ at the instant shown, be rotating about the Y axis
with constant angular velocity
1(8 rad/s) .
ω
= =jjΩ
Then the motion relative to the frame consists of a
rotation of the disk AB about the bent axle CD with constant angular velocity
22
(12 rad/s) .
ω
= =kkω
Motion of the coinciding Point
B′
in the frame.
/
22
8 (0.15 0.18 0.36 )
(2.88 m/s) (1.2 m/s)
8 ( 2.88 1.2 )
(9.6 m/s ) (23.04 m/s )
B BD
BB
′
′′
= ×
=× −−
=−−
= ×
= ×− −
=−+
vr
j i jk
ik
av
j ik
ik
Ω
Ω
Motion of Point B relative to the frame.
/ 2/
/ 2/
2
12 ( 0.18 )
(2.16 m/s)
12 2.16
(25.92 m/s )
BF BD
BF BF
= ×
= ×−
=
= ×
= ×
=
vr
kj
i
av
ki
j
ω
ω
Velocity of Point B.
/
2.88 1.2 2.16
B B BF
B
′
= +
=−−+
vvv
v ik i
(0.720 m/s) (1.200 m/s)
B=−−v ik
PROBLEM 15.243 (Continued)
B
PROBLEM 15.244
A square plate of side 2r is welded to a vertical shaft that
rotates with a constant angular velocity
1
.ω
At the same
time, rod AB of length r rotates about the center of the
plate with a constant angular velocity
2
ω
with respect to
the plate. For the position of the plate shown, determine
the acceleration of end B of the rod if (a)
0,
θ
=
(b)
90 ,
θ
= °
(c)
180 .
θ
= °
SOLUTION
Use a frame of reference moving with the plate.
2
2
2 12
0 (sin 30 cos30 ) 2
rr
ω ωω
= + °− ° −
jk k
22
2 2 12
sin 30 ( cos 30 2 )
Brrr
ω ω ωω
= ° − °+aj k
ω
PROBLEM 15.244 (Continued)
(b)
90
θ
= °
/
BA
r
=
ri
PROBLEM 15.245
Two disks, each of 130-mm radius, are welded to the 500-mm rod
CD. The rod-and–disks unit rotates at the constant rate
ω
2
3 rad/s=
with respect to arm AB. Knowing that at the instant shown
14 rad/s,
ω
=
determine the velocity and acceleration of (a) Point
E, (b) Point F.
SOLUTION
Let the frame of reference BXYZ be rotating about the Y axis with angular velocity
2(4 rad/s)
ω
= =Ωj j
.
The motion relative to this frame is a rotation about the X axis with angular velocity
(3 rad/s) .
x
ω
=ii
PROBLEM 15.245 (Continued)
PROBLEM 15.246
In Problem 15.245, determine the velocity and acceleration of
(a) Point G, (b) Point H.
PROBLEM 15.245 Two disks, each of 130-mm radius, are welded
to the 500-mm rod CD. The rod-and–disks unit rotates at the
constant rate
ω
2
3 rad/s=
with respect to arm AB. Knowing that at
the instant shown
14 rad/s,
ω
=
determine the velocity and
acceleration of (a) Point E, (b) Point F.
SOLUTION
Let the frame of reference BXYZ be rotating about the Y axis with angular velocity
2(4 rad/s)
ω
= =Ωj j
.
The motion relative to this frame is a rotation about the X axis with angular velocity
(3 rad/s) .
x
ω
=ii
PROBLEM 15.246 (Continued)
H
H
PROBLEM 15.247
The position of the stylus tip A is controlled by the
robot shown. In the position shown the stylus moves at
a constant speed
180 mm/su=
relative to the solenoid
BC. At the same time, arm CD rotates at the constant
rate
2
1.6 rad/s
ω
=
with respect to component DEG.
Knowing that the entire robot rotates about the X axis at
the constant rate
1
1.2 rad/s,
ω
=
determine (a) the
velocity of A, (b) the acceleration of A.
SOLUTION
PROBLEM 15.247 (Continued)
2
/
(480 mm/s ) [ (720 mm/s) (400 mm/s) (960 mm/s) ]
480 1.2 1.6 0
A D CD
ω
′
=− + ×− + +
=−+
a j jki
i jk
j
PROBLEM 15.247 (Continued)
PROBLEM 15.248
A wheel moves in the xy plane in such a way that the location of its
center is given by the equations =123 and == 2, where xO
and yO are measured in feet and t is measured in seconds. The angular
displacement of a radial line measured from a vertical reference line is
= 84, where is measured in radians. Determine the velocity of the
point P located on the horizontal diameter of the wheel at t = 1 s.
P