PROBLEM 15.249
Two blocks and a pulley are connected by inextensible cords as
shown. The relative velocity of block A with respect to block B
is 2.5 ft/s to the left at time t = 0 and 1.25 ft/s to the left when
t = 0.25 s. Knowing that the angular acceleration of the pulley
is constant, find (a) the relative acceleration of block A with
respect to block B, (b) the distance block A moves relative to
block B during the interval
0 0.25t≤≤
s.
SOLUTION
( ) For the pulley, a
( )
11 1
, 12 6 in. ft
22 2
A
rd r= = = =
( )
11
8 8 in. ft
23
B
r= = =
( )
/
/
,
AA BB AB AB AB
AB
AB
vr vr v vv rr
v
rr
ωω ω
ω
= = =−= −
=−
At
0,t=
011
23
2.5 15 rad/s
ω
= =
−
At
111
23
1.25 7.5 rad/s
ω
= =
−
2
12
7.5 15 30 rad/s
0.25t
ωω
α
−−
= = = −
2
30 rad/s=
( )
2
130 15 ft/s
2
AA
r
α

= = =


a
( )
2
130 10 ft/s
3
BB
r
α

= = =


a
2
/
5 ft/s
AB A B
aa=−=a
2
/
5.00 ft/s
AB
a=
()b
( ) ( )
2
// /
0
1
2
AB AB AB
tt= +xv a
( )( ) ( )( )
2
1
2.5 0.25 5 0.25 0.46875 ft
2
=−+ =−
/
5.63 in.
AB
=x
PROBLEM 15.250
A baseball pitching machine is designed to deliver a
baseball with a ball speed of 70 mph and a ball rotation
of 300 rpm clockwise. Knowing that there is no
slipping between the wheels and the baseball during the
ball launch, determine the angular velocities of wheels
A and B.
Given:
ball
m 1 h 5280 ft
70 102.667 ft/s 1232 in./s
h 3600 s mi
2 rad 1 min
(300 rpm) 10 rad/s
rev 60 s
G
v
π
ωπ

  
= = =

  
  



= =




B
PROBLEM 15.251
Knowing that inner gear A is stationary and outer gear C starts from rest
and has a constant angular acceleration of 4 rad/s2 clockwise, determine
at t = 5 s (a) the angular velocity of arm AB, (b) the angular velocity of
gear B, (c) the acceleration of the point on gear B that is in contact with
gear A.
SOLUTION
Angular velocity of gear C at t = 5 s.
2
B
PROBLEM 15.251 (Continued)
Calculate tangential accelerations:
1
() 0
t
a
=
PROBLEM 15.252
Knowing that at the instant shown bar AB has an angular velocity of
10 rad/s clockwise and it is slowing down at a rate of 2 rad/s2,
determine the angular accelerations of bar BD and bar DE.
SOLUTION
22
(0.4 m/s ) (20 m/s )
= +
ij
PROBLEM 15.252 (Continued)
2
//
D B BD D B BD D B
ω
=+×−
aaαr r
PROBLEM 15.253
Knowing that at the instant shown rod AB has zero angular
acceleration and an angular velocity of 15 rad/s
counterclockwise, determine (a) the angular acceleration of
arm DE, (b) the acceleration of Point D.
SOLUTION
3
tan , 36.87
4
45 in.
cos
45 in.
AB
ββ
β
= = °
= =
/Resolve into components.
D B DB
PROBLEM 15.253 (Continued)
()a
:
5 sin 1125cos 1125cos 1440
DE
αβ β β
+ =−−
2
1080 rad/s
DE
α
= −
2
1080 rad/s
DE =α
()b
[(5)( 1080)
D
= −a
2
] [1125 in./s
β
+
]
β
2
[5400 in./s=
2
] [1125 in./s
β
+
]
β
22
2
2
1125
tan 5400
11.77
5400 1125
5516 in./s
460 ft/s
90 64.9
D
a
γ
γ
βγ
=
= °
= +
=
=
°− + = °
2
460 ft/s
D
=a
64.9°
PROBLEM 15.254
Rod AB is attached to a collar at A and is fitted with a
wheel at B that has a radius r = 15 mm. Knowing that
when
θ
= 60° the collar has velocity of 250 mm/s
upward and the speed of the collar is decreasing at a
rate of 150 mm/s2, determine (a) the angular
acceleration of rod AB, (b) the angular acceleration of
the wheel.
PROBLEM 15.254 (Continued)
B t AB
PROBLEM 15.254 (Continued)
PROBLEM 15.255
Water flows through a curved pipe AB that rotates with a constant
clockwise angular velocity of 90 rpm. If the velocity of the water
relative to the pipe is 8 m/s, determine the total acceleration of a
particle of water at Point P.
SOLUTION
Let the curved pipe be a rotating frame of reference. Its angular velocity is
90 rpm
or
9.4248 rad/s=ω
.
Motion of the frame of reference at Point P
′
.
PROBLEM 15.256
A disk of 0.15-m radius rotates at the constant rate
2
ω
with respect to plate BC, which itself rotates at the
constant rate
1
ω
about the y axis. Knowing that
12
3 rad/s,
ωω
= =
determine, for the position shown,
the velocity and acceleration (a) of Point D, (b) of
Point F.
SOLUTION
Frame AXYZ is fixed.
Moving frame, Exyz, rotates about
y
axis at
PROBLEM 15.256 (Continued)
/
2
/
2 22
2
2(3 rad/s) ( 0.45 m/s)
(2.7 m/s )
(2.7 m/s ) (1.35 m/s ) (2.7 m/s )
c FF
F F FF c
′
= ×
= ×−
= −
=++
=−− −
av
jk
i
aaa a
i ii
Ω
2
(6.75 m/s )
F
= −ai
PROBLEM 15.257
Two rods AE and BD pass through holes drilled into a hexagonal
block. (The holes are drilled in different planes so that the rods will
not touch each other.) Knowing that rod AE has an angular velocity
of 20 rad/s clockwise and an angular acceleration of 4 rad/s2
counterclockwise when
θ
= 90°, determine, (a) the relative velocity
of the block with respect to each rod, (b) the relative acceleration
of the block with respect to each rod.
sin 60 sin 30 sin 90
° °°
1
2
57.735 mm
115.470 mm
r
r
=
=
The angle at H remains at 60° so that rods AE and BD have a common angular velocity
ω
=ω
and a
common angular acceleration
α
=α
, where