PROBLEM 15.112
The 18-in.–radius flywheel is rigidly attached to a
1.5-in.
–radius
shaft that can roll along parallel rails. Knowing that at the instant
shown the center of the shaft has a velocity of
1.2 in./s
and an
acceleration of
2
0.5 in./s ,
both directed down to the left, determine
the acceleration (a) of Point A, (b) of Point B.
SOLUTION
Velocity analysis.
Let Point G be the center of the shaft and Point C be the point of contact with the rails. Point C is the
instantaneous center of the wheel and shaft since that point does not slip on the rails.
1.2
G
v
B=a
PROBLEM 15.113
A 3–in.–radius drum is rigidly attached to a 5–in.–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 at the instant shown end D of
the cord has a velocity of 8 in./s and an acceleration of
2
30 in./s ,
both
directed to the left, determine the accelerations of Points A, B, and C of the
drums.
PROBLEM 15.113 (Continued)
C=a
PROBLEM 15.114
A 3–in.–radius drum is rigidly attached to a 5-in.-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 at the instant shown end D of the cord
has a velocity of 8 in./s and an acceleration of
2
30 in./s ,
both directed to the
left, determine the accelerations of Points A, B, and C of the drums.
B=a
PROBLEM 15.114 (Continued)
[45=
] [5
α
+
2
] [5
ω
+
]
[45=
] [75+
] [80+
]
2
[35 in./s=
2
] [75 in./s+
]
2
82.8 in./s
C=a
65.0°
PROBLEM 15.115
A heavy crate is being moved a short distance using three
identical cylinders as rollers. Knowing that at the instant shown
the crate has a velocity of 200 mm/s and an acceleration of
400
2
mm/s ,
both directed to the right, determine (a) the angular
acceleration of the center cylinder, (b) the acceleration of point
A on the center cylinder.
SOLUTION
PROBLEM 15.115 (Continued)
2
A
A
PROBLEM 15.116
A wheel rolls without slipping on a fixed cylinder. Knowing that at
the instant shown the angular velocity of the wheel is 10 rad/s
clockwise and its angular acceleration is 30 rad/
2
s
counterclockwise,
determine the acceleration of (a) Point A, (b) Point B, (c) Point C.
SOLUTION
.Velocity analysis
0.04 m 10 rad/sr= =ω
Point C is the instantaneous center of the wheel.
[( )
Ar
ω
=v
] [(0.04)(10)=
] 0.4 m/s=
]
.Acceleration analysis
2
30 rad/s
α
=
Point moves on a circle of radius A
0.16 0.04 0.2 m.Rr
r
= += + =
Since the wheel does not slip,
CC
a=a
//
()()
C A CA t CA n
=++aa a a
[
C
a
] [( )
At
a=
2
]A
v
r
+
[r
α
+
2
][r
ω
+
]
[( )
At
a=
2
(0.4)
]0.2
+
[(0.04)(30)
+
2
] [(0.04)(10)+
]
[( )
At
a=
] [0.8+
] [1.2+
] [4+
]
Components.
2
: ( ) 1.2 0 ( ) 1.2 m/s
At At
aa− += =
2
: 0.8 4.0 3.2 m/s
CC
aa=−+ =
(a) Acceleration of Point A.
2
[1.2 m/s
A
=a
2
] [0.8 m/s+
]
2
1.442 m/s
A
=a
33.7 °
PROBLEM 15.116 (Continued)
( ) b Acceleration of Point B.
//
()()
B A BA t BA n
=++aa a a
[1.2
B
=a
] [0.8+
][r
α
+
2
][r
ω
+
]
[1.2=
] [0.8+
] [(0.04)(30)+
2
] [(0.04)(10)+
]
2
[2.8 m/s=
2
] [2 m/s+
]
2
3.44 m/s
B
=a
35.5 °
( ) c Acceleration of Point C.
CC
a=a
2
3.20 m/s
C
=a
PROBLEM 15.117
The 100 mm radius drum rolls without slipping on a portion of a
belt that moves downward to the left with a constant velocity of
120 mm/s. Knowing that at a given instant the velocity and
acceleration of the center A of the drum are as shown, determine
the acceleration of Point D.
SOLUTION
Velocity analysis.
= +vvv
PROBLEM 15.117 (Continued)
.Acceleration of Point D
PROBLEM 15.118
In the planetary gear system shown the radius of gears A, B, C, and D is
3 in. and the radius of the outer gear E is 9 in. Knowing that gear A has a
constant angular velocity of 150 rpm clockwise and that the outer gear E is
stationary, determine the magnitude of the acceleration of the tooth of gear
D that is in contact with (a) gear A, (b) gear E.
SOLUTION
D
PROBLEM 15.118 (Continued)
Gear D:
T
(b) Tooth E in contact with gear E.
2
E=a
E
PROBLEM 15.119
The 200-mm–radius disk rolls without sliding on the
surface shown. Knowing that the distance BG is 160
mm and that at the instant shown the disk has an
angular velocity of 8 rad/s counterclockwise and an
angular acceleration of 2 rad/s2 clockwise, determine
the acceleration of A.
SOLUTION
2
2
/ //
10.64 0.32
A B AB B AB AB AB AB
αω
=+ =+ ×−
= +
a a a a kr r
ij
2
( 0.600 0.200 ) (1.6525) ( 0.600 0.200 )
10.64 0.32 0.77460 0.2 2.115 0.54615
12.755 0.86615 0.2 0.77460
AB
AB AB
A AB AB
a
α
αα
αα
+×−−− −−
= +− + + +
=+ +−
k ij ij
ij j i i j
ii ji j
PROBLEM 15.119 (Continued)
Resolve into components and transpose terms.
PROBLEM 15.120
Knowing that crank AB rotates about Point A with a constant angular velocity of
900 rpm clockwise, determine the acceleration of the piston P when
60 .
θ
= °
SOLUTION
Law of sines.
sin sin 60 16.779
0.05 0.15
ββ
°
= = °
Velocity analysis.
900 rpm 30 rad/s
AB
ωπ
= =
0.05 1.5 m/s
B AB
ωπ
= =v
60°
DD
v=v
BD BD
ωω
=
/0.15
D B BD
ω
=v
β
/D B DB
= +v vv
[
D
v
] [1.5
π
=
60 ] [0.15
BD
ω
°+
]
β
Components
:
0 1.5 cos 60 0.15 cos
BD
π ωβ
= °−
1.5 cos 60 16.4065 rad/s
0.15 cos
BD
π
ωβ
°
= =
Acceleration analysis.
0
AB
α
=
2 22
0.05 (0.05)(30 ) 444.13 m/s
B AB
ωπ
= = =a
30°
DD
a=a
BD BD
αα
=
/
[0.15
D B AB
α
=a
2
] [0.15 BD
βω
+
]
β
[0.15
BD
α
=
] [40.376
β
+
]
β
/
Resolve into components.
D B DB
= +a aa
PROBLEM 15.120 (Continued)
P
PROBLEM 15.121
Knowing that crank AB rotates about Point A with a constant angular velocity of
900 rpm clockwise, determine the acceleration of the piston P when
120 .
θ
= °
SOLUTION
sin sin120 , 16.779
ββ
°
/Resolve into components.
D B DB
PROBLEM 15.121 (Continued)
PD
P=a
PROBLEM 15.122
In the two–cylinder air compressor shown, the connecting rods
BD and BE are each 190 mm long and crank AB rotates about the
fixed Point A with a constant angular velocity of 1500 rpm
clockwise. Determine the acceleration of each piston when
θ
= 0.
SOLUTION