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CHAPTER 15
PROBLEM 15.1
The brake drum is attached to a larger flywheel that is not shown. The
motion of the brake drum is defined by the relation
2
36 1.6 ,tt
θ
= −
where
θ
is expressed in radians and t in seconds. Determine (a) the
angular velocity at t = 2 s, (b) the number of revolutions executed by the
brake drum before coming to rest.
SOLUTION
2
2
PROBLEM 15.2
The motion of an oscillating flywheel is defined by the relation
3
0
cos 4 ,
t
et
π
θθ π
−
=
where
θ
is expressed in radians and t in seconds. Knowing
that
0
θ
= 0.5 rad, determine the angular coordinate, the angular velocity, and the
angular acceleration of the flywheel when (a) t = 0, (b) t = 0.125 s.
SOLUTION
3
0.5 cos 4
t
et
π
θπ
−
=
PROBLEM 15.3
The motion of an oscillating flywheel is defined by the relation
7 /6
0
sin 4 ,
t
et
π
θθ π
−
=
where
θ
is expressed in radians and t in seconds. Knowing
that
0
θ
= 0.4 rad, determine the angular coordinate, the angular velocity, and the
angular acceleration of the flywheel when (a) t = 0.125 s, (b) t =
.∞
SOLUTION
7 /6
0
sin 4
t
et
π
θθ π
−
=
7 /6 7 /6
0
7sin 4 4 cos 4
6
tt
de te t
dt
ππ
θπ
ω θ ππ π
−−
==−+
22 2
7 /6 7 /6 7 /6 2 7 /6
0
7 /6 2 2
0
49 28 28
sin 4 cos 4 cos 4 16 sin 4
36 6 6
49 28
16 sin 4 cos 4
36 3
t t tt
t
de te te te t
dt
e tt
π π ππ
π
ωπ π π
αθπππππ
θ ππ ππ
− − −−
−
== −−−
=−− +
()a
0
0.4 rad,
θ
=
0.125 st=
7 (0.125)/6 0.63245, 4 , sin 1, cos 0
22 2
et
πππ π
π
−= = = =
( )( )( )
0.4 0.63245 1 0.25298 radians
θ
= =
0.253 rad
θ
=
( )( ) ( )
7
0.4 0.63245 1 0.92722 rad/s
6
π
ω
= −=−
0.927 rad/s
ω
= −
( )( ) ( )
22
49
0.4 0.63245 16 1 36.551 rad/s
36
απ
=− −=−
2
36.6 rad/s
α
= −
()b
,t= ∞
7 /6
0
t
e
π
−
=
0
θ
=
0
ω
=
0
α
=
PROBLEM 15.4
The rotor of a gas turbine is rotating at a speed of 6900 rpm when the turbine is shut down. It is observed that
4 min is required for the rotor to coast to rest. Assuming uniformly accelerated motion, determine (a) the
angular acceleration, (b) the number of revolutions that the rotor executes before coming to rest.
SOLUTION
0
6900 rpm
722.57 rad/s
4 min 240 st
ω
=
=
= =
PROBLEM 15.5
A small grinding wheel is attached to the shaft of an electric motor
which has a rated speed of 3600 rpm. When the power is turned on, the
unit reaches its rated speed in 5 s, and when the power is turned off, the
unit coasts to rest in 70 s. Assuming uniformly accelerated motion,
determine the number of revolutions that the motor executes (a) in
reaching its rated speed, (b) in coasting to rest.
PROBLEM 15.6
A connecting rod is supported by a knife-edge at Point A. For small oscillations the
angular acceleration of the connecting rod is governed by the relation
6
αθ
= −
where
α
is
expressed in rad/s2 and
θ
in radians. Knowing that the connecting rod is released from rest
when
20 ,
θ
= °
determine (a) the maximum angular velocity, (b) the angular position
when
2 s.t=
00
2 22 22
00
2
6( ) 6( )
ω θθ ω θθ
=−=−
(a)
ω
is maximum when
0.
θ
=
PROBLEM 15.7
When studying whiplash resulting from rear end collisions, the rotation of the head is of
primary interest. An impact test was performed, and it was found that the angular
acceleration of the head is defined by the relation
700cos 70sin
α θθ
= +
where
α
is
expressed in rad/s2 and
θ
in radians. Knowing that the head is initially at rest, determine the
angular velocity of the head when
θ
= 30°.
PROBLEM 15.8
The angular acceleration of an oscillating disk is defined by the relation
.k
αθ
= −
Determine (a) the value of
k for which
12
ω
=
rad/s when
0
θ
=
and
6
θ
=
rad when
0,
ω
=
(b) the angular velocity of the disk when
3
θ
=
rad.
PROBLEM 15.9
The angular acceleration of a shaft is defined by the relation
0.5 ,
αω
= −
where
α
is expressed in
2
rad/s
and
ω
in rad/s. Knowing that at
0t=
the angular velocity of the shaft is 30 rad/s, determine (a) the number
of revolutions the shaft will execute before coming to rest, (b) the time required for the shaft to come to rest,
(c) the time required for the angular velocity of the shaft to reduce to 2 percent of its initial value.
ω
0
2 ln 30
t=−=∞
t= ∞
()c
( )( )
0.02 30 0.6 rad/s
ω
= =
0.6
0 30
2
t
d
dt
ω
ω
= −
∫∫
0.6
2 ln 2 ln 50
30
t=−=
7.82 st=
Copyright © McGraw-Hill Education. Permission required for reproduction or display.
PROBLEM 15.10
The bent rod ABCDE rotates about a line joining Points A and
E with a constant angular velocity of 9 rad/s. Knowing that
the rotation is clockwise as viewed from E, determine the
velocity and acceleration of corner C.
SOLUTION
2222
/
0.4 0.4 0.2
0.6 m
(0.4 m) (0.15 m)
(0.4 m) (0.4 m) (0.2 m)
11
( 0.4 0.4 0.2 ) ( 2 2 )
CE
EA
EA
EA
EA
EA
=++
=
=−+
=−+ +
= = − + + = −+ +
r ij
i jk
i j k i jk
C
C
λ
C
PROBLEM 15.11
In Problem 15.10, determine the velocity and acceleration of
corner B, assuming that the angular velocity is 9 rad/s and
increases at the rate of
2
45 rad/s .
PROBLEM 15.10 The bent rod ABCDE rotates about a line
joining Points A and E with a constant angular velocity of 9
rad/s. Knowing that the rotation is clockwise as viewed from
E, determine the velocity and acceleration of corner C.
SOLUTION
2222
/
0.4 0.4 0.2
0.6 m
(0.25 m)
(0.4 m) (0.4 m) (0.2 m)
BA
EA
EA
EA
=++
=
= −
=−+ +
rj
i jk
C
C
B=++a i jk
PROBLEM 15.12
The rectangular block shown rotates about the diagonal OA with a constant
angular velocity of 6.76 rad/s. Knowing that the rotation is counterclockwise
as viewed from A, determine the velocity and acceleration of point B at the
instant shown.
SOLUTION
PROBLEM 15.13
The rectangular block shown rotates about the diagonal OA with an
angular velocity of 3.38 rad/s that is decreasing at the rate of 5.07
2
rad/s .
Knowing that the rotation is counterclockwise as viewed from A,
determine the velocity and acceleration of point B at the instant shown.
SOLUTION
2
( ) ( ) ( )
/5 in. 31.2 in. 12 in.
AO =++r i jk
( ) ( )
/
5 in. 15.6 in.
BO
= +rij
( ) ( ) ( )
2 22
5 31.2 12 33.8 in.
OA
l=+ +=
Angular velocity:
( )
/
3.38 5 31.2 12
33.8
AO
OA
l
ω
= = ++r i jkω
( ) ( ) ( )
0.5 rad/s 3.12 rad/s 1.2 rad/s=++i jkω
Velocity of point B:
/B BO
= ×vrω
0.5 3.12 1.2 18.72 6 7.80
5 15.6 0
B
= =− +−
i jk
v ij k
PROBLEM 15.13 (Continued)
Acceleration of point B:
/B BO B
=× +×ar vαω
0.75 4.68 1.8 0.5 3.12 1.2
5 15.6 0 18.72 6 7.8
B
=− −−+
−−
i j k i jk
a
28.08 9 11.7 31.536 18.564 61.406= −+ − − +ij k i j k
( ) ( ) ( )
222
3.46 in./s 27.6 in./s 73.1 in./s
B
=−− +a i jk
PROBLEM 15.14
A circular plate of 120 mm radius is supported by two
bearings A and B as shown. The plate rotates about the
rod joining A and B with a constant angular velocity of
26 rad/s. Knowing that, at the instant considered, the
velocity of Point C is directed to the right, determine the
velocity and acceleration of Point E.
SOLUTION
(100 mm) (240 mm) 260 mm
(100) (240) ,0
BA BA
BA
=−+ =
−+
jk
jk
C
C
E=−+ +a i jk
PROBLEM 15.15
In Problem 15.14, determine the velocity and acceleration of Point E, assuming that the angular velocity is
26 rad/s and increases at the rate of 65 rad/s2.
SOLUTION
PROBLEM 15.16
The earth makes one complete revolution around the sun in 365.24 days. Assuming that the orbit of the earth
is circular and has a radius of 93,000,000 mi, determine the velocity and acceleration of the earth.
SOLUTION
( )
( )
3600 s
24 h
day h
9
2 rad
(365.24 days)
199.11 10 rad/s
π
ω
−
=
= ×
