18 solution q. E. D



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Bog'liq
Chapter 18

Kinetic Energy and Work: Since the pulley rotates about a fixed axis,
. The mass moment of inertia of the 
pulley about point is 
slug ft
2
. Thus, the
kinetic energy of the system is
Thus,
. Referring to the FBD of the system shown
in Fig. a, we notice that 
, and 
do no work while 
does positive work and
does negative work. When moves 2 ft downward, the pulley rotates
Thus, the work of 
are
Principle of Work and Energy:
Ans.
v
=
20.4 rad
>
s
282.61
+
[40
+
(
-
30)]
=
0.7065
v
2
T
1
+
U
1
-
2
=
T
2
U
W
B
= -
W
B
S
B
= -
30(1)
= -
30 ft
#
lb
U
W
A
=
W
A
S
A
=
20(2)
=
40 ft
#
lb
W
A
and 
W
B
S
B
=
2(0.5)
=
1 ft 
c
2
1
=
S
B
0.5
u
=
S
A
r
A
=
S
B
r
B
W
B
W
A
W
p
O
x

O
y
T
1
=
0.7065(20
2
)
=
282.61 ft
#
lb
=
0.7065
v
2
=
1
2
(0.5590)
v
2
+
1
2
¢
20
32.2

[
v
(1)]
2
+
1
2
¢
30
32.2

[
v
(0.5)]
2
T
=
1
2
I
O
v
2
+
1
2
m
A
v
A
2
+
1
2
m
B
v
B
2
#
=
0.5590 
I
O
=
mk
O
2
=
¢
50
32.2

(0.6
2
)
v
A
=
v
r
A
=
v
(1) and 
v
B
=
v
r
B
=
v
(0.5)
Ans:
v
=
20.4 rad
>
s


920
© 2016 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently 
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 
18–9.
The disk, which has a mass of 20 kg, is subjected to the 
couple moment of M
=
(2
u
+
4) N
#
m, where 
u
is in 
radians. If it starts from rest, determine its angular velocity 
when it has made two revolutions.
Solution
Kinetic Energy. Since the disk starts from rest, T
1
=
0. The mass moment of inertia 
of the disk about its center O is I
0
=
1
2
mr
2
=
1
2
(
20
)(
0.3
2
)
=
0.9 kg
#
m
2
. Thus
T
2
=
1
2
I

v
2
=
1
2
(0.9) 
v
2
=
0.45 
v
2
Work. Referring to the FBD of the disk, Fig. a, only couple moment M does work, 
which it is positive
U
M
=
L
M
d
u
=
L
2(2
p
)
0
(2
u
+
4)d
u
=
u
2
+
4
u
`
0
4
p
=
208.18 J
Principle of Work and Energy.
T
1
+
Σ
U
1
-
2
=
T
2
0
+
208.18
=
0.45 
v
2
v
=
21.51 rad
>
s
=
21.5 rad
>

Ans.
O
M
300 mm
Ans:
v
=
21.5 rad
>
s


921
© 2016 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently 
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 
18–10.
The spool has a mass of 40 kg and a radius of gyration of 
k
O
=
0.3 m. If the 10-kg block is released from rest, 
determine the distance the block must fall in order for the 
spool to have an angular velocity 
v
=
15 rad
>
s. Also, what 
is the tension in the cord while the block is in motion? 
Neglect the mass of the cord.
Solution
Kinetic Energy. Since the system is released from rest, T
1
=
0. The final velocity 
of the block is 
v
b
=
v
r
=
15(0.3)
=
4.50 m
>
s. The mass moment of inertia of the 
spool about O is I
0
=
mk
0
2
=
40
(
0.3
2
)
=
3.60 Kg
#
m
2
. Thus
T
2
=
1
2
I

v
2
+
1
2
m
b
v
b
2
=
1
2
(3.60)
(
15
2
)
+
1
2
(10)
(
4.50
2
)
=
506.25 J
For the block, T
1
=
0 and T
2
=
1
2
m
b
v
b
2
=
1
2
(
10
)(
4.50
2
)
=
101.25 J
Work. Referring to the FBD of the system Fig. a, only W
b
does work when the block 
displaces s vertically downward, which it is positive.
U
W
b
=
W
b
s
=
10(9.81)s
=
98.1 s
Referring to the FBD of the block, Fig. bW
b
does positive work while T does 
negative work.
U
T
=
-
Ts
U
W
b
=
W
b
s
=
10(9.81)(s)
=
98.1 s
Principle of Work and Energy. For the system,
T
1
+
Σ
U
1
-
2
=
T
2
0
+
98.1s
=
506.25
s
=
5.1606 m
=
5.16 m 
Ans.
For the block using the result of s,
T
1
+
Σ
U
1
-
2
=
T
2
0
+
98.1(5.1606)
-
T
(5.1606)
=
101.25
T
=
78.48 N
=
78.5 N 
Ans.
500 mm
300 mm
O
Ans:
s
=
5.16 m
T
=
78.5 N


922
© 2016 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently 
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 
18–11.
The force of T
=
20 N is applied to the cord of negligible 
mass. Determine the angular velocity of the 20-kg wheel 
when it has rotated 4 revolutions starting from rest. The 
wheel has a radius of gyration of k
O
=
0.3 m.
Solution
Kinetic Energy. Since the wheel starts from rest, T
1
=
0. The mass moment of 
inertia of the wheel about point O is I
0
=
mk
0
2
=
20
(
0.3
2
)
=
1.80 kg
#
m
2
. Thus,
T
2
=
1
2
I

v
2
=
1
2
(1.80) 
v
2
=
0.9 
v
2
Work. Referring to the FBD of the wheel, Fig. a, only force T does work. 
This work is positive since T is required to displace vertically downward, 
s
T
=
u
r
=
4(2
p
)(0.4)
=
3.2
p
m.
U
T
=
Ts
T
=
20(3.2
p
)
=
64
p
J
Principle of Work and Energy.
T
1
+
Σ
U
1
-
2
=
T
2
0
+
64
p
=
0.9 
v
2
v
=
14.94 rad
>
s
=
14.9 rad
>

Ans.
T
20 N
O
0.4 m
Ans:
v
=
14.9 rad
>
s


923
© 2016 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently 
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 
*18–12.
75 mm
A
Determine the velocity of the 50-kg cylinder after it has
descended a distance of 2 m. Initially, the system is at rest.
The reel has a mass of 25 kg and a radius of gyration about its
center of mass of 
.
k
A
=
125 mm
SOLUTION
Ans.
v
=
4.05 m
>
s
+
1
2
(50) 
v
2
0
+
50(9.81)(2)
=
1
2
[(25)(0.125)
2
]
¢
v
0.075

2
T
1
+ ©
U
1
-
2
=
T
2
Ans:
v
=
4.05 m
>
s


924
© 2016 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently 
exist. No portion of this material may be reproduced, in any form or by any means, without permission in writing from the publisher. 
Solution
Kinetic Energy. Since the rod starts from rest, T
1
=
0. The mass moment of inertia 
of the rod about O is I
0
=
1
12
(10)
(
3
2
)
+
10
(
1.5
2
)
=
30.0 kg
#
m
2
. Thus, 
T
2
=
1
2
I

v
2
=
1
2
(30.0) 
v
2
=
15.0 
v
2
Work. Referring to the FBD of the rod, Fig. a, when the rod undergoes an angular 
displacement 
u
, force F does positive work whereas W does negative work. When 
u
=
90
°
S
W
=
1.5 m and S
F
=
u
r
=
a
p
2
b
(3)
=
3
p
2
m. Thus
U
F
=
150 
a
3
p
2
b
=
225
p
J
U
W
=
-
10(9.81)(1.5)
=
-
147.15 J
Principle of Work and Energy.
T
1
+
Σ
U
1
-
2
=
T
2
0
+
225
p
+
(
-
147.15)
=
15.0 
v
2
v
=
6.1085 rad
>
s
=
6.11 rad
>

Ans.
18–13.
The 10-kg uniform slender rod is suspended at rest when 
the force of F
=
150 N is applied to its end. Determine the 
angular velocity of the rod when it has rotated 90° clockwise 
from the position shown. The force is always perpendicular 
to the rod.
O
3 m

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