© 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.

18–1.
At a given instant the body of mass m has an angular
velocity V and its mass center has a velocity vG. Show that
its kinetic energy can be represented as T = 12IICv2, where
IIC is the moment of inertia of the body determined about
the instantaneous axis of zero velocity, located a distance
rG>IC from the mass center as shown.

IC
rG/IC

G
vG

SOLUTION
T =

1
1
my2G + IG v2
2
2

=

1
1
m(vrG>IC)2 + IG v2

2
2

=

1
A mr2G>IC + IG B v2
2

=

1
I v2
2 IC

where yG = vrG>IC

However mr2G>IC + IG = IIC
Q.E.D.

912

V


© 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.

18–2.
The wheel is made from a 5-kg thin ring and two 2-kg

slender rods. If the torsional spring attached to the wheel’s
center has a stiffness k = 2 N # m>rad, and the wheel is
rotated until the torque M = 25 N # m is developed,
determine the maximum angular velocity of the wheel if it
is released from rest.

0.5 m
O
M

SOLUTION
Kinetic Energy and Work: The mass moment of inertia of the wheel about point O is
IO = mRr 2 + 2 ¢

1
m l2 ≤
12 r

= 5(0.52) + 2 c

1
(2)(12) d
12

= 1.5833 kg # m2
Thus, the kinetic energy of the wheel is
T =

1
1

I v2 = (1.5833) v2 = 0.79167 v2
2 O
2

Since the wheel is released from rest, T1 = 0. The torque developed is M = ku = 2u.
Here, the angle of rotation needed to develop a torque of M = 25 N # m is
2u = 25

u = 12.5 rad

The wheel achieves its maximum angular velocity when the spacing is unwound that
M
is when the wheel has rotated u = 12.5 rad. Thus, the work done by q
is
12.5 rad

UM =

L

Mdu =

2u du

L0
12.5 rad
2

= u †


= 156.25 J
0

Principle of Work and Energy:
T1 + © u 1 - 2 = T2
0 + 156.25 = 0.79167 v2
Ans.

v = 14.0 rad/s

Ans:
v = 14.0 rad>s
913


© 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.

18–3.
The wheel is made from a 5-kg thin ring and two 2-kg slender
rods. If the torsional spring attached to the wheel’s center has
a stiffness k = 2 N # m>rad, so that the torque on the center
of the wheel is M = 12u2 N # m, where u is in radians,
determine the maximum angular velocity of the wheel if it is
rotated two revolutions and then released from rest.

0.5 m
O
M


SOLUTION
Io = 2 c

1
(2)(1)2 d + 5(0.5)2 = 1.583
12

T1 + ©U1 - 2 = T2
4p

0 +

L0

2u du =

1
(1.583) v2
2

(4p)2 = 0.7917v2
Ans.

v = 14.1 rad/s

Ans:
v = 14.1 rad>s
914



© 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.

*18–4.
A force of P = 60 N is applied to the cable, which causes
the 200-kg reel to turn since it is resting on the two rollers
A and B of the dispenser. Determine the angular velocity of
the reel after it has made two revolutions starting from rest.
Neglect the mass of the rollers and the mass of the cable.
Assume the radius of gyration of the reel about its center
axis remains constant at kO = 0.6 m.

P
O

0.75 m
1m

A

B
0.6 m

Solution
Kinetic Energy. Since the reel is at rest initially, T1 = 0. The mass moment of inertia
of the reel about its center O is I0 = mk 20 = 200 ( 0.62 ) = 72.0 kg # m2. Thus,
T2 =

1 2
1

I v = (72.0)v2 = 36.0 v2
20
2

Work. Referring to the FBD of the reel, Fig. a, only force P does positive work. When
the reel rotates 2 revolution, force P displaces S = ur = 2(2p)(0.75) = 3p m. Thus
Up = Ps = 60(3p) = 180p J
Principle of Work and Energy.
T1 + ΣU1 - 2 = T2
  0 + 180p = 36.0 v2
Ans.

v = 3.9633 rad>s = 3.96 rad>s

Ans:
v = 3.96 rad>s
915


© 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.

18–5.
A force of P = 20 N is applied to the cable, which causes
the 175-kg reel to turn since it is resting on the two rollers
A and B of the dispenser. Determine the angular velocity of
the reel after it has made two revolutions starting from rest.
Neglect the mass of the rollers and the mass of the cable.
The radius of gyration of the reel about its center axis is
kG = 0.42 m.


P
30°

250 mm
G
500 mm
A

B
400 mm

SOLUTION
T1 + ΣU1 - 2 = T2
0 + 20(2)(2p)(0.250) =
v = 2.02 rad>s

1
3 175(0.42)2 4 v2
2

Ans.

Ans:
v = 2.02 rad>s
916


© 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.


18–6.
A force of P = 20 N is applied to the cable, which causes
the 175-kg reel to turn without slipping on the two rollers A
and B of the dispenser. Determine the angular velocity of
the reel after it has made two revolutions starting from rest.
Neglect the mass of the cable. Each roller can be considered
as an 18-kg cylinder, having a radius of 0.1 m. The radius of
gyration of the reel about its center axis is kG = 0.42 m.

P
30°

250 mm
G
500 mm
A

B
400 mm

SOLUTION
System:
T1 + ΣU1 - 2 = T2
[0 + 0 + 0] + 20(2)(2p)(0.250) =
v = vr (0.1) = v(0.5)

1
1
3 175(0.42)2 4 v2 + 2c (18)(0.1)2 d v2r

2
2

vr = 5v
Solving:
Ans.

v = 1.78 rad>s

Ans:
v = 1.78 rad>s
917


© 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.

18–7.
The double pulley consists of two parts that are attached to
one another. It has a weight of 50 lb and a radius of gyration
about its center of kO = 0.6 ft and is turning with an angular
velocity of 20 rad>s clockwise. Determine the kinetic energy
of the system. Assume that neither cable slips on the pulley.

v

20 rad/s

0.5ft


1 ft
O

SOLUTION
T =

1
1
1
I v2O + mA v2A + mB v2B
2 O
2
2

T =

1 50
1 20
1 30
a
(0.6)2 b(20)2 + a
b C (20)(1) D 2 + a
b C (20)(0.5) D 2
2 32.2
2 32.2
2 32.2

= 283 ft # lb

B 30 lb

A 20 lb

Ans.

Ans:
T = 283 ft # lb
918


© 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.

*18–8.
The double pulley consists of two parts that are attached to
one another. It has a weight of 50 lb and a centroidal radius
of gyration of kO = 0.6 ft and is turning with an angular
velocity of 20 rad> s clockwise. Determine the angular
velocity of the pulley at the instant the 20-lb weight moves
2 ft downward.

v ϭ 20 rad/s

0.5 ft

1 ft
O

SOLUTION
Kinetic Energy and Work: Since the pulley rotates about a fixed axis,
vA = vrA = v(1) and vB = vrB = v(0.5). The mass moment of inertia of the

pulley about point O is IO = mkO 2 = ¢

kinetic energy of the system is
T =
=

50
≤ (0.62) = 0.5590 slug # ft2. Thus, the
32.2

B 30 lb
A 20 lb

1
1
1
I v2 + mAvA2 + mBvB2
2 O
2
2
1
1 20
1 30
(0.5590)v2 + ¢
≤ [v(1)]2 + ¢
≤ [v(0.5)]2
2
2 32.2
2 32.2


= 0.7065v2
Thus, T1 = 0.7065(202) = 282.61 ft # lb. Referring to the FBD of the system shown
in Fig. a, we notice that Ox, Oy, and Wp do no work while WA does positive work and
WB does negative work. When A moves 2 ft downward, the pulley rotates
u =

SA
SB
=
rA
rB

SB
2
=
1
0.5
SB = 2(0.5) = 1 ft c
Thus, the work of WA and WB are
UWA = WA SA = 20(2) = 40 ft # lb
UWB = - WB SB = - 30(1) = -30 ft # lb
Principle of Work and Energy:
T1 + U1 - 2 = T2
282.61 + [40 + ( - 30)] = 0.7065 v2
Ans.

v = 20.4 rad>s

Ans:
v = 20.4 rad>s

919


© 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 = (2u + 4) N # m, where u is in
radians. If it starts from rest, determine its angular velocity
when it has made two revolutions.

300 mm

M

O

Solution
Kinetic Energy. Since the disk starts from rest, T1 = 0. The mass moment of inertia
1
1
of the disk about its center O is I0 = mr 2 = ( 20 )( 0.32 ) = 0.9 kg # m2. Thus
2
2
T2 =

1
1
I v2 = (0.9) v2 = 0.45 v2

2 0
2

Work. Referring to the FBD of the disk, Fig. a, only couple moment M does work,
which it is positive
UM =

L

M du =

L0

Principle of Work and Energy.

2(2p)

(2u + 4)du = u 2 + 4u `

4p
0

= 208.18 J

T1 + ΣU1 - 2 = T2
  0 + 208.18 = 0.45 v2
Ans.

  v = 21.51 rad>s = 21.5 rad>s


Ans:
v = 21.5 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.

18–10.
The spool has a mass of 40 kg and a radius of gyration of
kO = 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.

300 mm 500 mm
O

Solution
Kinetic Energy. Since the system is released from rest, T1 = 0. The final velocity
of the block is vb = vr = 15(0.3) = 4.50 m>s. The mass moment of inertia of the
spool about O is I0 = mk 20 = 40 ( 0.32 ) = 3.60 Kg # m2. Thus
T2 =
=

1 2
1
I v + mbv2b
20

2
1
1
(3.60) ( 152 ) + (10) ( 4.502 )
2
2

= 506.25 J
For the block, T1 = 0 and T2 =

1
1
m v2 = ( 10 )( 4.502 ) = 101.25 J
2 b b
2

Work. Referring to the FBD of the system Fig. a, only Wb does work when the block
displaces s vertically downward, which it is positive.
UWb = Wb s = 10(9.81)s = 98.1 s
Referring to the FBD of the block, Fig. b. Wb does positive work while T does
negative work.
UT = - Ts
UWb = Wbs = 10(9.81)(s) = 98.1 s
Principle of Work and Energy. For the system,
T1 + ΣU1 - 2 = T2
  0 + 98.1s = 506.25
 

Ans.


s = 5.1606 m = 5.16 m

For the block using the result of s,
T1 + ΣU1 - 2 = T2
0 + 98.1(5.1606) - T(5.1606) = 101.25
     T = 78.48 N = 78.5 N

Ans.

Ans:
s = 5.16 m
T = 78.5 N
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.

18–11.
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 kO = 0.3 m.

0.4 m
O

Solution
Kinetic Energy. Since the wheel starts from rest, T1 = 0. The mass moment of
inertia of the wheel about point O is I0 = mk 20 = 20 ( 0.32 ) = 1.80 kg # m2. Thus,

1
1
I v2 = (1.80) v2 = 0.9 v2
2 0
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,
sT = ur = 4(2p)(0.4) = 3.2p m.
T2 =

T ϭ 20 N

UT = TsT = 20(3.2p) = 64p J
Principle of Work and Energy.
  T1 + ΣU1 - 2 = T2
 

0 + 64p = 0.9 v2
Ans.

   v = 14.94 rad>s = 14.9 rad>s

Ans:
v = 14.9 rad>s
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.


*18–12.
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 A of kA = 125 mm.

A

75 mm

SOLUTION
T1 + ©U1 - 2 = T2
0 + 50(9.81)(2) =

2
1
v
[(25)(0.125)2] ¢
≤
2
0.075

+

1
(50) v2
2
Ans.

v = 4.05 m>s


Ans:
v = 4.05 m>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.

18–13.
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

3m

Solution
Kinetic Energy. Since the rod starts from rest, T1 = 0. The mass moment of inertia
1
of the rod about O is I0 =
(10) ( 32 ) + 10 ( 1.52 ) = 30.0 kg # m2. Thus,
12
F

1
1

T2 = I0 v2 = (30.0) v2 = 15.0 v2
2
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
p
3p
m. Thus
u = 90°, SW = 1.5 m and SF = ur = a b(3) =
2
2
UF = 150 a

3p
b = 225p J
2

UW = - 10(9.81)(1.5) = -147.15 J
Principle of Work and Energy.
T1 + ΣU1 - 2 = T2
0 + 225p + ( - 147.15) = 15.0 v2
  v = 6.1085 rad>s = 6.11 rad>s

Ans.

Ans:
v = 6.11 rad>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.

18–14.
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 180°
clockwise from the position shown. The force is always
perpendicular to the rod.

O

3m

Solution
Kinetic Energy. Since the rod starts from rest, T1 = 0. The mass moment of inertia
1
of the rod about O is I0 =
(10) ( 32 ) + 10 ( 1.52 ) = 30.0 kg # m2. Thus,
12
F

1
1
T2 = I0 v2 = (30.0) v2 = 15.0 v2
2
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 = 180°, SW = 3 m and SF = ur = p(3) = 3p m. Thus

UF = 150(3p) = 450p J
UW = - 10(9.81)(3) = - 294.3 J
Principle of Work and Energy. Applying Eq. 18,
T1 + ΣU1 - 2 = T2
0 + 450p + ( -294.3) = 15.0 v2
v = 8.6387 rad>s = 8.64 rad>s

Ans.

Ans:
v = 8.64 rad>s
925


© 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.

18–15.
The pendulum consists of a 10-kg uniform disk and a 3-kg
uniform slender rod. If it is released from rest in the position
shown, determine its angular velocity when it rotates
clockwise 90°.

A

B
0.8 m

M ϭ 30 N и m
D


2m

Solution
Kinetic Energy. Since the assembly is released from rest, initially,
T1 = 0. The mass moment of inertia of the assembly about A is
IA = c

1
1
(3) ( 22 ) + 3 ( 12 ) d + c (10) ( 0.42 ) + 10 ( 2.42 ) d = 62.4 kg # m2. Thus,
12
2
T2 =

1
1
I v2 = (62.4) v2 = 31.2 v2
2A
2

Work. Referring to the FBD of the assembly, Fig. a. Both Wr and Wd do positive
work, since they displace vertically downward Sr = 1 m and Sd = 2.4 m, respectively.
Also, couple moment M does positive work
UWr = Wr Sr = 3(9.81)(1) = 29.43 J
UWd = WdSd = 10(9.81)(2.4) = 235.44 J
p
UM = Mu = 30 a b = 15p J
2


Principle of Work and Energy.
T1 + ΣU1 - 2 = T2

0 + 29.43 + 235.44 + 15p = 31.2 v2
v = 3.1622 rad>s = 3.16 rad>s

Ans.

Ans:
v = 3.16 rad>s
926


© 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.

*18–16.
A motor supplies a constant torque M = 6 kN # m to the
winding drum that operates the elevator. If the elevator has a
mass of 900 kg, the counterweight C has a mass of 200 kg, and
the winding drum has a mass of 600 kg and radius of gyration
about its axis of k = 0.6 m, determine the speed of the
elevator after it rises 5 m starting from rest. Neglect the mass
of the pulleys.

SOLUTION
vE = vC

M


5
s
u = =
r
0.8

C
D

T1 + ©U1 - 2 = T2
0 + 6000(

0.8 m

1
5
1
) - 900(9.81)(5) + 200(9.81)(5) = (900)(v)2 + (200)(v)2
0.8
2
2
+

1
v
[600(0.6)2]( )2
2
0.8
Ans.


v = 2.10 m s

Ans:
v = 2.10 m>s
927


© 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.

18–17.
The center O of the thin ring of mass m is given an angular
velocity of v0. If the ring rolls without slipping, determine
its angular velocity after it has traveled a distance of s down
the plane. Neglect its thickness.

v0
s
r
O

SOLUTION
u

T1 + ©U1-2 = T2
1
1
(mr2 + mr2)v0 2 + mg(s sin u) = (mr2 + mr2)v2
2
2

g
v = v0 2 + 2 s sin u
A
r

Ans.

Ans:
v =
928

A

v20 +

g
r2

s sin u


© 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.

18–18.
The wheel has a mass of 100 kg and a radius of gyration
of
motor
supplies
a

torque
kO = 0.2 m. A
M = (40u + 900) N # m, where u is in radians, about the
drive shaft at O. Determine the speed of the loading car,
which has a mass of 300 kg, after it travels s = 4 m. Initially
the car is at rest when s = 0 and u = 0°. Neglect the mass of
the attached cable and the mass of the car’s wheels.

M
s

0.3 m
O

Solution
s = 0.3u = 4
30Њ

u = 13.33 rad
T1 + ΣU1 - 2 = T2
[0 + 0] +

L0

13.33

vC = 7.49 m>s

(40u + 900)du - 300(9.81) sin 30° (4) =


vC 2
1
1
(300)v2C + c 100(0.20)2 d a b
2
2
0.3
Ans.

Ans:
vC = 7.49 m>s
929


© 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.

18–19.
The rotary screen S is used to wash limestone. When empty
it has a mass of 800 kg and a radius of gyration of
kG = 1.75 m. Rotation is achieved by applying a torque of
M = 280 N # m about the drive wheel at A. If no slipping
occurs at A and the supporting wheel at B is free to roll,
determine the angular velocity of the screen after it has
rotated 5 revolutions. Neglect the mass of A and B.

S

2m


Solution

B

A

0.3 m
M ϭ 280 N и m

TS + ΣU1 - 2 = T2
0 + 280(uA) =

1
[800(1.75)2] v2
2

uS(2) = uA(0.3)
5(2p)(2) = uA(0.3)
uA = 209.4 rad
Thus
Ans.

v = 6.92 rad>s

Ans:
v = 6.92 rad>s
930


© 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.

*18–20.
If P = 200 N and the 15-kg uniform slender rod starts from
rest at u = 0°, determine the rod’s angular velocity at the
instant just before u = 45°.

600 mm

A
45°

u

P ϭ 200 N
B

SOLUTION
Kinetic Energy and Work: Referring to Fig. a,
rA>IC = 0.6 tan 45° = 0.6 m
Then
rG>IC = 30.32 + 0.62 = 0.6708 m
Thus,
(vG)2 = v2rG>IC = v2(0.6708)
1
The mass moment of inertia of the rod about its mass center is IG =
ml2
12
1
=

(15)(0.62) = 0.45 kg # m2. Thus, the final kinetic energy is
12
T2 =
=

1
1
m(vG)22 + IG v22
2
2
1
1
(15)[w2(0.6708)]2 + (0.45) v2 2
2
2

= 3.6v22
Since the rod is initially at rest, T1 = 0. Referring to Fig. b, NA and NB do no work,
while P does positive work and W does negative work. When u = 45°, P displaces
through a horizontal distance sP = 0.6 m and W displaces vertically upwards
through a distance of h = 0.3 sin 45°, Fig. c. Thus, the work done by P and W is
UP = PsP = 200(0.6) = 120 J
UW = - Wh = - 15(9.81)(0.3 sin 45°) = -31.22 J
Principle of Work and Energy:
T1 + ©U1 - 2 = T2
0 + [120 - 31.22] = 3.6v22
Ans.

v2 = 4.97 rad>s


Ans:
v2 = 4.97 rad>s
931


© 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.

18–21.
A yo-yo has a weight of 0.3 lb and a radius of gyration
kO = 0.06 ft. If it is released from rest, determine how far it
must descend in order to attain an angular velocity
v = 70 rad>s. Neglect the mass of the string and assume
that the string is wound around the central peg such that the
mean radius at which it unravels is r = 0.02 ft.

SOLUTION
r

vG = (0.02)70 = 1.40 ft>s

O

T1 + ©U1 - 2 = T2
0 + (0.3)(s) =

1 0.3
1
0.3
a

b (1.40)2 + c (0.06)2 a
b d(70)2
2 32.2
2
32.2
Ans.

s = 0.304 ft

Ans:
s = 0.304 ft
932


© 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.

18–22.
If the 50-lb bucket is released from rest, determine its
velocity after it has fallen a distance of 10 ft. The windlass A
can be considered as a 30-lb cylinder, while the spokes are
slender rods, each having a weight of 2 lb. Neglect the
pulley’s weight.

B

3 ft

4 ft
0.5 ft


A

0.5 ft

SOLUTION
Kinetic Energy and Work: Since the windlass rotates about a fixed axis, vC = vArA
vC
vC
=
or vA =
= 2vC. The mass moment of inertia of the windlass about its
rA
0.5
mass center is
IA =

C

2
1 30
1
2
a
b A 0.52 B + 4 c
a
b A 0.52 B +
A 0.752 B d = 0.2614 slug # ft2
2 32.2
12 32.2

32.2

Thus, the kinetic energy of the system is
T = TA + T C
=

1
1
I v2 + mCvC 2
2 A
2

=

1
1 50
(0.2614)(2vC)2 + a
bv 2
2
2 32.2 C

= 1.2992vC 2
Since the system is initially at rest, T1 = 0. Referring to Fig. a, WA, Ax, Ay, and RB
do no work, while WC does positive work. Thus, the work done by WC, when it
displaces vertically downward through a distance of sC = 10 ft, is
UWC = WCsC = 50(10) = 500 ft # lb
Principle of Work and Energy:
T1 + ©U1-2 = T2
0 + 500 = 1.2992vC 2
Ans.


vC = 19.6 ft>s

Ans:
vC = 19.6 ft>s
933


© 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.

18–23.
The coefficient of kinetic friction between the 100-lb disk
and the surface of the conveyor belt is mA
0.2. If the
conveyor belt is moving with a speed of vC = 6 ft>s when
the disk is placed in contact with it, determine the number
of revolutions the disk makes before it reaches a constant
angular velocity.

0.5 ft
v C = 6 ft/s

B

A

SOLUTION
Equation of Motion: In order to obtain the friction developed at point A of the
disk, the normal reaction NA must be determine first.

+ c ©Fy = m(aG)y ;

N - 100 = 0

N = 100 lb

Work: The friction Ff = mk N = 0.2(100) = 20.0 lb develops a constant couple
moment of M = 20.0(0.5) = 10.0 lb # ft about point O when the disk is brought in
contact with the conveyor belt. This couple moment does positive work of
U = 10.0(u) when the disk undergoes an angular displacement u. The normal
reaction N, force FOB and the weight of the disk do no work since point O does not
displace.
Principle of Work and Energy: The disk achieves a constant angular velocity
when the points on the rim of the disk reach the speed of that of the conveyor
i.e; yC = 6 ft>s. This constant angular velocity is given by
yC
6
=
v =
= 12.0 rad>s. The mass moment inertia of the disk about point O is
r
0.5

belt,

IO =

1
1 100
mr2 = a

b A 0.52 B = 0.3882 slug # ft2. Applying Eq.18–13, we have
2
2 32.2
T1 + a U1 - 2 = T2
0 + U =
0 + 10.0u =
u = 2.80 rad *

1
I v2
2 O

1
(0.3882) A 12.02 B
2
1 rev
2p rad

Ans.

= 0.445 rev

Ans:
u = 0.445 rev
934


© 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.


*18–24.
The 30-kg disk is originally at rest, and the spring is
unstretched. A couple moment of M = 80 N # m is then
applied to the disk as shown. Determine its angular velocity
when its mass center G has moved 0.5 m along the plane.
The disk rolls without slipping.

M ϭ 80 N и m

G

k ϭ 200 N/m

A

0.5 m

Solution
Kinetic Energy. Since the disk is at rest initially, T1 = 0. The disk rolls without
slipping. Thus, vG = vr = v(0.5). The mass moment of inertia of the disk about its
1
1
center of gravity G is IG = mr = (30) ( 0.52 ) = 3.75 kg # m2. Thus,
2
2
T2 =
=

1
1

I v2 + Mv2G
2G
2
1
1
(3.75)v2 + (30)[v(0.5)]2
2
2

= 5.625 v2
Work. Since the disk rolls without slipping, the friction Ff does no work. Also when
sG
0.5
the center of the disk moves SG = 0.5 m, the disk rotates u =
=
= 1.00 rad.
r
0.5
Here, couple moment M does positive work whereas the spring force does negative
work.
UM = Mu = 80(1.00) = 80.0 J
1
1
UFsp = - kx2 = - (200) ( 0.52 ) = - 25.0 J
2
2
Principle of Work and Energy.
T1 + ΣU1 - 2 = T2
0 + 80 + ( -25.0) = 5.625 v2
Ans.


v = 3.127 rad>s = 3.13 rad>s

Ans:
v = 3.13 rad>s
935


© 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.

18–25.
The 30-kg disk is originally at rest, and the spring is
unstretched. A couple moment M = 80 N # m is then
applied to the disk as shown. Determine how far the center
of mass of the disk travels along the plane before it
momentarily stops. The disk rolls without slipping.

M ϭ 80 N и m

G

k ϭ 200 N/m

A

0.5 m

Solution
Kinetic Energy. Since the disk is at rest initially and required to stop finally,

T1 = T2 = 0.
Work. Since the disk rolls without slipping, the friction Ff does no work. Also, when
sG
sG
=
= 2 sG. Here, couple
the center of the disk moves sG, the disk rotates u =
r
0.5
moment M does positive work whereas the spring force does negative work.
UM = Mu = 80(2 sG) = 160 sG
1
1
2
2
UFsp = - kx2 = - (200) sG
= - 100 sG
2
2
Principle of Work and Energy.
T1 + ΣU1 - 2 = T2
0 + 160 sG +

( - 100 sG2 ) = 0

2
160 sG - 100 sG
= 0

sG(160 - 100 sG) = 0

Since sG ≠ 0, then
160 - 100 sG = 0
Ans.

sG = 1.60 m

Ans:
sG = 1.60 m
936