Engineering Mechanics - Statics Episode 1 Part 7 pptx

Engineering Mechanics - Statics Chapter 4
b 4m=
c 1.5 m=
d 10.5 m=
Solution:
r
AB
b
a−
cd−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= r
OA
0
0
d
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠

=
r
OB
b
a−
c
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= F
v
F
r
AB
r
AB
=
M
O1
r
OA
F
v
×= M
O1

61.2
81.6
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
Nm⋅=
M
O2
r
OB
F
v
×= M
O2
61.2
81.6
0−
⎛
⎜
⎜
⎝
⎞
⎟
⎟

⎠
Nm⋅=
Problem 4-43
Determine the smallest force F that must be applied along the rope in order to cause the curved
rod, which has radius r, to fail at the support C. This requires a moment to be developed at C of
magnitude M.
241
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Engineering Mechanics - Statics Chapter 4
Given:
r 5ft=
M 80 lb ft⋅=
θ
60 deg=
a 7ft=
b 6ft=
Solution:
r
AB
b
arsin
θ
()
−
r− cos
θ
()
⎛

⎜
⎜
⎝
⎞
⎟
⎟
⎠
= u
AB
r
AB
r
AB
= r
CB
b
a
r−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=
Guess F 1lb=
Given r
CB

F u
AB
()
× M= F Find F()= F 18.6 lb=
Problem 4-44
The pipe assembly is subjected to the

force
F
. Determine the moment of this force about
point A.
242
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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.
Engineering Mechanics - Statics Chapter 4
Given:
F 80 N=
a 400 mm=
b 300 mm=
c 200 mm=
d 250 mm=
θ
40 deg=
φ
30 deg=
Solution:
r
AC
bd+

a
c−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= F
v
F
cos
φ
()
sin
θ
()
cos
φ
()
cos
θ
()
sin
φ
()
−
⎛

⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠
=
M
A
r
AC
F
v
×= M
A
5.385−
13.093
11.377
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
Nm⋅=

Problem 4-45
The pipe assembly is subjected to the force
F
. Determine the moment of this force about
point B.
243
© 2007 R. C. Hibbeler. Published by 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.
Engineering Mechanics - Statics Chapter 4
Given:
F 80 N=
a 400 mm=
b 300 mm=
c 200 mm=
d 250 mm=
θ
40 deg=
φ
30 deg=
Solution:
r
BC
bd+
0
c−
⎛
⎜
⎜
⎝

⎞
⎟
⎟
⎠
= r
BC
550
0
200−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
mm=
F
v
F
cos
φ
()
sin
θ
()
cos
φ
()

cos
θ
()
sin
φ
()
−
⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠
= F
v
44.534
53.073
40−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠

N=
M
B
r
BC
F
v
×= M
B
10.615
13.093
29.19
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
Nm⋅=
Problem 4-46
The x-ray machine is used for medical diagnosis. If the camera and housing at C have mass M
and a mass center at G, determine the moment of its weight about point O when it is in the
position shown.
Units Used:
kN 10
3
N=
244

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be reproduced, in any form or by any means, without permission in writing from the publisher.
Engineering Mechanics - Statics Chapter 4
Given:
M 150 kg=
a 1.2 m=
b 1.5 m=
θ
60 deg=
g 9.81
m
s
2
=
Solution:
M
O
b()− cos
θ
()
a
b( )sin
θ
()
⎡
⎢
⎢
⎣
⎤

⎥
⎥
⎦
0
0
M− g
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
×= M
O
1.77−
1.1−
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
kN m⋅=
Problem 4-47
Using Cartesian vector analysis, determine the resultant moment of the three forces about the base

of the column at A.
Units Used
:
kN 10
3
N=
Given:
F
1
400
300
120
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
N=
F
2
100
100−
60−
⎛
⎜
⎜
⎝

⎞
⎟
⎟
⎠
N=
F
3
0
0
500−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
N=
a 4m=
b 8m=
245
© 2007 R. C. Hibbeler. Published by 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.
Engineering Mechanics - Statics Chapter 4
c 1m=
Solution:
r
AB

0
0
ab+
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= r
A3
0
c−
b
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=
The individual moments
M
A1
r
AB

F
1
×= M
A2
r
AB
F
2
×= M
A3
r
A3
F
3
×=
M
A1
3.6−
4.8
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
kN m⋅= M
A2

1.2
1.2
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
kN m⋅= M
A3
0.5
0
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
kN m⋅=
The total moment
M
A
M
A1

M
A2
+ M
A3
+= M
A
1.9−
6
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
kN m⋅=
Problem 4-48
A force
F
produces a moment
M
O
about the origin of coordinates, point O. If the force acts at a
point having the given x coordinate, determine the y and z coordinates.
Units Used
:
kN 10
3

N=
Given
:
F
6
2−
1
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
kN=
M
O
4
5
14−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
kN m⋅=

x 1m=
Solution:
The initial guesses:
y 1m= z 1m=
246
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be reproduced, in any form or by any means, without permission in writing from the publisher.
Engineering Mechanics - Statics Chapter 4
Given
x
y
z
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
F× M
O
=
y
z
⎛
⎜
⎝
⎞

⎟
⎠
Find yz,()=
y
z
⎛
⎜
⎝
⎞
⎟
⎠
2
1
⎛
⎜
⎝
⎞
⎟
⎠
m=
Problem 4-49
The force
F
creates a moment about point O of
M
O
. If the force passes through a point
having the given x coordinate, determine the y and z coordinates of the point. Also, realizing
that M
O

= Fd, determine the perpendicular distance d from point O to the line of action of
F
.
Given:
F
6
8
10
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
N=
M
O
14−
8
2
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠

Nm⋅=
x 1m=
Solution:
The initial guesses:
y 1m= z 1m=
Given
x
y
z
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
F× M
O
=
y
z
⎛
⎜
⎝
⎞
⎟
⎠
Find yz,()=
y

z
⎛
⎜
⎝
⎞
⎟
⎠
1
3
⎛
⎜
⎝
⎞
⎟
⎠
m=
d
M
O
F
= d 1.149 m=
Problem 4-50
The force
F
produces a moment
M
O
about the origin of coordinates, point O. If the force acts
at a point having the given x-coordinate, determine the y and z coordinates.
247

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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.
Engineering Mechanics - Statics Chapter 4
Units Used:
kN 10
3
N=
Given:
x 1m=
F
6
2−
1
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
kN=
M
O
4
5
14−
⎛
⎜

⎜
⎝
⎞
⎟
⎟
⎠
kN m⋅=
Solution:
Initial Guesses:
y 1m= z 1m=
Given M
O
x
y
z
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
F×=
y
z
⎛
⎜
⎝
⎞

⎟
⎠
Find yz,()=
y
z
⎛
⎜
⎝
⎞
⎟
⎠
2
1
⎛
⎜
⎝
⎞
⎟
⎠
m=
Problem 4-51
Determine the moment of the force
F
about the Oa axis. Express the result as a Cartesian vector.
Given:
F
50
20−
20
⎛

⎜
⎜
⎝
⎞
⎟
⎟
⎠
N=
a 6m=
b 2m=
c 1m=
d 3m=
e 4m=
248
© 2007 R. C. Hibbeler. Published by 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.
Engineering Mechanics - Statics Chapter 4
r
OF
c
b−
a
⎛
⎜
⎜
⎝
⎞
⎟
⎟

⎠
= r
Oa
0
e
d
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= u
Oa
r
Oa
r
Oa
=
M
Oa
r
OF
F×
()
u
Oa
⋅

⎡
⎣
⎤
⎦
u
Oa
= M
Oa
0
217.6
163.2
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
Nm⋅=
Problem 4-52
Determine the moment of the force
F
about the aa axis. Express the result as a Cartesian vector.
Given:
F 600 lb=
a 6ft=
b 3ft=
c 2ft=
d 4ft=

e 4ft=
f 2ft=
Solution:
F
v
F
c
2
d
2
+ e
2
+
d−
e
c
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= r
d
0
c−
⎛
⎜

⎜
⎝
⎞
⎟
⎟
⎠
= u
aa
1
a
2
b
2
+ f
2
+
b−
f−
a
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=
M
aa

rF
v
×
()
u
aa
⋅
⎡
⎣
⎤
⎦
u
aa
= M
aa
441−
294−
882
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
lb ft⋅=
249
Solution:
© 2007 R. C. Hibbeler. Published by 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.
Engineering Mechanics - Statics Chapter 4
Problem 4-53
Determine the resultant moment of the two forces about the Oa axis. Express the result as a
Cartesian vector.
Given:
F
1
80 lb=
F
2
50 lb=
α
120 deg=
β
60 deg=
γ
45 deg=
a 5ft=
b 4ft=
c 6ft=
θ
30 deg=
φ
30 deg=
Solution:
F
1v
F

1
cos
α
()
cos
β
()
cos
γ
()
⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠
= F
2v
0
0
F
2
⎛
⎜
⎜
⎜

⎝
⎞
⎟
⎟
⎟
⎠
=
r
1
b( )sin
θ
()
b( )cos
θ
()
c
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
= r
2
0
a()− sin
φ
()

0
⎡
⎢
⎢
⎣
⎤
⎥
⎥
⎦
=
u
aa
cos
φ
()
sin
φ
()
−
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= M
aa

r
1
F
1v
× r
2
F
2v
×+
()
u
aa
⎡
⎣
⎤
⎦
u
aa
=
M
aa
26.132
15.087−
0
⎛
⎜
⎜
⎝
⎞
⎟

⎟
⎠
lb ft⋅=
250
© 2007 R. C. Hibbeler. Published by 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.
Engineering Mechanics - Statics Chapter 4
The force
F
is applied to the handle of the box wrench. Determine the component of the
moment of this force about the z axis which is effective in loosening the bolt.
Given:
a 3in=
b 8in=
c 2in=
F
8
1−
1
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
lb=
Solution:

k
0
0
1
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= r
c
b−
a
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= M
z
rF×
()
k⋅= M
z

62lb in⋅=
Problem 4-55
The force
F
acts on the gear in the direction shown. Determine the moment of this force about
the y axis.
Given:
F 50 lb=
a 3in=
θ
1
60 deg=
θ
2
45 deg=
θ
3
120 deg=
251
Problem 4-54
© 2007 R. C. Hibbeler. Published by 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.
Engineering Mechanics - Statics Chapter 4
j
0
1
0
⎛
⎜

⎜
⎝
⎞
⎟
⎟
⎠
= r
0
0
a
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= F
v
F
cos
θ
3
( )
−
cos
θ
2
()

−
cos
θ
1
()
−
⎛
⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎠
= M
y
rF
v
×
()
j⋅= M
y
75lb in⋅=
Problem 4-56
The RollerBall skate is an in-line tandem skate that uses two large spherical wheels on each
skate, rather than traditional wafer-shape wheels. During skating the two forces acting on
the wheel of one skate consist of a normal force
F

2
and a friction force
F
1
. Determine the
moment of both of these forces about the axle AB of the wheel.
Given:
θ
30 deg=
F
1
13 lb=
F
2
78 lb=
a 1.25 in=
Solution:
F
F
1
F
2
0
⎛
⎜
⎜
⎜
⎝
⎞
⎟

⎟
⎟
⎠
= r
0
a−
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=
ab
cos
θ
()
sin
θ
()
−
0
⎛
⎜
⎜
⎝
⎞

⎟
⎟
⎠
= M
ab
rF×
()
ab⋅= M
ab
0lb in⋅=
Problem 4-57
The cutting tool on the lathe exerts a force
F
on the shaft in the direction shown. Determine the
moment of this force about the y axis of the shaft.
252
Solution:
© 2007 R. C. Hibbeler. Published by 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.
Engineering Mechanics - Statics Chapter 4
Units Used:
kN 10
3
N=
Given:
F
6
4−
7−

⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
kN=
a 30 mm=
θ
40 deg=
Solution:
r a
cos
θ
()
0
sin
θ
()
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= j

0
1
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= M
y
rF×
()
j⋅= M
y
0.277 kN m⋅=
Problem 4-58
The hood of the automobile is supported by the strut AB, which exerts a force
F
on the hood.
Determine the moment of this force about the hinged axis y.
Given:
F 24 lb= a 2ft= b 4ft= c 2ft= d 4ft=
253
© 2007 R. C. Hibbeler. Published by 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.
Engineering Mechanics - Statics Chapter 4

Solution:
r
A
b
0
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= r
AB
b− c+
a
d
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= F
v
F

r
AB
r
AB
= F
v
9.798−
9.798
19.596
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
lb=
j
0
1
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠

= M
y
r
A
F
v
×
()
j⋅= M
y
78.384− lb ft⋅=
Problem 4-59
The lug nut on the wheel of the automobile is to be removed using the wrench and applying the
vertical force
F
at A. Determine if this force is adequate, provided a torque M about the x axis is
initially required to turn the nut. If the force
F
can be applied at A in any other direction, will it
be possible to turn the nut?
254
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Engineering Mechanics - Statics Chapter 4
Given:
F 30 N=
M 14 N m⋅=
a 0.25 m=
b 0.3 m=

c 0.5 m=
d 0.1 m=
Solution:
M
x
Fc
2
b
2
−=
M
x
12N m⋅=
M
x
M<
No
For M
xmax
, apply force perpendicular to the handle and the x-axis.
M
xmax
Fc=
M
xmax
15N m⋅=
M
xmax
M>
Yes

Problem 4-60
The lug nut on the wheel of the automobile is to be removed using the wrench and applying the
vertical force
F
. Assume that the cheater pipe AB is slipped over the handle of the wrench and
the
F
force can be applied at any point and in any direction on the assembly. Determine if this
force is adequate, provided a torque M about the x axis is initially required to turn the nut.
Given:
F
1
30 N= M 14 N m⋅= a 0.25 m= b 0.3 m= c 0.5 m= d 0.1 m=
255
© 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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Engineering Mechanics - Statics Chapter 4
Solution:
M
x
F
1
ac+
c
c
2
b
2
−=

M
x
18N m⋅=
M
x
M>
Yes
M
xmax
occurs when force is applied perpendicular to both the handle and the x-axis.

M
xmax
F
1
ac+()=
M
xmax
22.5 N m⋅=
M
xmax
M>
Yes
Problem 4-61
The bevel gear is subjected to the force
F
which is caused from contact with another
gear. Determine the moment of this force
about the y axis of the gear shaft.
Given:

a 30 mm=
b 40 mm=
F
20
8
15−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
N=
256
© 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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Engineering Mechanics - Statics Chapter 4
r
b−
0
a
⎛
⎜
⎜
⎝
⎞
⎟

⎟
⎠
= j
0
1
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= M
y
rF×
()
j⋅= M
y
0N m⋅=
Problem 4-62
The wooden shaft is held in a lathe.
The cutting tool exerts force
F
on the
shaft in the direction shown.
Determine the moment of this force
about the x axis of the shaft. Express
the result as a Cartesian vector. The

distance OA is a.
Given:
a 25 mm=
θ
30 deg=
F
5−
3−
8
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
N=
Solution:
r
0
a( )cos
θ
()
a( )sin
θ
()
⎡
⎢
⎢

⎣
⎤
⎥
⎥
⎦
= i
1
0
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= M
x
rF×
()
i⋅
⎡
⎣
⎤
⎦
i= M
x
0.211
0

0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
Nm⋅=
Problem 4-63
Determine the magnitude of the moment of the force
F
about the base line CA of the tripod.
Given:
F
50
20−
80−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
N=
257
Solution:

© 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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Engineering Mechanics - Statics Chapter 4
a 4m=
b 2.5 m=
c 1m=
d 0.5 m=
e 2m=
f 1.5 m=
g 2m=
Solution:
r
CA
g−
e
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= u
CA
r
CA
r

CA
= r
CD
bg−
e
a
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= M
CA
r
CD
F×
()
u
CA
⋅= M
CA
226N m⋅=
Problem 4-64
The flex-headed ratchet wrench is
subjected to force
P
, applied

perpendicular to the handle as shown.
Determine the moment or torque this
imparts along the vertical axis of the
bolt at A.
Given:
P 16 lb= a 10 in=
θ
60 deg= b 0.75 in=
Solution:
MPba( )sin
θ
()
+
⎡
⎣
⎤
⎦
= M 150.564 lb in⋅=
258
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be reproduced, in any form or by any means, without permission in writing from the publisher.
Engineering Mechanics - Statics Chapter 4
Problem 4-65
If a torque or moment M is
required to loosen the bolt at A,
determine the force
P
that must be
applied perpendicular to the handle

of the flex-headed ratchet wrench.
Given:
M 80 lb in⋅=
θ
60 deg=
a 10 in=
b 0.75 in=
Solution:
MPb a( )sin
θ
()
+
⎡
⎣
⎤
⎦
= P
M
ba( )sin
θ
()
+
= P 8.50 lb=
Problem 4-66
The A-frame is being hoisted into
an upright position by the vertical
force
F
. Determine the moment of
this force about the y axis when

the frame is in the position shown.
Given:
F 80 lb=
a 6ft=
b 6ft=
θ
30 deg=
φ
15 deg=
Solution:
Using the primed coordinates we have
259
© 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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be reproduced, in any form or by any means, without permission in writing from the publisher.
Engineering Mechanics - Statics Chapter 4
j
sin
θ
()
−
cos
θ
()
0
⎛
⎜
⎜
⎝
⎞

⎟
⎟
⎠
= F
v
F
0
0
1
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
= r
AC
b− cos
φ
(
)
a
2
bsin
φ
()
⎛
⎜

⎜
⎜
⎜
⎝
⎞
⎟
⎟
⎟
⎟
⎠
=
M
y
r
AC
F
v
×
()
j⋅= M
y
281.528 lb ft⋅=
Problem 4-67
Determine the moment of each force acting on the handle of the wrench about the a axis.
Given:
F
1
2−
4
8−

⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
lb= F
2
3
2
6−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
lb=
b 6in=
c 4in=
d 3.5 in=
θ
45 deg=
Solution:
u
a

cos
θ
()
0
sin
θ
()
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=
r
1
b
cos
θ
()
0
sin
θ
()
⎛
⎜
⎜
⎝

⎞
⎟
⎟
⎠
cd+()
sin
θ
()
0
cos
θ
()
−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
+= r
2
b
cos
θ
()
0
sin
θ

()
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
c
sin
θ
()
0
cos
θ
()
−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
+=
M
1a
r

1
F
1
×
()
u
a
⋅= M
1a
30lb in⋅=
260
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be reproduced, in any form or by any means, without permission in writing from the publisher.
Engineering Mechanics - Statics Chapter 4
M
2a
r
2
F
2
×
()
u
a
⋅= M
2a
8lb in⋅=
Problem 4-68
Determine the moment of each force acting on the handle of the wrench about the z axis.

Given:
F
1
2−
4
8−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
lb= F
2
3
2
6−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
lb=
b 6in=
c 4in=

d 3.5 in=
θ
45 deg=
Solution:
r
1
b
cos
θ
()
0
sin
θ
()
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
cd+()
sin
θ
()
0
cos
θ
()

−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
+= r
2
b
cos
θ
()
0
sin
θ
()
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
c
sin
θ

()
0
cos
θ
()
−
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
+= k
0
0
1
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
=
M
1z
r

1
F
1
×
()
k⋅= M
1z
38.2 lb in⋅=
M
2z
r
2
F
2
×
()
k⋅= M
2z
14.1 lb in⋅=
Problem 4-69
Determine the magnitude and sense of the couple moment.
Units Used:
kN 10
3
N=
261
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be reproduced, in any form or by any means, without permission in writing from the publisher.
Engineering Mechanics - Statics Chapter 4

Given:
F 5kN=
θ
30 deg=
a 0.5 m=
b 4m=
c 2m=
d 1m=
Solution:
M
C
F cos
θ
()
ac+()F sin
θ
()
bd−()+=
M
C
18.325 kN m⋅=
Problem 4-70
Determine the magnitude and sense of the couple moment. Each force has a magnitude F
.
Given:
F 65 lb=
a 2ft=
b 1.5 ft=
c 4ft=
d 6ft=

e 3ft=
Solution:
M
c
=

Σ
M
B;

M
C
F
c
c
2
e
2
+
⎛
⎜
⎝
⎞
⎟
⎠
da+()
⎡
⎢
⎣
⎤

⎥
⎦
F
e
c
2
e
2
+
⎛
⎜
⎝
⎞
⎟
⎠
ca+()
⎡
⎢
⎣
⎤
⎥
⎦
+=
M
C
650lb ft⋅=
(Counterclockwise)
Problem 4-71
Determine the magnitude and sense of the couple moment.
262

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Engineering Mechanics - Statics Chapter 4
Units Used:
kip 10
3
lb=
Given:
F 150 lb=
a 8ft=
b 6ft=
c 8ft=
d 6ft=
e 6ft=
f 8ft=
Solution:
M
C
=
Σ
M
A
;
M
C
F
d
d
2

f
2
+
af+()F
f
d
2
f
2
+
cd+()+=
M
C
3120lb ft⋅= M
C
3.120 kip ft⋅=
Problem 4-72
If the couple moment has magnitude
M, determine the magnitude F of the
couple forces.
Given:
M 300 lb ft⋅=
a 6ft=
b 12 ft=
c 1ft=
d 2ft=
e 12 ft=
f 7ft=
263
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be reproduced, in any form or by any means, without permission in writing from the publisher.
Engineering Mechanics - Statics Chapter 4
Solution:
MF
ef a+()
fd−()
2
e
2
+
fd−()be+()
fd−()
2
e
2
+
−
⎡
⎢
⎣
⎤
⎥
⎦
=
F
M
ef a+()
fd−()
2

e
2
+
fd−()be+()
fd−()
2
e
2
+
−
= F 108lb=
Problem 4-73
A clockwise couple M is resisted by the shaft of the electric motor. Determine the magnitude of
the reactive forces
R−
and
R
which act at supports A and B so that the resultant of the two
couples is zero.
Given:
a 150 mm=
θ
60 deg=
M 5Nm⋅=
Solution:
M
C
M−
2Ra
tan

θ
()
+= 0= R
M
2
tan
θ
()
a
= R 28.9 N=
Problem 4-74
The resultant couple moment created by the two couples acting on the disk is
M
R
. Determine
the magnitude of force
T
.
264
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Engineering Mechanics - Statics Chapter 4
Units Used:
kip 10
3
lb=
Given:
M
R

0
0
10
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
kip in⋅=
a 4in=
b 2in=
c 3in=
Solution:
Initial Guess
T 1 kip=
Given
a
0
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠

0
T
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
×
b−
0
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
0
T−
0
⎛
⎜
⎜
⎝

⎞
⎟
⎟
⎠
×+
b− c−
0
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
0
T−
0
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
×+ M
R
=

T Find T()= T 0.909kip=
Problem 4-75
Three couple moments act on the pipe assembly. Determine the magnitude of M
3
and the
bend angle
θ
so that the resultant couple moment is zero.

Given:
θ
1
45 deg=
M
1
900 N m⋅=
M
2
500 N m⋅=
265
© 2007 R. C. Hibbeler. Published by Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved.
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