Problem 4.107* The yaxis points upward. The weight
of the 4-kg rectangular plate acts at the midpoint Gof
the plate. The sum of the moments about the straight
line through the supports Aand Bdue to the weight of
the plate and the force exerted on the plate by the cable
CD is zero. What is the tension in the cable?
C(200, 55, 390) mm
(0, 180, 360) mm
z
y
Solution: Note that the coordinates of point Gare
(150, 152.5, 195).
We calculate the moment about the line BA due to the two forces
as follows.
eBA D0.1iC0.07j0.36k
p0.1445
r1D⊲0.2i0.125jC0.03k⊳m,
MBA D3.871 N-m ⊲0.17793 m⊳TCD D0)TCD D21.8N
Problem 4.108 In Active Example 4.9, suppose that
the point of application of the force Fis moved from
(8, 3, 0) m to (8, 8, 0) m. Draw a sketch showing the new
position of the force. From your sketch, will the moment
y
(6, 6, 0) m
ⴚF
Problem 4.109 The forces are contained in the xy
plane.
(a) Determine the moment of the couple and represent
it as shown in Fig. 4.28c.
(b) What is the sum of the moments of the two forces
about the point (10, 40, 20) ft?
Solution: The right hand force is
FD[1000 ⊲lb⊳]⊲cos 60°isin 60°j⊳
FDC500i867jlb.
The vector from the xintercept of the left force to that of the right
Problem 4.110 The moment of the couple is 600 k
(N-m). What is the angle ˛?
100 N
100 N
(0, 4) m
(5, 0) m
a
a
x
Solution:
Problem 4.111 Point Pis contained in the xy
plane, jFjD100 N, and the moment of the couple is
500k(N-m). What are the coordinates of P?
Solution: The force is
FD100⊲icos⊲30°⊳Cjsin⊲30°⊳⊳ D86.6i50j.
98.48 D5.077 m. From above,
xD5.077⊲0.3420⊳D1.74 m,yD5.077⊲0.9397⊳D4.77 m
220
c
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Problem 4.112 Three forces of equal magnitude are
applied parallel to the sides of an equilateral triangle.
(a) Show that the sum of the moments of the forces is
the same about any point. (b) Determine the magnitude
L
F
Solution:
Problem 4.113 In Example 4.10, suppose that the 200
ft-lb couple is counterclockwise instead of clockwise.
Draw a sketch of the beam showing the forces and
couple acting on it. What are the forces Aand B?
y
AB
200 ft-lb
Problem 4.115 Determine the sum of the moments
exerted on the plate by the two couples.
30 lb
y
Solution: The moment due to the 30 lb couple, which acts in a
clockwise direction is
M30 D3⊲30⊳kD90kft lb.
Problem 4.116 Determine the sum of the moments
exerted about Aby the couple and the two forces.
100 lb
400 lb
Solution: Let the xaxis point to the right and the yaxis point
upward in the plane of the page. The moments of the forces are
M100 D⊲3i⊳ð⊲100j⊳D300k(ft-lb),
Problem 4.117 Determine the sum of the moments
exerted about Aby the couple and the two forces.
0.2 m 0.2 m0.2 m
300 N-m
Solution:
Problem 4.118 The sum of the moments about point
Adue to the forces and couples acting on the bar is zero.
(a) What is the magnitude of the couple C?
Solution:
(a) MAD20 kN-m ⊲2kN⊳⊲5m⊳⊲4kN⊳⊲3m⊳
222
Problem 4.119 In Example 4.11, suppose that instead
of acting in the positive zdirection, the upper 20-N force
acts in the positive xaxis direction. Instead of acting in
the negative zaxis direction, let the lower 20-N force
y
20 N 30 N
Problem 4.120 (a) What is the moment of the couple?
y
FD18.97 kN-m
3kN D6.32 m
Problem 4.121 Determine the sum of the moments
exerted on the plate by the three couples. (The 80-lb
forces are contained in the xzplane.)
z60°80 lb 60°80 lb
40 lb
40 lb
8 ft
x
y
3 ft
20 lb 20 lb
3 ft
M2D⊲8⊳⊲40⊳jD320jft lb
FD⊲80⊳⊲isin 60°Ckcos 60°⊳D69.282iC40k
The two forces in the third couple are separated by the vector
r3D⊲6iC8k⊳⊲8k⊳D6i
69.282 0 40
D240j.
The sum of the moments due to the couples:
Problem 4.122 What is the magnitude of the sum of
the moments exerted on the T-shaped structure by the
two couples?
z
y
x
50i + 20j – 10k (lb)
–50i – 20j + 10k (lb)
50j (lb)
–50j (lb)
3 ft
3 ft
3 ft
224
Problem 4.123 The tension in cables AB and CD is
500 N.
(a) Show that the two forces exerted by the cables on
the rectangular hatch at Band Cform a couple.
(b) What is the moment exerted on the plate by the
cables?
3 m
A(0, 2, 0) m
y
z
B
Problem 4.124 The cables AB and CD exert a couple
z
y
(⫺1.6, 2.2, ⫺1.2) m
(1.6, 0, 1.2) m
B
Problem 4.125 The bar is loaded by the forces
FBD2iC6jC3k(kN),
FCDi2jC2k(kN),
and the couple
Determine the sum of the moments of the two forces
and the couple about A.
1 m
y
x
B
F
B
A
C
M
C
Problem 4.126 In Problem 4.125, the forces
FBD2iC6jC3k(kN),
Solution: From the solution to Problem 4.125, the sum of the
moments of the two forces about Ais
226
Problem 4.127 Two wrenches are used to tighten an
elbow fitting. The force FD10k(lb) on the right
wrench is applied at (6, 5, 3) in, and the force F
on the left wrench is applied at (4, 5, 3) in.
(a) Determine the moment about the xaxis due to the
forces exerted on the two wrenches.
(c) Based on the results of (a) and (b), explain why
two wrenches are used.
y
jMRjDeXÐ⊲rRðF⊳D
653
00 10
D50 in lb
(a) The moment about the xaxis is
MRDjMRjeXD50i(in lb).
of the elbow and the pipe. This is demonstrated by calculating
the moment about the xaxis due to the left wrench:
jMXjDeXÐ⊲rLðFL⊳D
10 0
453
right wrench. The left wrench force is applied 2 in nearer the
absorbed by the space between, where it is wanted.
Problem 4.128 Two systems of forces act on the beam.
Are they equivalent?
228
Problem 4.129 Two systems of forces and moments
act on the beam. Are they equivalent?
50 ft-lb
System 1
10 lb
x
y
20 lb
Solution: The sums of the forces are:
Problem 4.130 Four systems of forces and moments
act on an 8-m beam. Which systems are equivalent?
Solution: For equivalence, the sum of the forces and the sum of
the moments about some point (the left end will be used) must be
4.130 can be made equivalent by adding a
couple to one of the systems. Which system is it, and
what couple must be added?
FD10jkN
Problem 4.132 System 1 is a force Facting at a point
O. System 2 is the force Facting at a different point O0
along the same line of action. Explain why these systems
230
Problem 4.133 The vector sum of the forces exerted
on the log by the cables is the same in the two cases.
Show that the systems of forces exerted on the log are
12 m
16 m
A
D
C
E
T1DjTj⊲icos 143.13°Cjsin 143.13°⊳
DjTj⊲0.8iC0.6j⊳.
0.4191j⊳DjT1j⊲0.6500iC0.3000⊳. The position vector of the right
end of the log is rRD20im relative to the left end. The moments
about the left end of the log for the second system are
ijk
Problem 4.134 Systems 1 and 2 each consist of a
couple. If they are equivalent, what is F?
System 1 System 2
y
20°
y
200 N
The moment due to the couple in System 1 is
For System 2, the positions of the forces are r21 D2i, and r22 D
4.7848 D264.6N
Problem 4.135 Two equivalent systems of forces and
moments act on the L-shaped bar. Determine the forces
FAand FBand the couple M.
50 N
60 N
System 1
120 N-m
40 N
System 2
3 m M
FA
FB
3 m
232
Problem 4.136 Two equivalent systems of forces and
moments act on the plate. Determine the force Fand
the couple M.
5 in
30 lb 10 lb
System 1
5 in
30 lb
System 2
100 in–lb
234
Problem 4.138 Three forces and a couple are applied
to a beam (system 1).
(a) If you represent system 1 by a force applied at A
and a couple (system 2), what are Fand M?
(b) If you represent system 1 by the force F(system 3),
what is the distance D?
y
System 1
Problem 4.139 Represent the two forces and couple
mine where its line of action intersects the xaxis.
y
–40j (N)
3 m 3 m
x
Solution: We first represent the system by an equivalent system
consisting of a force Fat the origin and a couple M:
We then represent this system by an equivalent system consisting of
Falone:
For equivalence, MDd⊲Fy⊳,so
dDM
FyD80
20 D4m.
y
d
x
Problem 4.140 The bracket is subjected to three forces
and a couple. If you represent this system by a force F,
what is F, and where does its line of action intersect the
xaxis?
0.65 m
0.2 m
400 N
180 N
y
x
Solution: We locate a single equivalent force along the xaxis a
distance dto the right of the origin. We must satisfy the following
236
Problem 4.141 The vector sum of the forces acting on
the beam is zero, and the sum of the moments about the
left end of the beam is zero.
(a) Determine the forces Axand Ay, and the couple
MA.
(b) Determine the sum of the moments about the right
end of the beam.
(c) If you represent the 600-N force, the 200-N force,
and the 30 N-m couple by a force Facting at the
left end of the beam and a couple M, what are F
and M?
Solution: (a) The sum of the forces is
FXDAXiD0 and
FYD⊲AY600 C200⊳jD0,
from which AYD400 N. The sum of the moments is
MLD⊲MA0.38⊲600⊳30 C0.560⊲200⊳⊳kD0,
from which MAD146 N-m. (b) The sum of the moments about the
Problem 4.142 The vector sum of the forces acting on
the truss is zero, and the sum of the moments about the
origin Ois zero.
(a) Determine the forces Ax,Ay, and B.
(b) If you represent the 2-kip, 4-kip, and 6-kip forces
by a force F, what is F, and where does its line of
action intersect the yaxis?
(c) If you replace the 2-kip, 4-kip, and 6-kip forces by
the force you determined in (b), what are the vector
sum of the forces acting on the truss and the sum
of the moments about O?
Solution: (a) The sum of the forces is
FXD⊲AX246⊳iD0,
from which AXD12 kip
FYD⊲AYCB⊳jD0.
The sum of the moments about the origin is
MOD⊲3⊳⊲6⊳C⊲6⊳⊲4⊳C⊲9⊳⊲2⊳C6⊲B⊳ D0,
Problem 4.143 The distributed force exerted on part
of a building foundation by the soil is represented by
five forces. If you represent them by a force F, what is
F, and where does its line of action intersect the xaxis?
y
Solution: The equivalent force must equal the sum of the forces
Problem 4.144 At a particular instant, aerodynamic
forces distributed over the airplane’s surface exert the
88-kN and 16-kN vertical forces and the 22 kN-m
counterclockwise couple shown. If you represent these
forces and couple by a system consisting of a force F
acting at the center of mass Gand a couple M, what are
Fand M?
5 m
88 kN
y
x
16 kN
G
Problem 4.145 If you represent the two forces and
couple acting on the airplane in Problem 4.144 by a
force F, what is F, and where does its line of action
intersect the xaxis?
238