Problem 4.75 The 200-kg slider at Ais held in place
on the smooth vertical bar by the cable AB. Determine
the moment about the bottom of the bar (point Cwith
coordinates xD2m,yDzD0) due to the force exerted
on the slider by the cable.
z
y
x
A
C
B
2 m
5 m
2 m
2 m
Solution: The slider is in equilibrium. The smooth bar exerts no
vertical forces on the slider. Hence, the vertical component of FAB
supports the weight of the slider.
FAB
Problem 4.76 To evaluate the adequacy of the design
of the vertical steel post, you must determine the moment
about the bottom of the post due to the force exerted on
the post at Bby the cable AB. A calibrated strain gauge
mounted on cable AC indicates that the tension in cable
AC is 22 kN. What is the moment?
y
x
A
B
D
O
C
(6, 2, 0) m
z
12 m
3 m
4 m
5 m
5 m
8 m
Solution: To find the moment, we must find the force in cable AB.
In order to do this, we must find the forces in cables AO and AD also.
This requires that we solve the equilibrium problem at A.
where itakes on the values B,C,D, and O.Weget
eAB D0.986iC0.164jC0k
We now write the forces as
TAB DTABeAB
TAD
C(0, 8, 5) m
D(0, 4, −5) m
O(0, 0, 0) m
In component form,
TABeABx CTACeACx CTADeADx CTAOeAOx D0
TABeABy CTACeACy CTADeADy CTAOeAOy D0
TAB D163.05 kN,T
AD D18.01 kN TAO D141.28 kN
We now know that TAB is given as
200
Problem 4.77 The force FD20iC40j10k(N). Use
both of the procedures described in Example 4.7 to deter-
mine the moment due to Fabout the zaxis.
y
x
F
Solution: First Method: We can use Eqs. (4.5) and (4.6)
rD⊲8i⊳m
Second Method: The y-component of the force is perpendicular to the
plane containing the zaxis and the position vector r. The perpendicular
distance from the zaxis to the y-component of the force is 8 m.
Problem 4.78 Use Eqs. (4.5) and (4.6) to determine
the moment of the 20-N force about (a) the xaxis,
(b) the yaxis, (c) the zaxis. (First see if you can write
Solution: The force is parallel to the zaxis. The perpendicular
distance from the xaxis to the line of action of the force is 4 m. The
perpendicular distance from the yaxis is 7 m and the perpendicular
Problem 4.79 Three forces parallel to the yaxis act
on the rectangular plate. Use Eqs. (4.5) and (4.6) to
determine the sum of the moments of the forces about
the xaxis. (First see if you can write down the result
without using the equations.)
3 kN
x
y
z
6 kN
2 kN
900 mm
600 mm
Solution: By inspection, the 3 kN force has no moment about the
xaxis since it acts through the xaxis. The perpendicular distances of
020
M6kND
10 0
00.6
06 0
iD3.6ikN
Problem 4.80 Consider the rectangular plate shown in
Problem 4.79. The three forces are parallel to the y
axis. Determine the sum of the moments of the forces
(a) about the yaxis, (b) about the zaxis.
Solution: (a) The magnitude of the moments about the yaxis is
MDeYÐ⊲rðF⊳. The position vectors of the three forces are given
(b) The magnitude of each moment about the zaxis is
202
Problem 4.81 The person exerts a force FD0.2i
0.4jC1.2k(lb) on the gate at C. Point Clies in the xy
plane. What moment does the person exert about the
gate’s hinge axis, which is coincident with the yaxis?
y
3.5 ft
A
C
Solution:
.2.41.2
Problem 4.82 Four forces act on the plate. Their
components are
FAD2iC4jC2k(kN),
FBD3j3k(kN),
FCD2jC3k(kN),
FDD2iC6jC4k(kN).
Determine the sum of the moments of the forces
(a) about the xaxis; (b) about the zaxis.
FA
FDFC
FB
x
y
z3 m
2 m
Solution: Note that FAacts at the origin so no moment is generated
about the origin. For the other forces we have
ij k
ijk
Now we find
Problem 4.83 The force FD30iC20j10k(lb).
(a) What is the moment of Fabout the yaxis?
(b) Suppose that you keep the magnitude of Ffixed,
but you change its direction so as to make the
moment of Fabout the yaxis as large as possible.
What is the magnitude of the resulting moment?
z
x
(4, 2, 2) ft
F
y
Solution:
(a) MyDjÐ[⊲4iC2jC2k⊳ft ð⊲30iC20j10k⊳lb]
01 0
Problem 4.84 The moment of the force Fshown in
Problem 4.83 about the xaxis is 80i(ft-lb), the moment
about the yaxis is zero, and the moment about the zaxis
is 160k(ft-lb). If FyD80 lb, what are Fxand Fz?
204
Problem 4.85 The robotic manipulator is stationary.
The weights of the arms AB and BC act at their
midpoints. The direction cosines of the centerline of arm
AB are cos xD0.500, cos yD0.866, cos zD0, and
the direction cosines of the centerline of arm BC are
cos xD0.707, cos yD0.619, cos zD0.342. What
total moment is exerted about the zaxis by the weights
of the arms?
y
600 mm
C
B
600 mm
160 N
The weight vectors acting at Gand Hare WGD200jN, and WHD
160jN. The moment vectors of these forces about the zaxis are of
the form
eXeyez
Problem 4.86 In Problem 4.85, what total moment is
exerted about the xaxis by the weights of the arms?
Problem 4.87 In Active Example 4.6, suppose that the
force changes to FD2iC3jC6k(kN). Determine
the magnitude of the moment of the force about the axis
of the bar BC.
y
F ⫽ ⫺2i ⫹ 6j ⫹ 3k (kN)
x
B
C
A (4, 2, 2) m
(0, 4, 0) m
Solution: We have the following vectors
rBA D⊲4iC2j1k⊳m
The moment of Fabout the axis of the bar is
00.80.6
206
Problem 4.88 Determine the moment of the 20-N force
about the line AB. Use Eqs. (4.5) and (4.6), letting the unit
vector epoint (a) from Atoward B, (b) from Btoward A.
(–4, 0, 0) m
B
A(0, 5, 0) m (7, 4, 0) m
20k (N)
y
x
z
rD⊲70⊳iC⊲45⊳jC⊲00⊳km
rD7i1jC0km
Problem 4.89 The force FD10iC5j5k(kip).
Determine the moment of Fabout the line AB. Draw
a sketch to indicate the sense of the moment.
(6, 6, 0) ft
B
y
Solution: The moment of Fabout pt. Ais
F
Problem 4.90 The force FD10iC12j6k(N).
What is the moment of Fabout the line OA? Draw a
sketch to indicate the sense of the moment.
y
(0, 6, 4) m
A
Solution: The strategy is to determine a unit vector parallel to OA
Problem 4.91 The tension in the cable AB is 1 kN.
Determine the moment about the xaxis due to the force
exerted on the hatch by the cable at point B. Draw a
sketch to indicate the sense of the moment.
y
x
A(400, 300, 0) mm
Solution: The vector parallel to BA is
The moment about Ois
ij k
The magnitude is
208
Problem 4.92 Determine the moment of the force app-
lied at Dabout the straight line through the hinges A
and B. (The line through Aand Blies in the yzplane.)
20i – 60j (lb)
A
E
y
6 ft
Solution: From the figure, we see that the unit vector along the
Problem 4.93 In Problem 4.92, the tension in the
cable CE is 160 lb. Determine the moment of the force
exerted by the cable on the hatch at Cabout the straight
line through the hinges Aand B.
Solution: From the figure, we see that the unit vector along the line
Problem 4.94 The coordinates of Aare (2.4, 0,
0.6) m, and the coordinates of Bare (2.2, 0.7,
1.2) m. The force exerted at Bby the sailboat’s main
sheet AB is 130 N. Determine the moment of the force
about the centerline of the mast (the yaxis). Draw a
sketch to indicate the sense of the moment.
y
x
B
Solution: The position vectors:
210
Problem 4.95 The tension in cable AB is 200 lb.
Determine the moments about each of the coordinate
axes due to the force exerted on point Bby the cable.
Draw sketches to indicate the senses of the moments.
(2, 5, –2) ft
A
(10, –2, 3) ftB
x
y
z
Solution: The position vector from Bto Ais
rOB ðFD
ijk
10 23
136.2 119.285.1
D187iC443jC919k⊲ft-lb⊳.
The moments about the x,y, and zaxes are
[⊲rOB ðF⊳Ði]iD187i⊲ft-lb⊳,
[⊲rOB ðF⊳Ðj]jD443j⊲ft-lb⊳,
[⊲rOB ðF⊳Ðk]kD919k⊲ft-lb⊳.
y
443 ft-lb
Problem 4.96 The total force exerted on the blades
of the turbine by the steam nozzle is FD20i120jC
100k(N), and it effectively acts at the point (100, 80,
300) mm. What moment is exerted about the axis of the
turbine (the xaxis)?
y
x
Fixed
Rotating
Solution: The moment about the origin is
ijk
Problem 4.97 The pneumatic support AB holds a trunk
lid in place. It exerts a 35-N force on the fixture at Bthat
points in the direction from Atoward B. Determine the
magnitude of the moment of the force about the hinge
axis of the lid, which is the zaxis.
Solution: The vector from Ato Bis
212
Problem 4.98 The tension in cable AB is 80 lb. What
is the moment about the line CD due to the force exerted
by the cable on the wall at B?
Solution: The strategy is to find the moment about the point C
exerted by the force at B, and then to find the component of that
moment acting along the line CD. The coordinates of the points B,
Problem 4.99 The magnitude of the force Fis 0.2 N
and its direction cosines are cos xD0.727,cos yD
x
z
P
A
Solution: We have
MAB D0.2N
p0.26
0.26 m 0.025 m 0.11 m
D0.0146 N-m
Problem 4.100 A motorist applies the two forces
shown to loosen a lug nut. The direction cosines of
Fare cos xD4
–FF
Solution: The unit vectors for the forces are the direction cosines.
The position vector of the force Fis rOF D1.333kft. The magnitude
of the moment due to Fis
The magnitude of the moment due to Fis
2.46 D13 lb
214
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Problem 4.101 The tension in cable AB is 2 kN. What
is the magnitude of the moment about the shaft CD due
to the force exerted by the cable at A? Draw a sketch to
3 m
Solution: The strategy is to determine the moment about Cdue
to A, and determine the component parallel to CD. The moment is
determined from the distance CA and the components of the tension,
eAB D0.4082i0.8165jC0.4082k.
The tension is
The moment about CD is
Problem 4.102 The axis of the car’s wheel passes
through the origin of the coordinate system and
What is the moment of Fabout the wheel’s axis?
y
Solution: We have to determine the moment about the axle where
a unit vector along the axle is
The vector from the origin to the point of contact with the road is
The moment of the force Fabout the axle is
MAXLE D[eÐ⊲rðF⊳]e
0.940 0 0.342
Problem 4.103 The direction cosines of the centerline
0.707, cos yD0.619, and cos zD0.342. What is the
moment about OA due to the 250-N weight? Draw a
x
z
O
Solution: By definition, the direction cosines are the scalar compo-
e1D0.5iC0.866j,and e2D0.707iC0.619j0.341k.
rWD0.600e1C0.750e2D0.8303iC0.9839j0.2565k
Problem 4.104 The radius of the steering wheel is
200 mm. The distance from Oto Cis 1 m. The center C
of the steering wheel lies in the xyplane. The driver
exerts a force FD10iC10j5k(N) on the wheel at A.
If the angle ˛D0, what is the magnitude of the moment
about the shaft OC? Draw a sketch to indicate the sense
Solution: The strategy is to determine the moment about C, and
then determine its component about OC. The radius vectors parallel
to OC and CA are:
rOC D1⊲icos 20°Cjsin 20°C0k⊳D0.9397iC0.3420j.
216
Problem 4.105* The magnitude of the force Fis 10 N.
Suppose that you want to choose the direction of the
force Fso that the magnitude of its moment about the
line Lis a maximum. Determine the components of F
Solution: The moment of the general force FDFxiCFyjCFzk
about the line is developed by
eBA D3iC6j6k
9D1
3⊲iC2j2k⊳,
Problem 4.106 The weight Wcauses a tension of
100 lb in cable CD.IfdD2 ft, what is the moment
about the zaxis due to the force exerted by the cable
CD at point C?
(3, 0, 10) ft
(12, 10, 0) ft
(0, 3, 0) ft
W
D
C
d
y
x
z
Solution: The strategy is to use the unit vector parallel to the bar
to locate point Crelative to the origin, and then use this location to
find the unit vector parallel to the cable CD. With the tension resolved
into components about the origin, the moment about the origin can be
resolved into components along the zaxis. Denote the top of the bar
by Tand the bottom of the bar by B. The position vectors of the ends
of the bar are:
The position vector of point Crelative to the origin is
218