Chapter
4
Forces applied to wrenches and wheels will produce rotation or a tendency for
rotation. This effect is called a moment, and in this chapter we will study how
to determine the moment of a system of forces and calculate their resultants.
Dr. Aslam Al-Omari
4.0) Introduction
Equilibrium of a particle (Chapter 3)
∑
G
F =0⇒
∑
G
Fx = 0 &
∑
G
Fy = 0 &
Equilibrium of a rigid body (Chapter 5)
∑
∑
G
F =0⇒
G
Mo = 0
∑
G
Fx = 0 &
∑
G
Fy = 0 &
∑
G
Fz = 0
∑
G
Fz = 0
Dr. Aslam Al-Omari
4.1) Introduction
(Scalar Formulation)
F ⊥d
Dr. Aslam Al-Omari
4.1) Introduction
(Scalar Formulation)
Dr. Aslam Al-Omari
4.1) Moment of a Force
(Scalar Formulation)
Dr. Aslam Al-Omari
4.1) Moment of a Force
(Scalar Formulation)
Dr. Aslam Al-Omari
4.1) Moment of a Force
(Scalar Formulation)
4.1
EXAMPLE
119
MOMENT OF A FORCE—SCALAR FORMULATION
4.1
For each case illustrated in Fig. 4–4, determine the moment of the
force about point O.
SOLUTION (SCALAR ANALYSIS)
The line of action of each force is extended as a dashed line in order to
establish the moment arm d. Also illustrated is the tendency of
rotation of the member as caused by the force. Furthermore, the orbit
of the force about O is shown as a colored curl. Thus,
Fig. 4–4a
Fig. 4–4b
Fig. 4–4c
M O = (100 N)(2 m) = 200 N # m b
100 N
Ans.
M O = (50 N)(0.75 m) = 37.5 N # mb
Ans.
M O = (40 lb)(4 ft + 2 cos 30⬚ ft) = 229 lb # ft b
4
Ans.
Fig. 4–4d
M O = (60 lb)(1 sin 45⬚ ft) = 42.4 lb # ft d
Ans.
Fig. 4–4e
M O = (7 kN)(4 m - 1 m)
Ans.
= 21.0 kN # m d
O
2m
(a)
2 ft
2m
30⬚ 40 lb
O
O
0.75 m
4 ft
2 cos 30⬚ ft
50 N
(b)
(c)
2m
1m
7 kN
3 ft
4m
O
1 ft
45⬚
1 sin 45⬚ ft
60 lb
(d)
O
Fig. 4–4
(e)
120
CHAPTER 4
EXAMPLE
FORCE SYSTEM RESULTANTS
4.2
Determine the resultant moment of the four forces acting on the rod
shown in Fig. 4–5 about point O.
SOLUTION
Assuming that positive moments act in the + k direction, i.e.,
counterclockwise, we have
y
50 N
2m
2m
60 N
x
30⬚
O
a+ (MR)O = ⌺Fd;
(MR)O = -50 N(2 m) + 60 N(0) + 20 N(3 sin 30⬚ m)
20 N
3m
-40 N(4 m + 3 cos 30⬚ m)
4
(MR)O = -334 N # m = 334 N # mb
40 N
Fig. 4–5
Ans.
For this calculation, note how the moment-arm distances for the 20-N
and 40-N forces are established from the extended (dashed) lines of
action of each of these forces.
FH
F
MA ⫽ FdA
A
dA
B
O
FN
As illustrated by the example problems, the moment of a
force does not always cause a rotation. For example, the
force F tends to rotate the beam clockwise about its support
at A with a moment MA = FdA. The actual rotation would
occur if the support at B were removed.
The ability to remove the nail will require the
moment of FH about point O to be larger than the
moment of the force FN about O that is needed to
pull the nail out.
Dr. Aslam Al-Omari
4.2) Cross Product
Dr. Aslam Al-Omari
4.2) Cross Product
Dr. Aslam Al-Omari
4.2) Cross Product
Dr. Aslam Al-Omari
4.2) Cross Product
Dr. Aslam Al-Omari
4.2) Cross Product
Dr. Aslam Al-Omari
4.2) Cross Product
Dr. Aslam Al-Omari
4.2) Cross Product
Dr. Aslam Al-Omari
4.2) Cross Product
Dr. Aslam Al-Omari
4.2) Cross Product
Dr. Aslam Al-Omari
4.2) Cross Product
Dr. Aslam Al-Omari
4.3) Moment of a Force
(Vector Formulation)
Dr. Aslam Al-Omari
4.3) Moment of a Force
(Vector Formulation)
Dr. Aslam Al-Omari
4.3) Moment of a Force
(Vector Formulation)
Dr. Aslam Al-Omari
4.3) Moment of a Force
(Vector Formulation)
Dr. Aslam Al-Omari
4.3) Moment of a Force
(Vector Formulation)
Dr. Aslam Al-Omari
4.3) Moment of a Force
(Vector Formulation)
126
CHAPTER 4
EXAMPLE
FORCE SYSTEM RESULTANTS
4.3
Determine the moment produced by the force F in Fig. 4–14a about
point O. Express the result as a Cartesian vector.
z
SOLUTION
As shown in Fig. 4–14b, either rA or rB can be used to determine the
moment about point O. These position vectors are
A
12 m
rA = {12k} m¬ and¬ rB = {4i + 12j} m
F ⫽ 2 kN
Force F expressed as a Cartesian vector is
4
uAB
O
F = FuAB = 2 kN C
x
12 m
B
{4i + 12j - 12k} m
2(4 m)2 + (12 m)2 + (-12 m)2
S
= {0.4588i + 1.376j - 1.376k} kN
4m y
Thus
(a)
MO = rA * F = 3
z
i
0
0.4588
j
0
1.376
k
12 3
-1.376
= [0(-1.376) - 12(1.376)]i - [0(-1.376) - 12(0.4588)]j
+ [0(1.376) - 0(0.4588)]k
= {-16.5i + 5.51j} kN # m
Ans.
or
A
MO = rB * F = 3
rA
F
O
x
MO
j
12
1.376
k
0 3
-1.376
= [12( -1.376) - 0(1.376)]i - [4(-1.376) - 0(0.4588)]j
+ [4(1.376) - 12(0.4588)]k
rB
B
i
4
0.4588
y
= {-16.5i + 5.51j} kN # m
Ans.
NOTE: As shown in Fig. 4–14b, MO acts perpendicular to the plane
(b)
Fig. 4–14
that contains F, rA, and rB. Had this problem been worked using
MO = Fd, notice the difficulty that would arise in obtaining the
moment arm d.
127
4.3 MOMENT OF A FORCE—VECTOR FORMULATION
EXAMPLE
4.4
Two forces act on the rod shown in Fig. 4–15a. Determine the resultant
moment they create about the flange at O. Express the result as a
Cartesian vector.
z
z
F1 {60i 40j 20k} lb
F1
A
O
y
5 ft
A
y
4
rB
4 ft
x
rA
O
2 ft
x
B
B
F2 {80i 40j 30k} lb
F2
(a)
(b)
SOLUTION
Position vectors are directed from point O to each force as shown in
Fig. 4–15b. These vectors are
z
(MR) {30i 40j 60k} lb·ft
O
rA = {5j} ft
g 39.8
rB = {4i + 5j - 2k} ft
a 67.4
The resultant moment about O is therefore
x
(c)
(MR)O = (r * F)
Fig. 4–15
= rA * F1 + rB * F2
i
= 3 0
-60
j
5
40
k
i
0 3 + 3 4
20
80
j
5
40
k
-2 3
-30
= [5(20) - 0(40)]i - [0]j + [0(40) - (5)(-60)]k
+ [5(-30) - (-2)(40)]i - [4(-30) - (-2)(80)]j + [4(40) - 5(80)]k
= {30i - 40j + 60k} lb # ft
Ans.
NOTE: This result is shown in Fig. 4–15c. The coordinate direction
angles were determined from the unit vector for (MR ) O. Realize that
the two forces tend to cause the rod to rotate about the moment axis
in the manner shown by the curl indicated on the moment vector.
b 121
O
y
Dr. Aslam Al-Omari
4.5) Moment of a Force about a
Specified Axis
Dr. Aslam Al-Omari
4.5) Moment of a Force about a
Specified Axis
or simply, its magnitude is:
Dr. Aslam Al-Omari
4.5) Moment of a Force about a
Specified Axis
141
4.5 MOMENT OF A FORCE ABOUT A SPECIFIED AXIS
Important Points
• The moment of a force about a specified axis can be determined
provided the perpendicular distance da from the force line of
action to the axis can be determined. Ma = Fda.
• If vector analysis is used, Ma = ua # (r * F), where ua defines the
direction of the axis and r is extended from any point on the axis
to any point on the line of action of the force.
• If Ma is calculated as a negative scalar, then the sense of direction
of Ma is opposite to ua.
4
• The moment Ma expressed as a Cartesian vector is determined
from Ma = Maua.
EXAMPLE
4.7
Determine the resultant moment of the three forces in Fig. 4–22 about
the x axis, the y axis, and the z axis.
z
F2 ⫽ 50 lb
SOLUTION
A force that is parallel to a coordinate axis or has a line of action that
passes through the axis does not produce any moment or tendency for
turning about that axis. Therefore, defining the positive direction of
the moment of a force according to the right-hand rule, as shown in
the figure, we have
Ans.
My = 0 - (50 lb)(3 ft) - (40 lb)(2 ft) = -230 lb # ft
Ans.
Mz = 0 + 0 - (40 lb)(2 ft) =
-80 lb # ft
F3 ⫽ 40 lb
C
A
2 ft
+ 0 = 220 lb # ft
Mx = (60 lb) (2 ft) + (50 lb) (2 ft)
B
O
2 ft
F1 ⫽ 60 lb
2 ft 3 ft
x
y
Ans.
The negative signs indicate that My and Mz act in the -y and -z
directions, respectively.
Fig. 4–22
142
CHAPTER 4
EXAMPLE
FORCE SYSTEM RESULTANTS
4.8
z
0.6 m
Determine the moment MAB produced by the force F in Fig. 4–23a,
which tends to rotate the rod about the AB axis.
0.3 m
A
y
C
0.4 m
F = 300 N
B
0.2 m
SOLUTION
A vector analysis using MAB = uB # (r * F) will be considered for the
solution rather than trying to find the moment arm or perpendicular
distance from the line of action of F to the AB axis. Each of the terms
in the equation will now be identified.
Unit vector uB defines the direction of the AB axis of the rod,
Fig. 4–23b, where
uB =
4
(a)
x
{0.4i + 0.2j} m
rB
=
= 0.8944i + 0.4472j
rB
2(0.4 m)2 + (0.2 m)2
Vector r is directed from any point on the AB axis to any point on the
line of action of the force. For example, position vectors rC and rD are
suitable, Fig. 4–23b. (Although not shown, rBC or rBD can also be used.)
For simplicity, we choose rD, where
z
rD = {0.6i} m
The force is
F = {-300k} N
A
rC
C
MAB
uB
rD
F
B
D
x
(b)
Fig. 4–23
y
Substituting these vectors into the determinant form and expanding,
we have
0.8944 0.4472
0
MAB = uB # (rD * F) = 3 0.6
0
0 3
0
0
-300
= 0.8944[0(-300) - 0(0)] - 0.4472[0.6( -300) - 0(0)]
+ 0[0.6(0) - 0(0)]
= 80.50 N # m
This positive result indicates that the sense of MAB is in the same
direction as uB.
Expressing MAB in Fig. 4-23b as a Cartesian vector yields
MAB = MABuB = (80.50 N # m)(0.8944i + 0.4472j)
= {72.0i + 36.0j} N # m
Ans.
NOTE: If axis AB is defined using a unit vector directed from B toward
A, then in the above formulation -uB would have to be used. This
would lead to MAB = -80.50 N # m. Consequently, MAB = MAB(-uB),
and the same result would be obtained.
143
4.5 MOMENT OF A FORCE ABOUT A SPECIFIED AXIS
EXAMPLE
4.9
z
Determine the magnitude of the moment of force F about segment
OA of the pipe assembly in Fig. 4–24a.
SOLUTION
The moment of F about the OA axis is determined from
MOA = uOA # (r * F), where r is a position vector extending from any
point on the OA axis to any point on the line of action of F. As
indicated in Fig. 4–24b, either rOD, rOC, rAD, or rAC can be used;
however, rOD will be considered since it will simplify the calculation.
The unit vector uOA, which specifies the direction of the OA axis, is
uOA
0.5 m
D
O
0.5 m
x
rOA
{0.3i + 0.4j} m
=
=
= 0.6i + 0.8j
rOA
2(0.3 m)2 + (0.4 m)2
F ⫽ 300 N
C
B
0.3 m
0.4 m
0.2 m
0.1 m y
A
(a)
and the position vector rOD is
z
rOD = {0.5i + 0.5k} m
The force F expressed as a Cartesian vector is
D
F = Fa
rCD
b
rCD
rOD
rAD
= (300 N) C
2(0.4 m) + (-0.4 m) + (0.2 m)
2
2
rAC
S
x
A
= {200i - 200j + 100k} N
(b)
Therefore,
Fig. 4–24
MOA = uOA # (rOD * F)
0.6
3
= 0.5
200
0.8
0
-200
0
0.5 3
100
= 0.6[0(100) - (0.5)(-200)] - 0.8[0.5(100) - (0.5)(200)] + 0
= 100 N # m
C
O
uOA
{0.4i - 0.4j + 0.2k} m
2
F
rOC
Ans.
y
4
Dr. Aslam Al-Omari
4.6) Moment of a Couple
Dr. Aslam Al-Omari
4.6) Moment of a Couple
Couple moment is the same at any point
This means that;
Dr. Aslam Al-Omari
4.6) Moment of a Couple
Dr. Aslam Al-Omari
4.6) Moment of a Couple
Dr. Aslam Al-Omari
4.6) Moment of a Couple
Dr. Aslam Al-Omari
4.6) Moment of a Couple
Dr. Aslam Al-Omari
4.6) Moment of a Couple
150
CHAPTER 4
FORCE SYSTEM RESULTANTS
Important Points
• A couple moment is produced by two noncollinear forces that
F
are equal in magnitude but opposite in direction. Its effect is to
produce pure rotation, or tendency for rotation in a specified
direction.
F
• A couple moment is a free vector, and as a result it causes the
same rotational effect on a body regardless of where the couple
moment is applied to the body.
4
Steering wheels on vehicles have been
made smaller than on older vehicles
because power steering does not require
the driver to apply a large couple moment
to the rim of the wheel.
• The moment of the two couple forces can be determined about
any point. For convenience, this point is often chosen on the line
of action of one of the forces in order to eliminate the moment of
this force about the point.
• In three dimensions the couple moment is often determined
using the vector formulation, M = r * F, where r is directed
from any point on the line of action of one of the forces to any
point on the line of action of the other force F.
• A resultant couple moment is simply the vector sum of all the
couple moments of the system.
EXAMPLE
4.10
F1 ⫽ 200 lb
Determine the resultant couple moment of the three couples acting
on the plate in Fig. 4–30.
F3 ⫽ 300 lb
d1 ⫽ 4 ft
F2 ⫽ 450 lb A
d3 ⫽ 5 ft
d2 ⫽ 3 ft
F2 ⫽ 450 lb
B
F1 ⫽ 200 lb
F3 ⫽ 300 lb
Fig. 4–30
SOLUTION
As shown the perpendicular distances between each pair of couple forces
are d1 = 4 ft, d2 = 3 ft, and d3 = 5 ft. Considering counterclockwise
couple moments as positive, we have
a+ M R = ⌺M; M R = -F1d1 + F2d2 - F3d3
= -(200 lb)(4 ft) + (450 lb)(3 ft) - (300 lb)(5 ft)
= -950 lb # ft = 950 lb # ft b
Ans.
The negative sign indicates that MR has a clockwise rotational sense.
152
CHAPTER 4
EXAMPLE
FORCE SYSTEM RESULTANTS
4.12
Determine the couple moment acting on the pipe shown in Fig. 4–32a.
Segment AB is directed 30° below the x–y plane.
z
z
O
O
rA
25 lb
25 lb
x
A
rB
x
8 in.
A
y
y
30⬚
25 lb
25 lb
6 in.
4
B
B
(b)
z
(a)
SOLUTION I (VECTOR ANALYSIS)
The moment of the two couple forces can be found about any point. If
point O is considered, Fig. 4–32b, we have
O
25 lb
x
A
y
25 lb
rAB
(c)
M = rAB * (25k)
= (6 cos 30⬚i - 6 sin 30⬚k) * (25k)
= { -130j} lb # in.
z
O
25 lb
x
A
6 in.
30⬚
d
B
(d)
Fig. 4–32
= { -130j} lb # in.
Ans.
It is easier to take moments of the couple forces about a point lying on
the line of action of one of the forces, e.g., point A, Fig. 4–32c. In this
case the moment of the force at A is zero, so that
B
25 lb
M = rA * (-25k) + rB * (25k)
= (8j) * (-25k) + (6 cos 30⬚i + 8j - 6 sin 30⬚k) * (25k)
= -200i - 129.9j + 200i
Ans.
SOLUTION II (SCALAR ANALYSIS)
Although this problem is shown in three dimensions, the geometry is
simple enough to use the scalar equation M = Fd. The perpendicular
distance between the lines of action of the couple forces is
y
d = 6 cos 30⬚ = 5.196 in., Fig. 4–32d. Hence, taking moments of the
forces about either point A or point B yields
M = Fd = 25 lb (5.196 in.) = 129.9 lb # in.
Applying the right-hand rule, M acts in the -j direction. Thus,
M = { -130j} lb # in.
Ans.
4.6
EXAMPLE
MOMENT OF A COUPLE
153
4.13
Replace the two couples acting on the pipe column in Fig. 4–33a by a
resultant couple moment.
5
z
125 N
5
M2
4
4
3
3
D
C
125 N
x
150 N
A
5
3
M2
MR
0.3 m
M1
y
4
4
B
M1
0.4 m
150 N
(c)
(b)
(a)
Fig. 4–33
SOLUTION (VECTOR ANALYSIS)
The couple moment M1, developed by the forces at A and B, can
easily be determined from a scalar formulation.
M1 = Fd = 150 N(0.4 m) = 60 N # m
By the right-hand rule, M1 acts in the +i direction, Fig. 4–33b. Hence,
M1 = {60i} N # m
Vector analysis will be used to determine M2, caused by forces at C
and D. If moments are calculated about point D, Fig. 4–33a,
M2 = rDC * FC, then
M2 = rDC * FC = (0.3i) *
3 125 1 54 2 j
- 125 1 53 2 k 4
= (0.3i) * [100j - 75k] = 30(i * j) - 22.5(i * k)
= {22.5j + 30k} N # m
Since M1 and M2 are free vectors, they may be moved to some
arbitrary point and added vectorially, Fig. 4–33c. The resultant couple
moment becomes
MR = M1 + M2 = {60i + 22.5j + 30k} N # m
Ans.
