Moment of a Force
Objects
a particle
a 3-D body
External Effects
translation
translation and rotation
The moment of a force about a point or an axis: a measure of
the tendency of the force to rotate a body about that point or
the axis.
Definition: Moment
Common-sense definition:
M  F d
Magnitude:
F
o
d
Direction: is perpendicular to the plane defined by the force and
the rotation center and can be determined by the
right-hand rule.
Vector representation:
M  r F
r is a position vector from the rotation center to any point
on the line of action of the force.
Example 1: Find the moments of F 1 , F2 and F3 about
point o.
o
Example 2: Determine
(a) the moment of force F about point A
(b) the perpendicular distance from point
A to the line of action of the force.
30o
Moment of a Force about a Line
Definition:
l
Ml   r  F   l  l
F
Example:
What is the moment about x axis?
z
M
y
x
Couples
• Definition
– two equal, noncollinear, parallel forces of opposite
senses are called a couple.
• Effects
– rotation only, no translation
Moment of a Couple
• Common sense definition
F
Magnitude:
d
M  F d
Direction: right-hand rule
-F
• Vector representation
F
r
-F
M  r F
Couples
• Characteristics
– The magnitude of the moment of the couple
– The direction of the moment of a couple
• Equivalent couples - same
and
F
d
F
d
F
F
F
F d
Equivalent Force/Couple Systems
• External effects
– translation
– rotation
• Conditions for two force/couple systems to be
equivalent:
F 
i 1

F 
i 2
M   M 
io 1
io 2
The Force-and-Couple Resultant of
a Force System
F2
F1
F3
R
o
C


( M Q )1  ( M Q ) 2
F5
F4
F  R
i
M
io
C
o
Equivalent Force / Couple Systems
Example 1: (a) Replace F by an equivalent force/couple system
applied at A.
(b) Replace F by an equivalent force/couple system
applied at B.
F=100N
5m
B
5m
A
30m
Equivalent Force / Couple Systems
Example 2: Reduce the force/couple system by an equivalent
single force
y
1000N
500N 500N
45o
x
3m
2m
1m
Equivalent Force / Couple Systems
Questions:
1. Can any general force/couple system be reduced to a single force/couple system?
2. Can any general force/couple system be reduced to a single force?
3. If two force/couple systems have the same moments about point P and


(
M
)

(
M
have the same net force, is  Q 1  Q )2 valid? Q is an arbitrary point
other than P.
Distributed Force Systems
• Force distributed over a line
f (x)
• Force distributed over a surface
y
f
(x,y)
x
• Force distributed over a volume
F(x,y,z)
x
Force Distributed Over a Line
F
f (x)
x
L
  F
1
F
d
M
o
 F  d
L
F   f  x dx
0
Examples:
f0
f (x)=f
0
O
L
x
O
L
x
Centroid of an Area
Definition:
xc 
 xdA
 dA
yc 
 ydA
 dA
Example 1: Find the centroid of a rectangle.
h
w
Centroid of an Area
xc 
Composite body:
xc1  A1  xc 2  A2
A1  A2
yc 
yc1  A1  yc 2  A2
A1  A2
Example 2: Find the centroid of the following area.
h
w
w
w
4h