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Chapter 3
Centroids and Center of Mass
3.1 First Moment and Centroid of a Set of Points
The position vector of a point
P
relative to a point
O
is
r
P
and a scalar associated
with
P
is
s
, e.g., the mass
m
of a particle situated at
P
. The
ﬁrst moment
of a point
P
with respect to a point
O
is the vector
M
=
s
r
P
. The scalar
s
is called the
strength
of
P
. The set of
n
points
P
i
,
i
=
1
,
2
,...,
n
, is
{
S
}
, Fig. 3.1(a)
{
S
}
=
{
P
1
,
P
2
P
n
}
=
{
P
i
}
i
=
1
,
2
,...,
n
.
The strengths of the points
P
i
are
s
i
,
i
=
1
,
2
n
, i.e.,
n
scalars, all having the same
dimensions, and each associated with one of the points of
{
S
}
.
The
centroid
of the set
{
S
}
is the point
C
with respect to which the sum of the
ﬁrst moments of the points of
{
S
}
is equal to zero. The centroid is the point deﬁning
the geometric center of system or of an object.
The position vector of
C
relative to an arbitrarily selected reference point
O
is
r
C
, Fig. 3.1(b). The position vector of
P
i
relative to
O
is
r
i
. The position vector of
P
i
relative to
C
is
r
i

r
C
. The sum of the ﬁrst moments of the points
P
i
with respect to
C
is
n
∑
i
=
1
s
i
(
r
i

r
C
)
. If
C
is to be centroid of
{
S
}
, this sum is equal to zero
{
S
}
(a)
(b)
{
S
}
P
1
()
s
1
P
i
s
i
P
n
s
n
P
i
(
s
i
)
O
r
i
r
C
C
P
2
s
2
Fig. 3.1
(a) Set of points and (b) centroid of a set of points
1
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3 Centroids and Center of Mass
n
∑
i
=
1
s
i
(
r
i

r
C
) =
n
∑
i
=
1
s
i
r
i

r
C
n
∑
i
=
1
s
i
=
0
.
The position vector
r
C
of the centroid
C
, relative to an arbitrarily selected reference
point
O
, is given by
r
C
=
n
∑
i
=
1
s
i
r
i
n
∑
i
=
1
s
i
.
If
n
∑
i
=
1
s
i
=
0 the centroid is not deﬁned. The centroid
C
of a set of points of given
strength is a unique point, its location being independent of the choice of reference
point
O
.
The cartesian coordinates of the centroid
C
(
x
C
,
y
C
,
z
C
)
of a set of points
P
i
,
i
=
1
,...,
n
, of strengths
s
i
,
i
=
1
n
, are given by the expressions
x
C
=
n
∑
i
=
1
s
i
x
i
n
∑
i
=
1
s
i
,
y
C
=
n
∑
i
=
1
s
i
y
i
n
∑
i
=
1
s
i
,
z
C
=
n
∑
i
=
1
s
i
z
i
n
∑
i
=
1
s
i
.
The
plane of symmetry
of a set is the plane where the centroid of the set lies, the
points of the set being arranged in such a way that corresponding to every point on
one side of the plane of symmetry there exists a point of equal strength on the other
side, the two points being equidistant from the plane.
A set
{
S
0
}
of points is called a
subset
of a set
{
S
}
if every point of
{
S
0
}
is a point
of
{
S
}
. The centroid of a set
{
S
}
may be located using the
method of decomposition
:
•
divide the system
{
S
}
into subsets;
•
ﬁnd the centroid of each subset;
•
assign to each centroid of a subset a strength proportional to the sum of the
strengths of the points of the corresponding subset;
•
determine the centroid of this set of centroids.
3.2 Centroid of a Curve, Surface, or Solid
The position vector of the centroid
C
of a curve, surface, or solid relative to a point
O
is
3.2 Centroid of a Curve, Surface, or Solid
3
r
C
=
Z
τ
r
d
τ
Z
τ
d
τ
,
(3.1)
where,
τ
is a curve, surface, or solid,
r
denotes the position vector of a typical point
of
τ
, relative to
O
, and
d
τ
is the length, area, or volume of a differential element of
τ
. Each of the two limits in this expression is called an “integral over the domain
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This note was uploaded on 08/29/2011 for the course MECH 2110 taught by Professor Clark,b during the Spring '08 term at Auburn University.
 Spring '08
 Clark,B
 Statics

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