Chapter 5 Review

1
Bölüm 5 Yayili Kuvvetler Merkez ve Agirlik
Merkezi
The center of gravity of a rigid body is the
point G where a single force W, called the
weight of the body, can be applied to represent
the effect of the earths attraction on the body.
Consider two-dimensional bodies, such as flat
plates and wires contained in the xy plane. By
adding forces in the vertical z direction and
moments about the horizontal y and x axes, the
following relations were derived
W dW xW x dW yW y dW
which define the weight of the body and the
coordinates x and y of its center of gravity.
2
For a homogeneous flat plate of uniform
thickness, the center of gravity G of the plate
coincides with the centroid C of the area A of
the plate, the coordinates of which are
defined by the relations
These integrals are referred to as the first
moments of area A with respect to the y and x
axes, and are denoted by Qy and Qx , respectively
Qy xA Qx yA
3
Similarly, the determination of the center of
gravity of a homogeneous wire of uniform cross
section contained in a plane reduces to the
determination of the centroid C of the line L
representing the wire we have
xL x dL yL y dL
y
C
x
L
y
O
x
4
W3
z
y
y
z
W1
W2
S W
G3
G
G2
O
O
G1
x
x
The areas and the centroids of various common
shapes are tabulated. When a flat plate can be
divided into several of these shapes, the
coordinates X and Y of its center of gravity G
can be determined from the coordinates x1, x2 .
. . and y1, y2 . . . of the
centers of gravity of the various parts using
5
y
z
G
O
x
If the plate is homogeneous and of uniform
thickness, its center of gravity coincides with
the centroid C of the area of the plate, and the
first moments of the composite area are
Qy X SA S xA Qx Y SA S yA
6
When the area is bounded by analytical curves,
the coordinates of its centroid can be determined
by integration. This can be done by evaluating
either double integrals or a single integral
which uses a thin rectangular or pie-shaped
element of area. Denoting by xel and yel the
coordinates of the centroid of the element dA, we
have
Qy xA xel dA Qx yA yel dA
7
L
The theorems of Pappus-Guldinus relate the
determination of the area of a surface of
revolution or the volume of a body of revolution
to the determination of the centroid of the
generating curve or area. The area A of the
surface generated by rotating a curve of length L
about a fixed axis is
C
y
x
2py
A 2pyL
where y represents the distance from the
centroid C of the curve to the fixed axis.
8
C
A
y
x
2py
The volume V of the body generated by rotating an
area A about a fixed axis is
V 2pyA
where y represents the distance from the
centroid C of the area to the fixed axis.
9
dW
W
w
x
w
w
W A
C
x
B
O
x
B
O
P
dx
x
L
L
The concept of centroid of an area can also be
used to solve problems other than those dealing
with the weight of flat plates. For example, to
determine the reactions at the supports of
a beam, we replace a distributed load w by a
concentrated load W equal in magnitude to the
area A under the load curve and passing through
the centroid C of that area. The same approach
can be used to determine the resultant of the
hydrostatic forces exerted on a rectangular plate
submerged in a liquid.
10
The coordinates of the center of gravity G of a
three- dimensional body are determined from
xW x dW yW y dW zW z
dW
For a homogeneous body, the center of gravity G
coincides with the centroid C of the volume V of
the body the coordinates of C are defined by
the relations
xV x dV yV y dV zV z
dV
If the volume possesses a plane of symmetry, its
centroid C will lie in that plane if it
possesses two planes of symmetry, C will be
located on the line of intersection of the two
planes if it possesses three planes of symmetry
which intersect at only one point, C will
coincide with that point.
11
The volumes and centroids of various common
three- dimensional shapes are tabulated. When a
body can be divided into several of these shapes,
the coordinates X, Y, Z of its center of gravity
G can be determined from the corresponding
coordinates of the centers of gravity of the
various parts, by using
X SW S xW Y SW S yW Z SW S zW
If the body is made of a homogeneous material,
its center of gravity coincides with the centroid
C of its volume, and we write
X SV S xV Y SV S yV Z SV S
zV
12
z
xel x
When a volume is bounded by analytical surfaces,
the coordinates of its centroid can be determined
by integration. To avoid the computation of
triple integrals, we can use elements of
volume in the shape of thin filaments (as shown).
P(x,y,z)
yel y
zel z
z
dV z dx dy
y
xel
dx
yel
x
dy
Denoting by xel , yel , and zel the coordinates
of the centroid of the element dV, we write
xV xel dV yV yel dV zV
zel dV
If the volume possesses two planes of symmetry,
its centroid C is located on their line of
intersection.
13
y
xel x
dV pr 2 dx
x
z
dx
If the volume possesses two planes of symmetry,
its centroid C is located on their line of
intersection. Choosing the x axis to lie along
that line and dividing the volume into thin
slabs parallel to the xz plane, the centroid C
can be determined from
xV xel dV
For a body of revolution, these slabs are
circular.