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CE 201- Statics
Chapter 9 – Lecture 1
CENTER OF GRAVITY AND CENTROID
The following will be studiedLocation of center of gravity (C. G.) and center of
mass for discrete particlesLocation of C. G. and center of mass for an
arbitrary-shaped bodyLocation of centroid or geometric center
Center of Gravity and Center of Mass for a System of Particles
• Consider ( n ) particles• Weight of particles are
parallel forces• Weight can be replaced by
a single resultant weight• The point of application of
the resultant weight is called the center of gravity (C. G.)
Resultant weight, WR = w
w1
w4wn
w3w2
WR
y
z
x
Moment of all weights about x, y, and z is equal to the moment of the resultant weight about the same axes.
My xRWR = x1w1 + x2w2 + ..+xnwn Mx yRWR = y1w1 + y2w2 + ..+ynwn
w1
w4wn
w3w2
WR
y
z
x
To find z, imagine rotating the system coordinates by 90 with particles are fixed in it.
w1
w4wn
w3w2
WR
y
z
x
w1
w4w
n
w3w2
WR
y
z
x
Mx zRWR = z1w1 + z2w2 + ……..+znwn
then, x = ( x W) / WR
y = ( y W) / WR
Z = ( z W) / WR
Note:x, y, and z for C. G. of the systemx, y, and z for C. G. of each particle
Center of Mass
W = mg
x = ( x m) / mR
y = ( y m) / mR
x = ( z m) / mR
The location of the center of gravity coincides with that of the center of mass
CENTER OF GRAVITY, CENTER OF MASS AND CETROID FOR A BODY
Center of Gravity
A rigid body is composed of a system of particles, where each particle has a differential weight (dW). Applying the same principles that were used with discrete particles, the following is obtained:
x = ( x dW) / ( dW)
y = ( y dW) / ( dW)
z = ( z dW) / ( dW)
x
y
z
dW
here, we use integration rather than summation due to differential weight (dW).
If dW = dV
Where
= specific weight (weight / volume)
V = volume of body
Then,
x = ( x dV) / ( dV)
y = ( y dV) / ( dV)
z = ( z dV) / ( dV)
x
y
z
dW
Center of Mass
Substitute = g into the previous equations ( = density (mass/volume), then:
x = ( x g dV) / ( g dV)
y = ( y g dV) / ( g dV)
z = ( z g dV) / ( g dV)
Centroid
Centroid is the geometric center of the object Centroid is independent of the weight Centroid is dependent of the body's geometry
Volume Centroid
•Subdivide the object into volume elements (dV)
•Compute the moments of the volume elements about the coordinate axes
x = ( x dV) / ( dV)
y = ( y dV) / ( dV)
z = ( z dV) / ( dV)
x
y
z
dV
y
x
c
z
Area Centroid
Subdivide the object into area elements (dA)
Compute the moments of the area elements about the coordinate axes
x = ( x dA) / ( dA)
y = ( y dA) / ( dA)
z = ( z dA) / ( dA)
x
y
z
dV
y
x
c
z
Line Centroid
Subdivide the line into elements (dA)
Compute the moments of the line elements about the coordinate axes
x = ( x dL) / ( dL) y = ( y dL) / ( dL) z = ( z dL) / ( dL)
Centroid could be located off the object in space
Centroid of some shapes may be specified by using the conditions of symmetry
x
y
z
dL
y
x
c
z
Line Centroid
If the shape has an axis of symmetry, then the centroid will be located along that axis
For every element dL having a distance ( y ), there is an element dL having a distance ( -y )
So, y = 0
x
y
c
dL
dL
The same thing can apply if the shape has more than one axis of symmetry. The centroid lies at the intersection of the axe.
Procedure for Analysis
To determine the Center of Gravity or the Centroid: select an appropriate coordinate axes select an appropriate differential element for
integration (dL, dA, dV) express he differential element (dL, dA, dV) in
terms of the coordinates (x, y, z) determine the coordinate (x, y, z) or moment
arms for the centroid or center of gravity of the element
integrate