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MER212: Chapter 9MER212: Chapter 9Center of Gravity and Centroid
Center of Gravity center of Mass • Center of Gravity, center of Mass, and the Centriod of a Body (9.1)
• Composite Bodies (9 3)• Composite Bodies (9.3)• Fluid Pressure (9.5)
Union CollegeMechanical Engineering
ESC020: Rigid Body Mechanics 1
Resultant Weight and Center of Gravity Resultant Weight Resultant Weight
R iW W W dV Resultant of Weight=>Center of
Gravity
d
dWxW
WxW
Wxx iiii
dW
dWyW
WyW
Wyy
dWWW
iiii
iR
dW
dWzW
WzW
Wzz
dWWW
iiii
iR
Union CollegeMechanical Engineering
ESC020: Rigid Body Mechanics 3
dWWW iR
Centers of Mass
dV
dVx
dV
dVx
dm
dmx
dmg
dmgxm
mxgm
gmxx
i
ii
i
ii
dV
dVy
dV
dVy
dm
dmy
dmg
dmgym
mygm
gmyy
i
ii
i
ii
dV
dVz
dV
dVz
dm
dmz
dmg
dmgzm
mzgm
gmzz
i
ii
i
ii
Weight Specific Weight: Weight per unit volume Density: Mass per unit volume
Vm g
W m g
Union CollegeMechanical Engineering
ESC020: Rigid Body Mechanics 4
y p
Centroid of a Volume
dV
dVxV
Vxx ii
dVyVy
y
dVV
ii
i
dVzVz
dVVy
ii
i
dVVz
i
ii
Union CollegeMechanical Engineering
ESC020: Rigid Body Mechanics 5
Centroid of an AreadA dA dAA A A i i i i i i
i i i
x dA y dA z dAx A y A z Ax y z
A A AdA dA dA
Union CollegeMechanical Engineering
ESC020: Rigid Body Mechanics 6
Centroid of Line
x dLx
xdL
y dLy
ydL
Union CollegeMechanical Engineering
ESC020: Rigid Body Mechanics 7
Example 1 Locate the centroid of the Locate the centroid of the
triangular area shown
Union CollegeMechanical Engineering
ESC020: Rigid Body Mechanics 8
Example 2 Locate the centroids of the area Locate the centroids of the area
bounded by the two curves shown
Union CollegeMechanical Engineering
ESC020: Rigid Body Mechanics 9
Example 3 Locate the centroid for the area Locate the centroid for the area
shown
Union CollegeMechanical Engineering
ESC020: Rigid Body Mechanics 11