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Finally, imagine that the body is fixed within the coordinate
system and this system is rotated 90° about the y axis, Fig. 1-77,c.
Then the sum of the moments about the y axis gives

∫
) =Σ
(M R y M ; zW = ɶ zdW .
y
Therefore, the location of the center of gravity G with respect to
the x, y, z axes becomes
∫ xdW ∫ ydW ∫ zdW
ɶ
ɶ
ɶ
x = , y = , z = . 1-56
∫ dW ∫ dW ∫ dW
Here ,,x yz are the coordinates of the center of gravity G, Fig.
1-77,b; ,,x yz ɶ are the coordinates of each particle in the body, Fig. 1-
ɶɶ
77,a.
Center of Mass of a Body. In order to study the dynamic
response or accelerated motion of a body, it becomes important to
locate the body’s center of mass Cm, Fig. 1-78. This location can be
determined by substituting dW = gdm into Eqs. 1-56. Since g is
constant, it cancels out, and so
∫ xdm ∫ ydm ∫ zdm
ɶ
ɶ
ɶ
x = , y = , z = . 1-57
∫ dm ∫ dm ∫ dm

Fig. 1-78.

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