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Last equations may be expressed in vector form with the aid of
Fig. 1-78, in which the elemental mass and the mass center G are
located by their respective position vectors r ɶ = x +i ɶ y +j ɶ zk ɶ and
r = x +i y +j zk . Thus, Eqs. 1-57 are the components of the single
vector equation

∫ r ɶ dm
r = . 1-58
∫ dm

Center of Mass of a Volume. If the body in Fig. 1-79 is made
from a homogeneous material, then its density ρ=const (rho) will be
constant. Therefore, a differential element of volume dV has a mass
dm ρ= dV Substituting this into Eqs. 1-57 and canceling out ρ, we
obtain formulas that locate the center of mass C of the body; namely
∫ xdV ∫ ydV ∫ zdV
ɶ
ɶ
ɶ
x = y = z = 1-59
∫ dV ∫ dV ∫ dV

Fig. 1-79.

Center of Mass of an Area. When a body of density ρ has a
small but constant thickness h, we can model it as a surface area S,
Fig. 1-80. The mass of an element becomes dm ρ= hdS . Again, if ρ
and h are constant over the entire area, the coordinates may be written

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