Game Development Reference
In-Depth Information
r ,
()
Mdm
=
(14.18)
V
()
where
dm r represents the differential mass at the position r , and V is the vol-
ume occupied by the object. If the density at the position r is described by the
function
()
ρ r , then this integral can be written as
r
()
M ρ dV
=
.
(14.19)
V
The center of mass for a solid object is then computed using the integral
1
()
.
(14.20)
C
=
V ρ dV
rr
M
Example 14.1. Calculate the center of mass of a cone of radius R , height h , and
constant density ρ , whose base is centered at the origin on the x - y plane (see
Figure 14.5).
z
h
R
x y
Figure 14.5. The cone used in Example 14.1.
()
Solution. We use cylindrical coordinates. The radius
rz of a cross section of
the cone at a height z above the x - y plane is given by
R
() (
)
rz
=−
h z h
.
(14.21)
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