Game Development Reference
InDepth 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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