Game Development Reference

In-Depth Information

(We have used
s
to represent the squared distance from the origin to avoid con-

fusion with the radial distance
r
in cylindrical coordinates.) The quantity

2

2

s

−

z

is equal to the squared distance from the
z
axis, which in cylindrical coordinates

is simply

r
. The differential volume
dV
in cylindrical coordinates is given by

dV

=

rdrdθdz

,

(14.61)

so Equation (14.60) becomes

h π R

22

3

=

ρ

rdrdθdz

33

−

h

20 0

4

(14.62)

=

1

2

πρhR

.

2

The volume of the cylinder is given by

V πhR

=

, so we can write the moment of

inertia as

2

2

=

1

ρVR

=

1

mR

,

(14.63)

33

2

2

where
m ρV

is the mass of the cylinder. Since a cylinder is symmetric about the

z
axis, we must have
11

=

. We can calculate the moment of inertia about the

x
axis by evaluating the integral

=

22

(

)

2

2

=

sxρdV

−

.

(14.64)

11

V

2

2

2

2

2

2

Making the substitutions

s

=+

rz

and

x

=

r

cos

θ

, we have

h π R

22

(

)

2

2

2

2

=

ρ

rzr θ rdrdθdz

+

−

cos

11

−

h

h π R

20 0

22

h π R

22

3

2

2

=

ρ

r θdr dθdz ρ

sin

+

zrdrdθdz

.

(14.65)

−

h

20 0

−

h

20 0

Evaluating the integrals for the variables
r
and
z
in the first term, and evaluating

all three integrals in the second term gives us

2

π

=

ρhR

4

sin

2

θdθ hR

+

3

2

.

(14.66)

1

1

11

4

12

0

Using the trigonometric identity

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