Game Development Reference
In-Depth Information
M
2
M
3
M
1
M
2
M
3
M
1
(A)
(B)
R
1
=3
R
2
=2
R
3
=1
M
1
=2
M
2
=2
M
3
=2
R
1
=3
R
2
=3
R
3
=3
M
1
=2
M
2
=2
M
3
=2
M
1
M
2
M
3
M
1
M
2
M
3
(C)
(D)
R
1
=3
R
2
=2
R
3
=1
M
1
=3
M
2
=2
M
3
=1
R
1
=3
R
2
=2
R
3
=1
M
1
=1
M
2
=2
M
3
=3
Figure 12.20.
Distribution of mass can affect the moment of inertia. Each of the examples
above has the same total mass.
Consider the four disks in Figure 12.20. Since each disk has three
masses, we start by expanding the sum in Equation (12.25) to J = m
1
r
1
2
+
m
2
r
2
2
+ m
3
r
3
2
:
(a) J = 2 3
2
+ 2 2
2
+ 2 1
2
= 18 + 8 + 2 = 28;
(b) J = 2 3
2
+ 2 3
2
+ 2 3
2
= 18 + 18 + 18 = 54;
(c) J = 3 3
2
+ 2 2
2
+ 1 1
2
= 27 + 8 + 1 = 36;
(d) J = 1 3
2
+ 2 2
2
+ 3 1
2
= 9 + 8 + 3 = 20.
Aside from the rote practice this example provides, it highlights a crucial
fact: the distribution of mass within the object can have a profound effect
on the moment of inertia. Notice the widely differing moments of inertia,
despite the fact that each of these disks has a total mass of 6. Compare
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