Game Development Reference
In-Depth Information
Torque is the rotational analog to linear force and induces an angular accelera-
tion. If the net torque acting on a particle is zero, then the angular momentum
remains constant because
()
()
L τ
t
=
t
.
(14.34)
14.2.3 The Inertia Tensor
Angular momentum is related to angular velocity in a much more complicated
way than linear momentum is related to linear velocity. In fact, the angular mo-
mentum vector and the associated angular velocity vector do not necessarily
point in the same direction. The relationship between these two quantities is the
topic of this section.
The angular momentum of a rigid body composed of a set of particles is
equal to the sum
()
()
()
L
t
=
r
t
×
p
t
,
(14.35)
k
k
k
()
()
where
k p represents the mo-
mentum of the k -th particle, and the summation is taken over all the particles be-
longing to the system. Since the linear momentum
t
represents the position of the k -th particle,
r
k
()
t
can be written as
p
k
()
()
()
()
p
t
=
m t
v
=
m t
ω r
×
t
,
(14.36)
k
k
k
k
k
the angular momentum becomes
]
()
()
[
()
()
L
t
=
mt
r
×
ω r
t
×
t
.
(14.37)
kk
k
k
Using the vector identity given by Theorem 2.9(f),
PQP PQP
××=××=
(
)
P
2
Q PQP ,
−⋅
(
)
(14.38)
the angular momentum can also be written as
(
)
]
()
2
() ()
[
()
()
()
L
t
=
mr
t
ω
t
r ω r
t
t
t
.
(14.39)
kk
k
k
k
Dropping the function-of- t notation for the moment, we can express the i -th
component of L by
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