Game Development Reference
In-Depth Information
a v is the acceleration of the particle in the rotating reference
frame. The total linear acceleration
()
()
where
t
=
t
r
r
()
a
t
of the particle is thus given by
()
()
()
()
()
()
()
a
t
= = ×+ ×+
=
v ω r ω r a
ω r ω r ω v
t
t
t
t
t
t
f
()
()
()
()
()
()
()
t
×
t
+
t
×
t
+
t
×
t
+
a
t
.
(14.11)
r
r
()
()
Since
r is equal to the linear velocity
t
v
t
of the particle, we can write
]
a ω r ωωr
()
t
=
()
t
×
()
t
+
()
t
×
()
t
×
()
t
+
2
ω v
()
t
×
()
t
+
a
()
t
. (14.12)
[
r
r
()
The force
F
t
experienced by the particle is therefore
()
()
()
()
()
()
()
]
t
=
mt
a
=
mt
ω r
×
t
+
mt
ωωr
×
t
×
t
F
[
()
()
()
+
2
mt
ω v
×
t
+
m t
a
.
(14.13)
r
r
()
In the reference frame of the rotating system, the force
t
on the object ap-
F
r
pears to be the following.
]
()
()
()
()
()
()
()
()
F
t
=
mt
a
=
F
t
mt
ω r
×
t
mt
ωωr
×
[
t
×
t
r
r
()
()
2
mt
ω v
×
t
(14.14)
r
As expected, the centrifugal force shows up again, but there is also a new term
called the Coriolis force that acts on the particle in a direction perpendicular to its
velocity in the rotating reference frame (see Figure 14.4). The Coriolis force,
given by
()
()
=−
2
mt
ω v
×
t
,
(14.15)
F
Coriolis
r
()
ω
t
() ()
2
mt
ωv
t
r
()
v
r t
()
r
t
α
O
Figure 14.4. The Coriolis force.