Game Development Reference
In-Depth Information
Mathematically, you can write these relations between angular displacement, angular
velocity, and angular acceleration as:
ω = dΩ/dt
α = dω/dt = d 2 Ω/dt 2
ω = ∫ α dt
Ω = ∫ ω dt
ω dω = α dΩ
In fact, you can substitute the angular properties Ω, ω, and α for the linear properties
s , v , and a in the equations derived earlier for particle kinematics to obtain similar
kinematic equations for rotation. For constant angular acceleration, you'll end up with
the following equations:
ω 2 = ω 1 + α t
ω 2 2 = ω 1 2 + 2 α (Ω 2 − Ω 1 )
Ω 2 = Ω 1 + ω 1 t + (1/2) α t 2
When a rigid body rotates about a given axis, every point on the rigid body sweeps out
a circular path around the axis of rotation. You can think of the body's rotation as causing
additional linear motion of each particle making up the body—that is, this linear motion
is in addition to the linear motion of the body's center of mass. To get the total linear
motion of any particle or point on the rigid body, you must be able to relate the angular
motion of the body to the linear motion of the particle or point as it sweeps its circular
path about the axis of rotation.
Before we show you how to do this, we'll explain why you would even want to perform
such a calculation. Basically, in dynamics, knowing that two objects have collided is not
always enough, and you'll often want to know how hard, so to speak, these two objects
have collided. When you're dealing with interacting rigid bodies that may at some point
make contact with one another or with other fixed objects, you need to determine not
only the location of the points of contact, but also the relative velocity or acceleration
between the contact points. This information will allow you to calculate the interaction
forces between the colliding bodies.
The arc length of the path swept by a particle on the rigid body is a function of the
distance from the axis of rotation to the particle and the angular displacement, Ω. We'll
use c to denote arc length and r to denote the distance from the axis of rotation to the
particle, as shown in Figure 2-9 .