Game Development Reference

In-Depth Information

and acceleration are radians (rad), radians per sec (rad/s), and radians per second-

squared (rad/s
2
), respectively.

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
.