Game Development Reference
In-Depth Information
equal to the distance from the center of the parallelogram to the corresponding face, then a
collision has occurred.
Because determining whether a collision has occurred with a complicated, asymmetric
shaped object is difficult, for the purposes of collision modeling you might want to model the
objects in your simulations as simple shapes. A person, for example, could be modeled as a
sphere on top of a cylinder.
Angular Momentum and Impulse
In the first part of this chapter, you learned about how the linear momentum of an object is
equal to its translational velocity multiplied by its mass. We developed the equations that
describe linear momentum and explored the concepts of a force impulse and conservation of
linear momentum. As we saw in Chapter 3, translational motion is not the only type of motion
an object can have. In this section, we will turn our attention to the momentum due to rotation.
Because angular momentum is analogous to linear momentum, the same derivation process
is performed to obtain the equations that characterize angular momentum. You learned in
Chapter 4 that a net torque,
t
, on an object results in an angular acceleration,
a
, of the object.
(6.25)
τα
=
As was explained in Chapter 4, the moment of inertia,
I
, is a material and geometrical
property that resists a change in angular motion. Equation (6.25) can also be expressed in
terms of the derivative of angular velocity. If the moment of inertia is constant, it can be pulled
into the derivative as well.
dI
dt
ω
τ
=
(6.26)
The product of the moment of inertia and the angular velocity is known as the
angular
momentum
,
L
.
(6.27)
L ω
=
Looking at Equation (6.26), we see that if there is no net torque on an object, then the
derivative of angular velocity with respect to time is zero, which means that angular velocity is
constant. To determine the changse in angular momentum caused by an applied torque, the
left- and right-hand sides of Equation (6.27) can be integrated with respect to time.
∫
τ
=−
dt
L
L
(6.28)
1
0
In Equation (6.28),
L
0
is the initial value of angular momentum, and
L
1
is the angular
momentum at the end of the time interval being considered. Equation (6.28) indicates that a
change in angular momentum of an object is equal to the integral of the net external torque on
the object as a function of time. The integration of the torque with respect to time is known as
the
angular impulse
of torque.
∫
ττ
=
dt
(6.29)
