Game Development Reference
In-Depth Information
Since the bowling ball is initially sliding without rolling, the initial angular velocity,
w
0
, is
equal to zero. The moment of inertia for the ball can be found earlier in Table 4-2.
2
5
2
I
=
r
(4.45)
Inserting Equation (4.45) into Equation (4.44) results in an equation for angular velocity of
the bowling ball at any time,
t
.
5
2
μ
gt
ω
=
(4.46)
The next step in the analysis process is to determine an equation for the translational
velocity,
v
x
, of the ball as a function of time. The ball begins with an initial translational velocity
equal to
v
0
. As the ball slides down the lane, the friction force slows the ball down according to
Equation (4.15).
F
f
vv t
=+=−
v
t
=−
v t
μ
(4.47)
x
0
0
0
m
The time at which the bowling ball begins to roll without sliding can be found by multiplying
Equation (4.46) by the radius,
r
, and setting it equal to Equation (4.47).
5
2
v
−=
μ
gt
μ
gt
(4.48)
0
2
7
v
0
t
=
(4.49)
μ
g
The time at which the bowling ball rolls without sliding is proportional to the initial trans-
lational velocity of the ball and the coefficient of friction between the ball and the lane. The
translational velocity of the bowling ball when it begins to roll without sliding can be calculating
by inserting the results from Equation (4.49) into Equation (4.47).
v
vv g
2
5
=−
μ
0
=
v
(4.50)
x
0
0
7
μ
g
7
In looking at the results of Equation (4.50), an interesting conclusion appears. The velocity
at which the bowling ball begins to roll without sliding is independent of the coefficient of friction
between the ball and the lane. Whether the ball is sliding on a bowling lane or a slab of concrete,
it will begin to roll without sliding when the translational velocity decreases to 5/7 of its original
value. The time it takes the ball to roll without sliding, however, is a function of the coefficient
of friction. A ball traveling on a slippery surface, where the coefficient of friction is low, will
slide longer than will a ball traveling on a rough surface.
Summary
A lot of ground was covered in this chapter. You are now armed with the basic equations that
can be used to compute the translational and rotational motion of objects. These basic kinematic
relations will form the basis of almost every physical model we develop in the rest of the topic.
