Game Development Reference
In-Depth Information
Equations (4.32) and (4.35) can be combined to express the rotational acceleration,
a
, as a
function of the friction force.
(
)
t
==
F r
mg
sin
q
−
ma r
=
I
a
(4.36)
Because the object is rolling without slipping, the translational acceleration can be related
to the angular acceleration by the expression
a
=
ra
. Replacing
a
with
ra
in Equation (4.36), an
expression for the angular acceleration of the object can be obtained.
rmg
Imr
sin
q
a
=
(4.37)
2
+
The translational acceleration can be determined from Equation (4.37) once again using
the expression
a
=
ra
.
2
rmg
sin
q
ar
==
+
(4.38)
2
Imr
If the ramp angle,
q
, is constant, then the translational and rotational accelerations of the
object as it rolls down the ramp are constant as well. Expressions for the rotational and trans-
lational velocity of the object rolling down the ramp can be obtained by integrating Equations
(4.37) and (4.38).
rmg
sin
q
w
=
t
+
w
(4.39)
0
Imr
+
2
rmg
2
sin
q
v
=
t
+
v
(4.40)
0
2
Imr
+
The quantities
w
0
and
v
0
are the initial angular and translational velocities of the object. To
apply Equations (4.39) and (4.40) to any object, the mass, radius, and moment of inertia of the
object must be determined.
Exercise
4.
A cylinder and a sphere of equal diameter and mass are released at the top of a ramp at the same time.
Assuming that both objects roll without slipping, which object will reach the bottom of the ramp first?
Bowling Ball Kinematics
If you have ever been bowling, you know that if you throw the ball hard enough it will start off
sliding down the lane. As the ball slows down, it begins to roll rather than slide. The second
problem we will look at to demonstrate the rigid body kinematic analysis process is to analyze
the motion of a bowling ball as it travels down the lane. The information that we want to deter-
mine is how long the bowling ball will slide and what will be the translational and rotational
velocity of the ball when it begins to roll without sliding.
