Game Development Reference
In-Depth Information
()
()
z
0
0
=
=
B
z
ω
,
(14.111)
we can easily deduce
A ω
B
0
=
=
z
0 .
(14.112)
As discussed in Section 13.2.1, we may express Equation (14.108) in the form
()
(
)
zt
=
C ωt δ
sin
+
,
(14.113)
where
v
2
0
C
=
+
z
2
0
2
ω
z
1
0
δ
=
sin
.
(14.114)
C
The constant C represents the amplitude of the oscillations and corresponds to the
largest distance that the mass is ever displaced from its equilibrium position. The
constant δ represents the phase of the oscillations and corresponds to the initial
position of the mass.
Example 14.8. Determine the frequency and amplitude of a 2-kg mass attached
to a spring having a restoring constant of
. Suppose that the mass
was previously being pulled downward and that it is released at time
2
3kg s
k
=
=
with
t
0
an initial displacement of 0
z
=−
4 m
and an initial velocity of 0
=−
1m s
.
v
Solution. The angular frequency ω is given by
k
6
ω
==
.
(14.115)
2
Dividing by 2 π radians gives us the frequency f in oscillations per second:
6 0.195 Hz
24
ω
f
== ≈
.
(14.116)
ππ