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

==

rad s

.

(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

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