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in a graph that is horizontally “compressed”: the frequency of oscillation
is increased. Likewise, smaller values of K cause the spring to oscillate
more slowly, and the graph is expanded. Furthermore, we observe that
the frequency is proportional to the square root of K—when we increase
K by a factor of four, the frequency doubles. This gives us a hint as to
where K should appear, since all we are doing is scaling the time axis.
√
x(t) = cos(
K t),
A solution, but is it the
only one?
√
√
x(t) = −
K sin(
K t),
√
x(t) = −K cos(
K t).
One verifies that this is a solution to the differential equation by plugging
it into Equation (12.8). Remembering that a(t) = x(t), we have
a(t) = −Kx(t),
√
√
−K cos(
K t) = −K(cos(
K t)).
√
The quantity
K is the angular frequency and comes up often enough that
we find it helpful to introduce the notation
√
ω =
K =
k/m,
Angular frequency
and we can write the solution as
x(t) = cos(ωt);
(12.9)
hence the reason for the name “angular frequency” becomes apparent.
So we have found the kinematics equation for the spring. Or, perhaps
we should say that we have found a solution to the differential equation.
There are some degrees of freedom inherent in the motion of the spring that
are not accounted for in Equation (12.9). First, we are not accounting for
the maximum displacement, known as the amplitude of the oscillations and
denoted A. Our equation always has an amplitude of 1. Second, we are
assuming that x(0) = A, meaning the spring was initially stretched to the
maximum displacment A and released with zero initial velocity. However,
in general, we could have pulled it to some displacement x
0
= A and then
given it a shove so it has some initial velocity v
0
.
It would appear that we have three more variables which need to be
somehow accounted for in our equation if it is going to be completely gen-
eral. As it turns out, the three variables we have just discussed—the ampli-
tude, initial position, and initial velocity—are interrelated. If we pick any
two, the value for the third is locked in. We'll keep A as is, but we'll replace
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