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the unknown function x(t) that satisfies the equation. We have been able
to just barely scratch the surface of basic differential and integral calculus
in this topic, so we're not going to be able to cover the techniques of solving
differential equations. Luckily, you don't need to know differential equations
in order to verify that a proposed function x(t) is a solution—that requires
only the ability to differentiate the function x(t). As it turns out, this
will be su
cient in the few cases in which we bump up against differential
equations in this topic.
We can make a pretty good guess at the form of x(t) by looking at a
graph. We are not engaging in circular logic here; we don't need to know
x(t) in order to get a graph, all we need is a spring with some sort of
marking device attached to it.
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Such a graph is shown in Figure 12.5.
Figure 12.5
The graph of the motion of a spring. Hey,
that looks familiar. . .
This function ought to look familiar to you: it's the graph of the cosine
function. Let's see what happens if we just try x(t) = cos(t) as our position
function. Differentiating twice to get the velocity and acceleration functions
(remember, we learned about the derivative of the sine and cosine functions
in Section 11.4.6), we get
x(t) = cos(t),
Close, but not quite right
x(t) = − sin(t),
x(t) = − cos(t),
which is very close, but we're missing the factor of K.
To understand where K should appear in x(t), consider what happens to
the graph of x(t) when we change the value of K. In other words, we repeat
our physical experiment and vary the stiffness of the spring or the mass of
the marking device attached to the end of the spring. The result is that
larger values of K (stiffer springs or less massive marking devices) result
11
Professor Walter Lewin does this classroom demonstration in his physics class at
MIT. All of the lectures can be downloaded free through MIT OpenCourseWare at
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