Chapter 5
Modeling with Differential Equations
5.3 Periodic Motion
Section Summary
In this section we investigated oscillating spring motion as a simple example of periodic motion. Ignoring the decay of motion caused by friction and air resistance, we found that this motion may be described by a differential equation of the form
where \(x\) is the displacement of the spring, \(m\) is the mass of the weight on the spring (the "spring bob"), and \(k\) is the spring constant.
This problem is physically simple, but the mathematical formulation involves a second-order differential equation and thus represents a new problem. None of the functions we studied in our first four chapters can represent oscillating behavior, so we turned our attention to the sine and cosine functions -- familiar from prior mathematical study -- as possible building blocks for oscillations.
We reviewed the basic properties of these two trigonometric functions and the general notions of period and frequency. Then we determined the derivatives of these functions:
In the next section we will return to the spring motion differential equation, where we will see that the sine and cosine functions do indeed enable us to describe solutions.
