5.3.2 A Differential Equation for Spring Motion

By Newton's Second Law of Motion, the total force on the moving object of mass \(m\) is also its mass times its acceleration. Thus, we have a relationship between the displacement \(x\) and its second derivative:

Some features of this equation should look familiar and some should appear new. First, under the influence of gravity, our bouncing mass is a falling body — but with a second force acting on it, the spring. When the falling body had no other force acting on it, our differential equation model expressed the second derivative of the position function as a constant. Here the second derivative is proportional to the position function.

Second, as was the case with the falling body, we should expect to undo differentiation twice to find \(x\) as a function of \(t\). But, unlike the falling body model, it is not at all clear how to do that. We might also expect that two "undifferentiation" steps will need two initial conditions, probably an initial velocity and an initial position.

Finally, note that the phrase "proportional to the unknown function" has appeared before — in natural population growth. What's different here is that it is the second derivative, not the first, that is proportional to the unknown function. Of course, an exponential function, say, \(f(t)=e^{rt}\), has a second derivative that is proportional to the function itself, but it seems unlikely that such a function could tell us anything about bouncing up and down on a spring. Let's rule out exponential functions as solutions to the differential equation right away.

Activity 1

1. Calculate the second derivative of

   ${\displaystyle f\left(t\right)=e^{rt}\mathrm{.}}$

2. Show that \(f\,''\) is proportional to \(f\) and that the proportionality constant must be positive, no matter what sign \(r\) has — as long as \(r\) is not \(0\).

3. Explain why \(f\) cannot be a solution of an equation of the form

   ${\displaystyle m\frac{d^{2}x}{d{t}^2}=-kx\mathrm{.}}$

Comment on Activity 1

We have just eliminated a promising candidate — promising mathematically, but not physically — for a solution to the differential equation

In fact, as we will see, the functions we need to model spring motion and other oscillations and vibrations are sines and cosines. We turn now to investigating these fundamental functions.