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Unformatted text preview: 17 Springs: Part I Second-order differential equations arise in a number of applications. We saw one involving a falling object at the beginning of this text (the falling frozen duck example in section 1.2). In fact, since acceleration is given by the second derivative of position, any application requiring Newtons equation F = ma has the potential to be modeled by a second-order differential equation. In this chapter we will consider a class of applications involving masses bouncing up and down at the ends of springs. This is a particularly good class of examples for us to examine. For one thing, the basic model is relatively easy to derive, and is given by a second-order differential equation with constant coefficients. So we will be able to apply what we learned in the last chapter to derive reasonably accurate descriptions of the motion under a variety of situations. Moreover, most of us already have an intuitive idea of how these mass/spring systems behave. Hopefully, what we derive will correspond to what we expect, and may even refine our intuitive understanding. Another good point about the work we are about to begin is that many of the notions and results we will develop here can carry over to the analysis of other applications involving things that vibrate or oscillate in some manner. For example, the analysis of current in basic electric circuits is completely analogous to the analysis well carry out for masses on springs. 17.1 Modeling the Action The Mass/Spring System Imagine a horizontal spring with one end attached to an immobile wall and the other end attached to some object of interest (say, a box of frozen ducks) which can slide along the floor, as in figure 17.1. For brevity, this entire assemblage of spring, object, wall, etc. will be called a mass/spring system . Let us assume that: 1. The object can only move back and forth in the one horizontal direction. 2. Newtonian physics apply. 3. The total force acting on the object is the sum of: (a) The force from the spring responding to the spring being compressed and stretched. 357 358 Springs: Part I (a) (b) y ( t ) y ( t ) Y Y m m F spring F spring Natural length of the spring Natural length of the spring Figure 17.1: The mass/spring system with the direction of the spring force F spring on the mass (a) when the spring is extended ( y ( t ) > 0 ), and (b) when the spring is compressed ( y ( t ) < 0 ). (b) The forces resisting motion because of air resistance and friction between the box and the floor. (c) Any other forces acting on the object. (This term will usually be zero in this chapter. We include it here for use in later chapters, so we dont have to re-derive the equation for the spring to include other forces.) All forces are assumed to be directed parallel to the direction of the objects motion....
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