ABSTRACT
Consider a mechanical system consisting of a taut string that is stretched between two immovable walls. Suppose that N identical beads of mass m are attached to the string in such a manner that they cannot slide along it. Let the beads be equally spaced a distance a apart, and let the distance between the first and the last beads and the neighboring walls also be a. (See Figure 4.1.) Consider transverse oscillations of the string, that is, oscillations in which the string moves up and down (i.e., in the y-direction). It is assumed that the inertia of the string is negligible with respect to that of the beads. It follows that the sections of the string between neighboring beads, and between the outermost beads and the walls, are straight. (Otherwise, there would be a net tension force acting on the sections, and they would consequently suffer an infinite acceleration.) In fact, we expect the instantaneous configuration of the string to be a set of continuous straight-line segments of varying inclinations, as shown in the figure. Finally, assuming that the transverse displacement of the string is relatively small, it is reasonable to suppose that each section of the string possesses the same tension, T . (See p. 49.)
It is convenient to introduce a Cartesian coordinate system such that x measures distance along the string from the left wall and y measures the transverse displacement of the string from its equilibrium position. (See Figure 4.1.) Thus, when the
FIGURE 4.1 A beaded string.
