8

Segment response (Eq. ) -

There are two ways of looking at this equation. First, it basically just states Newton's law, : on each segment two tension forces pull along y in opposite directions (hence the derivative), measures the mass of the segment and is the acceleration. Another way is to rewrite it as , which says that the acceleration of each segment is in direct proportion to the rate of change in the y-component of the tension and in inverse proportion to the mass per unit length . The inverse proportion to comes directly from Newton's law F=ma. The smaller the mass of each segment the more acceleration we expect. The proportionality to the derivative comes from the fact that tension pulls in opposite directions on the ends of each segment (Figure 2). If the tension in the y-direction were constant, then the two tensions would cancel and the segment would not accelerate. The greater the rate of change of the tension , the greater the difference in tension on the two sides of a segment and the greater its acceleration. Thus, we do expect the acceleration to be in direct proportion to .