There are two questions: 1. Muscles as springs, and 2. Spring constant of DNA. 1. Muscles as springs. The muscles in animals are complex structures consisting of bundles of long fibers that contract in response to chemical signals. The force provided by these contractile fibers when they are lengthened and when they are bundled depends on the physics of how connected springs add their forces. In order to understand how this works, let's consider a simplified model of a muscle: many springs connected in different ways. Let's think about how springs respond to being pulled rather than to how they pull on other objects. We'll take as our basic element a single spring (model of a fiber) with rest length , and spring constant k. If it is pulled from opposite directions by a tension force, T, it will stretch an amount Af that satisfies Hooke's law T = kAl, where the total stretched length of the spring is now lo + Al. First let's consider the effect of linking fibers together end to end into a longer fiber. (In a biological muscle, multiple cells combine, creating single long cells with multiple nuclei.) This kind of connection is referred to as "series". Let's start by considering two identical springs each having spring constant k, linked together (shown by the black oval in the figure) and T pulled from opposite ends by equal tension forces T. We imagine that they are connected by molecules that are short and don't stretch significantly compared to the springs themselves. a) Consider this combination as a single "effective" spring. If one spring stretches a distance Al when pulled by the force T, what distance AL would this combined spring stretch when pulled by the same force T? Explain your answer. b) If we define the effective spring constant of the combined springs by the equation T = KeffAL, where AL is the total amount the series springs stretch (your answer to part a), how is keff related to k? Explain your