g. Now suppose that we have attached not just two springs in parallel, but / springs. Write an equation that expresses the effective spring constant of the combination using the spring constant of the original spring & and the number of springs N. h. Is the combination of many springs connected in parallel softer, stronger, or the same as a single spring? 1. In the human body, the distance that muscles can stretch is limited by the size and range of motion of the body . If we assume that the maximum stretching distance of a muscle is fixed, then which is more desirable for the ability to lift heavier objects: a large effective spring constant or a small effective spring constant ? What does this imply about the way that muscles are built in the body? 2. Over the past 20 years, the development of sensitive physical techniques like atomic force microscopy and optical tweezers has allowed us to characterize the mechanical properties of DNA molecules in great detail. This characterization is important because DNA is subjected to a range of mechanical manipulations within the cell: it must be coiled, zipped, twisted, and deformed in a variety of ways during the replication or transcription process. An understanding of the elastic properties of DNA can give scientists insight into how DNA and proteins interact in order to carry out essential cellular processes. In this problem, you will explore how one of these modern physical techniques - optical tweezers - has allowed us to model the spring-like properties of DNA. If someone asked you to measure the spring constant & for a DNA molecule, how might you do so? Well, as with any spring, you'd probably like to be able to pick it up and tug on it, to see how much force you must apply in order to stretch it a certain distance. This would give you a sense of how taut or tense the spring is, and therefore a sense of its spring constant . Unfortunately, a single DNA molecule is tiny, so we can't just go to the bathroom cabinet and get a pair of everyday tweezers to pick it up. We must devise a cleverer tweezer! The diagram shows the key features of one such clever device, the "optical tweezer." The important thing to know right now is that one end of the DNA molecule is chemically attached to a small polystyrene bead (the bead's radius is about 10 com ), which is "trapped" in space by one or more laser beams. The other end of the DNA molecule is fixed in place, such as by attaching it to a surface, as shown in the figure