114-20_Elastic_Pot_Energy_Resilience

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University of North Carolina, Chapel Hill *

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Biology

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Jan 9, 2024

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Physics Activities for the Life Sciences (PALS) © Physics and Astronomy Education Research Group The University of North Carolina at Chapel Hill 1 Elastic Potential Energy and Resilience Introduction W hen an object, such as an artery wall, is stretched, it can store energy in the form of elastic potential energy. This elastic potential energy can be converted into other forms of energy when the object relaxes. Ideally, all of the elastic potential energy is converted into a form of energy that is useful, such as kinetic energy, and the amount that is converted into thermal energy is minimized. The effectiveness of converting elastic potential energy into useful forms of energy is quantified by the resilience of the material. Resilience is defined as the work done by the material as it relaxes divided by the work done on the material as it stretches (or contracts). Larger values of resilience indicate that less elastic potential energy is coverted into thermal energy when the material relaxes. Learning Goals After completing this activity, you should be able to… Experimentally determine the resilience of a rubber band by determining both the work done in stretching the rubber band and the work done by the rubber band as it relaxes. A. Force versus extension curves Recall the “Stretching Rubber” activity earlier in the course in which you applied a force to a rubber band and observed how much its length increased depending on the applied force. We have also discussed how the extension and compression of a spring varies with an applied force. 1. Download the Excel file “Studio_18_Spring_Rubber_Band_Data” from Sakai and look at the data in the “Spring” tab. The data was collected by hanging a mass
2 Physics Activities for the Life Sciences (PALS) © Physics and Astronomy Education Research Group The University of North Carolina at Chapel Hill from a spring and measuring the corresponding extension as shown. Make a plot of the force applied to the spring versus the extension of the spring, for both the loading and unloading phase. Include both curves on the same plot. The two curves may appear to lie right on top of one another, but they are slightly different. In order to see that difference, reduce the size of the markers on your plot. 2. What mathematical function (linear, power law, or exponential) describes the relationship between the applied force and the change in length of the spring? 3. Determine the spring constant of the spring. 4. Consider the force versus extension data for the rubber band in the “Rubber_ Band” tab, and compare to the data for the spring. How does the behavior of the rubber band differ from that of the spring? Briefly explain. B. Work of extension Now we want to consider other aspects of the behavior of the rubber band that are relevant to biological function, specifically the energy involved in the extension and contraction. 1. When the rubber band is extended, is the net external work done on the rubber band positive, negative, or zero? Explain. 2. Did the elastic potential energy of the system increase , decrease , or stay the same as the rubber band was stretched? Explain. A metal spring is not a very good model for the behavior of a biological material such as a tendon—the behavior of a rubber band (which is of course a biological material) is much closer to that of a tendon, as you saw in the earlier activity. However, the metal spring has the virtue of having a simple relationship between the applied force and the change in length, which makes it easy to calculate the work done on it. Examine your plot of force vs. extension for the metal spring, looking at the curves for the loading phase and for the unloading phase extension separately. 3. What force did work on the spring as it extended from its initial length to its final length? Briefly explain. 4. How can you determine the amount of work done on the spring from the plot of force vs. extension? Briefly explain.How much work was done on the spring to l l mass hanger Elastic Potential Energy and Resilience
Physics Activities for the Life Sciences (PALS) © Physics and Astronomy Education Research Group The University of North Carolina at Chapel Hill 3 extend it from its shortest length to its longest length? Show your work. This is called the work of extension. 5. As the spring was contracting as mass was removed from the hanger, did the spring do work, and if so, on what? Explain your answer. 6. How much work was done by the spring as it contracted from its longest to its shortest length? Show your work. 7. Was the work done on the spring as it extended larger than, smaller than, or equal to the work done by the spring as it contracted? Explain. We now want to carry out a similar exercise for the force vs extension plot for the rubber band. However, as you can see, the force vs extension curve for the rubber band is non-linear, so calculating the area under the curve will require numerical integration. 8. To the right of the force and extension data, you will see columns referring to extension intervals, and blank cells for the work done. Determine the work done in each extension interval, and then populate the blank cells for both the loading and unloading phase. 9. Was the work done on the rubber band as it extended larger than, smaller than, or equal to the work done by the rubber band as it contracted? Explain. C. Resilience and its biological effects The work done on the rubber band as it is extended is not equal to the work done by the rubber band as it contracts. The process of recovering the stored elastic potential energy is not perfect — some of the work done has been converted into thermal energy. The ratio of the amount of work done by the object (spring or rubber band) as it relaxes to the work done on the object as it is extended is called the resilience . 1. What are the units of the resilience? 2. What are the maximum and minimum possible values of the resilience? What do those values mean, physically? Explain. 3. What is the resilience of the metal spring? 4. What is the resilience of the rubber band? The mantis shrimp ( Gonodactylaceus falcatus ) has a “raptorial appendage,” which it contracts before lashing out with sufficient force to smash its hard-shelled prey. When the raptorial appendage contracts, it stores energy in the form of elastic potential energy, which is later converted to kinetic energy during a hunting strike. Elastic Potential Energy and Resilience
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4 Physics Activities for the Life Sciences (PALS) © Physics and Astronomy Education Research Group The University of North Carolina at Chapel Hill 5. The graph 1 below shows the force exerted on the raptorial appendage as a function of its contraction distance, for both the contraction and release of the appendage. What is the resilience of the raptorial appendage? Use the area under the red dashed line to approximate the area under the “contraction” curve and the area of the green triangle to approximate the area under the “release” curve. 6. Use the graph and your answer to the previous question to determine the kinetic energy of the raptorial appendage when it is released from rest after being contracted by a distance of 0.3 mm. Assume there is no change in gravitational potential energy during this motion. You may ignore any friction and drag forces that the water exerts on the appendage. D. Applications and Extensions To assess your understandings of some of this studio’s key ideas, your group must answer the following questions together without help from the instructors or other groups. Artery walls include the proteins collagen , and elastin , which both have a resilience of ~0.90. 2 1. T. I. Zack, T. Claverie, and S. N. Patek, “Elastic energy storage in the mantis shrimp’s fast predatory strike,” J. Exp. Bio. 212, 4002-4009 (2009). 2. J. Goslin et al ., “Elastic proteins: biological roles and mechanical properties,” Phil. Trans. R. Soc. Lond. B 357, 121–132 (2002). Elastic Potential Energy and Resilience
Physics Activities for the Life Sciences (PALS) © Physics and Astronomy Education Research Group The University of North Carolina at Chapel Hill 5 1. How does your measured value of the resilience of rubber compare to the resilience of collagen and elastin? Comment briefly on the significance of having a larger or smaller resilience. 2. The dragline silk produced by spiders and used to make the frame of their webs is a protein that has much lower resilience than does elastin, typically ~ 0.35. 1 If elastin had a resilience similar to that of dragline silk, would more or less energy be lost as thermal energy as the arteries expand and contract? Explain. 3. If elastin had the resilience of dragline silk, would the heart have to pump harder or not as hard in order to maintain a normal human blood pressure? Explain. 4. Another elastic protein, resilin, is found in insect joints; it can have a resilience as high as ~ 0.97 (which is how it gets its name). 1 Why is it advantageous for an insect, such as a jumping grasshopper, to have joints with such high resilience? Explain. Elastic Potential Energy and Resilience