Basic Model for Task One.- We will also be introducing a new basic physical model from which we build denite integrals, that of gravitational attraction. Newton's law of gravitation is one example of what are called inverse square laws. For Newton's law. if two particles (called point-masses) in space have mass measures m1 and m2, then the force of their gravitational attraction is: M?) = G'";;;\"2 where G is Newton's gravitational constant (approximately 6.67 X 10711] and D is the xed distance between the particles. Importantly, this is a basic model, and only holds under certain assumptions. As in previous modules and contexts, when we break the assumptions of the basic model and introduce variation. there is a need for approximation and an eventual integration to calculate the desired quantity (in this case force). The underlying assumptions here are that the pointmasses are so small that their physical properties need only describing via their positions and their masses. Hence, Newton's Law only needs to measure the distance between the point masses lie. the distance between their locations) and to measure the points' masses to make the force calculation. We will break this assumption by replacing one of the masses with a thin rod, thus requiring extra attention to consider the new variation in distance. In this discussion, we will use Newton's law of gravitation as an opportunity to explore infinite accumulation processes. Determining whether or not an infinite process will yield a finite result (and how this is even possible) is an important skill that will remain central to much of our work in the rest of the course. Task One: You're creating a massless device that is meant to keep objects floating in space separate from each other. Your first goal is to separate a thin uniformly dense rod of length L meters and mass M kilograms from a particle of mass m kilograms. Your device will keep the particle a fixed initial distance Do meters from the closest point on the rod, and the rod will line up perfectly extending away from the particle. The following GeoGebra application simulates this effect specifically when Do = 4, L1 = 7, M = 10, and m2 = 6, and visualizes an approximation of the gravitational force (as if we were calculating a Riemann Sum). These initial measures will be helpful in your initial solving of the problem, but your group's final solution will need to keep Do, L1, M1, and my as unknowns (but fixed). The particle is shown in blue on the bottom left of the screen, and the rod is shown to the right of the particle. The applet breaks the rod into N segments. You may alter the "N=" slider to increase or decrease the number of segments into which the rod can be broken. You may alter the "Sample Distance=" slider to view the approximate sample force from the point mass to any particular rod section, which is calculated for you in the applet. Sample Distance = 5.75 N = 4 G . Sample Mass . Point Mass Sample Force = G . 2.5 . 6 D2 33.06 D = 5.75 Sample Mass = 2.5 Length = 7 Point Mass = 6 Mass = 10 Mass = 2.5Task: Answer the questions listed below. Again. be sure to label each piece of the required denite integrals to show how Newton's law of Gravitation is being applied locally. and to specify what each symbol in the integral formula means. 1. What is the force that your device will have to exert to keep the particle {of mass m) and the rod (of mass M. length L, and initial distance D0) from moving closer to each other. Be sure to label each piece of the denite integral to show how Newton's law of Gravitation is being applied locally. and to specify what each symbol in the integral formula means. (Hint: The integral is not too difcult to compute by hand. but if you want to check your work, you may use MATLAB to do so by setting up L, M. m, and Do as symbolic variables and integrating with respect to D) 2. If you copy the rod and fuse the copy to the side of the original rod farthest from the point, how much more force will your machine need to exert to keep the new rod from the particle? (Hint: Once you set up the integral. you may compute the integral in MATLAB if you wish) 3. You will make a rod of innite length by progressively copying and fusing the rod as described in Question 1. Is there a force that your machine can exert to keep the particle away from this innite length rod? Why or why not? Your answer must include a relevant limit. and must describe how the limiting process unfolds in conjunction with the progressive copying process. {Hint You may want to use a new variable, perhaps N. to represent the number of copies. You need to make the relevant computation by hand. but may want to check it in MATLAB.)