Project 2 This assignment follows the standard form for a project submission. You need to include an introduction, primary discussion, and summary. Include graphs, tables, and images, as necessary, to improve the clarity of your discussion. Your project needs to be both correct and well written. Communication remains a critical component of our modern, technological society. A few notes about format: you MUST use MS Word for your project and use Equation Editor for all mathematical symbols, e.g. () = sin() + 1 . ln() If you have any questions about the requirements for this project, ask before you submit. This project addresses modeling with Ordinary Differential Equations and solutions to those equations. You will solve a problem analytically and program an Improved Euler's method numerical solver. You are required to write your own numerical methods in either MATLAB or MS Excel. You are not allowed to use an analytic solution or numerical solver written by anyone else. The physical model and problems are provided in the MS PowerPoint presentation entitled: Project2Description.pptx. Open the PowerPoint file and hit the F5 button on your keyboard to activate the animation. As you review the required problems, ask questions if you encounter anything that is not clear. A *.pdf file has been provided if you cannot open the *.pptx file. Grading Rubric Projects provide you with an opportunity to improve your Mathematical skills as well as your communication. For this project you will need to correctly solve the problems and effectively communicate your ideas and solutions. This assignment will be evaluated across the areas of Validity, Readability, and Fluency. Validity - Validity corresponds to the validity of your arguments. It addresses the extent to which your method is appropriate, your calculations are correct, and your analysis is accurate. Readability - If your written work is not readable it cannot be assessed. Since the ability to communicate Mathematics is a focal point for this class, special attention will be paid to the readability of your work. Fluency - Mathematics is a concise and precise language, and we wish to enhance your fluency. Therefore, part of every assessment will focus on your ability to incorporate correct, established notation and terminology into your written work Evaluation criteria Validity Readability Fluency Descriptive adjectives quality methods, correct solutions, proper conclusions, complete reasoning organization, presentation, format, clarity, effectiveness proper notation, proper terminology, appropriate definitions, conciseness Scoring 40% 35% 25% Project 2 Course and Learning Objectives This Project supports the following Course Objectives: CO-3: Investigate linear systems of Ordinary Differential Equations (ODEs) and their associated solutions. CO-4: Formulate single variable models for time dependent systems. CO-5: Analyze the qualitative behavior of single variable, differential time models and their functional solutions. CO-6: Synthesize several solution techniques to determine their appropriate application. Project 2: A bead sliding along a rod Simple Harmonic Motion Free Sliding Bead A is constrained to slide along a rod of length . The rod is rotating in a vertical plane beadwith a constant angular speed, , about a pivot in the middle of the rod. The pivot allows the bead to freely slide along the rod, i.e. the pivot does not impede the movement of the bead. Let denote the distance of the bead away from the pivot where can be positive or negative. Equation of Motion Applying Newton's second law provides a balance of forces due to gravity, friction, centripetal acceleration, and linear acceleration. The equation resulting from these forces is where is the mass of the bead, is the coefficient of viscous damping, is the constant speed of angular rotation, is the acceleration due to gravity, and is the distance between the pivot and the bead. The rod is initially horizontal, and the initial conditions for the bead are and . Problem 1 Consider the frictionless rod, i.e. . The equation of motion becomes with and a constant angular speed . The rod is initially horizontal, and the initial conditions for the bead are and . A) Analytically solve this initial value problem for B) Consider the initial position to be zero, i.e. . Find the initial velocity, , that results in a solution, , which displays simple harmonic motion, i.e. a solution that does not tend toward infinity. C) Explain why any initial velocity besides the one you found in part B) causes the bead to fly off the rod. D) Given displays simple harmonic motion, i.e. part B), find the minimum required length of the rod, , as a function of the angular speed, . E) Suppose , graph the solutions, , for the initial conditions given here: and initial velocities of , and the initial velocity you found in part B). Use Problem 2 Consider the frictionless rod, i.e. . The equation of motion becomes with and a constant angular speed . The rod is initially horizontal, and the initial conditions for the bead are and . You will need to write an Improved Euler Method system solver to find and A) Numerically solve for when , , and . Solve in the time interval . Use step sizes and compare your results. Also, compare your best numerical answers with your analytic answers from Problem 1 part E). B) Numerically solve for when , , and is selected to give simple harmonic motion, i.e. Problem 1 part B. Use small step sizes, e.g. etc. Solve for the longest time interval that provides reasonable values for . Compare your results to the analytic solution that gives simple harmonic motion. What does this demonstrate about numerical solutions