Question

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. z(t) = sin(t) +   1/ln(t)
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.

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              quality methods, correct solutions, proper conclusions, complete reasoning
Readability       organization, presentation, format, clarity, effectiveness
Fluency              proper notation, proper terminology, appropriate definitions, conciseness

Project 2: A bead sliding along a rod

A bead is constrained to slide along a rod of length L. The rod is rotating in a vertical plane with a constant angular speed, W, 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 r(t) denote the distance of the bead away from the pivot where r(t) 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

M d²r/dt² + B dr/dt – mw²r = -mg sin(wt) where m is the mass of the bead, B is the coefficient of viscous damping, w is the constant speed of angular rotation, g = 9.81 m/s² is the acceleration due to gravity, and r is the distance between the pivot and the bead.The rod is initially horizontal, and the initial conditions for the bead are r(0) = r₀ and r’(0) = v₀.

Problem 1

Consider the frictionless rod, i.e. B = 0. The equation of motion becomes m d²r/dt² - mw²r = -mg sin (wt) with g = 9.81 m/s² and a constant angular speed w.
The rod is initially horizontal, and the initial conditions for the bead are r(0) = r₀ and r’(0) = v₀.

A) Analytically solve this initial value problem for r(t)
B) Consider the initial position to be zero, i.e. r₀ = 0. Find the initial velocity, v₀, that results in a solution, r(t) , 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 r(t) displays simple harmonic motion, i.e. part B), find the minimum required length
of the rod, L, as a function of the angular speed, W.
E) Suppose W = 2, graph the solutions, r(t) , for the initial conditions given here: r₀ = 0 and initial velocities of v₀ = 2.40, 2.45, 2.50, and the initial velocity you found in part B). Use 0 < t < 5

Problem 2

Consider the frictionless rod, i.e. B = 0. The equation of motion becomes m d²r/dt² - mw²r = -mg sin (wt) with g = 9.81 m/s² and a constant angular speed W. The rod is initially horizontal, and the initial conditions for the bead are r(0) = r₀ and r’(0) = v₀. You will need to write an Improved Euler Method system solver to find r(t) and v(t)

A) Numerically solve for r(t) when W = 2, r₀ = 0, and v₀ = 2.40, 2.45, 2.50. Solve in the time interval t E [0,5] . Use step sizes h = 1/32, 1/128, 1/512 and compare your results. Also, compare your best numerical answers with your analytic answers from Problem 1 part E).
B) Numerically solve for r(t) when W = 2, r₀ = 0, and v₀ is selected to give simple harmonic motion, i.e. Problem 1 part B. Use small step sizes, e.g. h =  1/512, 1/2048, 1/8192, etc. Solve for the longest time interval that provides reasonable values for r(t) . Compare your results to the analytic solution that gives simple harmonic motion. What does this demonstrate about numerical solutions?

Answer 1

Problem 1) By trial and error method we can guess the solution of the equation is found to be

A) Substituting the initial conditions we get and . Solve them to get c1 and c2.

B) If  .

So

This solution doesn't tend to infinity only if c1=0. So

C) As the term in increases rapidly and reaches infinity at large time, the bead will eventually fly off.

D) From B for simple harmonic motion of bead the equation is . The maximum amplitude of the bead is . Hence the rod should be minimum