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Problem 1 Consider the frictionless rod, i.e. ß = 0. The equation of motion becomes d²r dt² m - mw²r= -mg sin(wt) with 1 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) = r。 and r¹(0) = vo. A) Analytically solve this initial value problem for r(t)

Problem 1 Consider the frictionless rod, i.e. ß = 0. The equation of motion becomes d²r dt² m - mw²r= -mg sin(wt) with 1 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) = r。 and r¹(0) = vo. A) Analytically solve this initial value problem for r(t)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I am trying to work through the problem attached.  I am struggling with part A.  Please give me guidance or help walk me through it. 

Problem 1 Consider the frictionless rod, i.e. ß = 0. The equation of motion becomes d²r dt² mw²r = -mg sin(wt) m 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) = r and r'(0) = vo. A) Analytically solve this initial value problem for r(t) B) Consider the initial position to be zero, i.e. r = 0. Find the initial velocity, vo, 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: ro = 0 and initial velocities of vo = 2.40, 2.45, 2.50, and the initial velocity you found in part B). Use 0 ≤ t ≤ 5
Transcribed Image Text:Problem 1 Consider the frictionless rod, i.e. ß = 0. The equation of motion becomes d²r dt² mw²r = -mg sin(wt) m 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) = r and r'(0) = vo. A) Analytically solve this initial value problem for r(t) B) Consider the initial position to be zero, i.e. r = 0. Find the initial velocity, vo, 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: ro = 0 and initial velocities of vo = 2.40, 2.45, 2.50, and the initial velocity you found in part B). Use 0 ≤ t ≤ 5
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