Consider the frictionless rod, i.e. β=0. The equation of motion becomes
m (d^2 r)/(dt^2 )-mω^2 r=-mg sin(ωt)
with g=9.81 m/s^2 and a constant angular speed ω.
The rod is initially horizontal, and the initial conditions for the bead are r(0)=r_0 and r^′ (0)=v_0.
A)Analytically solve this initial value problem for r(t) B)Consider the initial position to be zero, i.e. r_0=0. Find the initial velocity, v_0, 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, ω. E)Suppose ω=2, graph the solutions, r(t), for the initial conditions given here: r_0=0 and initial velocities of v_0=2.40, 2.45, 2.50, and the initial velocity you found in part B). Use 0≤t≤5
Project 2: A bead sliding along a rod
Simple Harmonic Motion
A bead is constrained to slide along a rod of length . The rod is rotating in a vertical plane with 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.
Free Sliding Bead
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 .
Equation of Motion
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 .
Analytically solve this initial value problem for
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.
Explain why any initial velocity besides the one you found in part B) causes the bead to fly off the rod.
Given displays simple harmonic motion, i.e. part B), find the minimum required length of the rod, , as a function of the angular speed, .
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 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 .
You will need to write an Improved Euler Method system solver to find and
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).
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?
Problem 2
