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