Consider the frictionless rod, i.e. beta = 0. The equation of motion becomes m d²r/dt² - m ω² r = -mg sin (ωt) with g = 9.81 m/s² and a constant angular speed ω. The rod is initial 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.