Consider a bead threaded on a circular hoop rotating at a constant angular velocity $\backslash (\backslash$Omega $\backslash ),$ about an axis parallel to a diameter.

$1.$ Working with Newton's laws in a frame rotating with the hoop, find the equation of motion of the bead.

$2.$ Find the equilibrium positions of the bead, and explain them using a free$-$body diagram

Hint: We solved this problem in class using Lagrangian mechanics, so you should be able to find the solutions. It should be interesting to compare the two methods $($but it is not required to do this in the problem$).$