Figure 7.16 is a bird's-eye view of a smooth horizontal wire hoop that is forced to rotate at a fixed angular velocity co about a vertical axis through the point A. A bead of mass m is threaded on the hoop and is free to move around it, with its position specified by the angle [itex]\phi[/itex] that it makes at the center with the diameter AB. Find the Lagrangian for this system using as your generalized coordinate. (Read the hint in Problem 7.29.) Use the Lagrange equation of motion to show that the bead oscillates about the point B exactly like a simple pendulum. What is the frequency of these oscillations if their amplitude is small? I am sort of confused as to exactly what the motion is. Does the hoop rotate around the point A, and does this hoop then exert some force on the bead, causing it to rotate through angles around the loop, the angles being measured relative to the fixed point B? If this is true, is there someway of relating the angular velocity of the loop to the angular velocity of the bead?