A square hoop ABCD is made of fine smooth wire and has side length 2a. The hoop is horizontal and rotating with constant angular speed ω about a vertical axis through A. A small bead which can slide on the wire is initially at rest, relative to the hoop, at the midpoint of the side BC. Choose axes fixed relative to the hoop, and let y be the distance of the bead from the vertex B on the side BC. Write down the position vector of the bead in the rotating frame.
Using the standard expression for acceleration in a rotating frame, show that y ̈ − ω^2 y = 0
Hence show that the time which the bead takes to reach a corner of the hoop is ω^−1 cosh^−1 2. Using dimensional analysis, explain why this time is independent of a.
Obtain an expression for the magnitude of the force exerted by the hoop on the bead.