Question

A thin circular loop of radius R rotates about its vertical diameter with an angular frequency ω. Show that a small bead on the wire loop remains at its lowermost point for ω ≤√(g/R). What is the angle made by the radius vector joining the centre to the bead with the vertical downward direction for ω=√(g/R)? Neglect friction.

Answer 1

Let the radius vector joining the bead with the centre make an angle θ, with the vertical downward direction.

OP = R = Radius of the circle
N = Normal reaction

The respective vertical and horizontal equations of forces can be written as:

Since cosθ ≤ 1, the bead will remain at its lowermost point for g/(Rω²)≤ 1, i.e.,for ω≤ √(g/R)