A small bead of mass \(m\) is threaded on a frictionless circular wire hoop of radius \(R\). The hoop rotates about a vertical axis through its center with a constant angular velocity \(\omega\), as shown in the figure. The bead is observed to remain at a constant position on the hoop at an angle \(\theta\) relative to the vertical axis, where \(\theta\) is measured from the bottom of the hoop. Which of the following expressions represents the angular velocity \(\omega\) required to maintain the bead at this fixed position?