Two identical particles of mass \(\displaystyle m\) move in a circle of radius \(\displaystyle R\), \(\displaystyle 180^ o\) out of phase, at an angular velocity \(\displaystyle \vec{\omega } = \omega _ z\hat{k}\) in a plane parallel to but a distance \(\displaystyle h\) above the x-y plane. Treat the two particles as a system.
Calculate \(\displaystyle \vec{L}_ S\), the angular momentum of the system about point \(\displaystyle S\). Express your answer in terms of \(\displaystyle m\), \(\displaystyle R\), \(\displaystyle h\), \(\displaystyle \omega _ z\), \(\displaystyle \hat{k}\), and \(\displaystyle \hat{r}\) as needed.