This work addresses the observability analysis for a cooperative range-based navigation system that uses a beacon with circular motion installed on board the support platform to find the location of an underwater vehicle. A nonlinear model is first obtained in order to describe the motion of the vehicle and the beacon, that is attached to the tip of a 1 DoF rotating arm (deployed from a stationary support platform or vessel). A state-space realization is obtained and augmented in order to convert the nonlinear system into a Linear Time Varying (LTV) system. Then, the observability analysis is performed for the new system considering its equivalence since the state trajectories of the LTV system have one-to-one correspondence to the state trajectories of the nonlinear system. We show that, when the vehicle moves in circles or in straight lines, it is possible to ensure observability by rotating the arm that has the beacon attached. This result is important from a theoretical point of view, and represents a new way of implementing range-based navigation systems, without relying on deviation of the vehicle's mission or the need of another vehicle to ensure observability.