In this dissertation, I model the effects of inertial forces on the dynamics of weaklyinteracting dilute atomic gas Bose-Einstein condensates (BECs) in rotating frames of reference through the use of numerical simulations. The dynamics of BECs are modeled here in the mean-field limit of the time-dependent Gross-Pitaevskii equation (GPE). When the GPE is considered in a rotating frame of reference, the inertial forces acting on a BEC manifest themselves as an effective coupling between the angular momentum of the condensate itself and the angular velocity vector that defines the rotational motion of the reference frame. Using a method-of-lines implementation of a generalized 4th-order Runge-Kutta method with finite differences to numerically approximate solutions of the GPE, I study how this effective coupling of the inertial forces to the condensate arise in practice by simulating the dynamics of BECs in three quantum-mechanical analogues of classical rotating systems. These systems are (1) a Foucault pendulum-like harmonic oscillator, (2) a mechanical gyroscope, and (3) a Compton generator. When possible, the observed dynamics of the condensates are compared via the Ehrenfest theorem to the dynamics predicted for the corresponding classical system. In general, I find that the observed dynamics for each of these systems agrees quite well with their classical counterparts. However, in a few cases, there appears to be some unaccounted for internal dynamics of the BECs that is not well-explained by the classical equations of motion. But further study will be required to elucidate the source of these possible discrepancies. In addition to simulating the quantum analogues of these classical rotating systems, I also extend some of my earlier work on the interference of matter waves in a Sagnac interferometer. In particular, I extend this work to consider how the observed Sagnac phase shift accumulates in time when the wave packets propagate within an “off-axis” ring potential that orbits the centre of rotation in the system. Here, once again, I find that the accumulation of the Sagnac phase shift may occur in discrete phase jumps under certain conditions. This unexpected behavior in the time-dependence of the Sagnac phase shift is ascribed to the initial angular momentum the wave packets acquire starting at rest with respect to the interferometer’s uniformly rotating ring potential, which prevents the center of mass between the wave packets from drifting linearly about the ring.