In this paper the initial “spin-up” stage of the flow field generated by the slow axial motion of a symmetric particle or drop in a rotating fluid in a “short cylinder” of length2His considered for small Rossby and Ekman numbers,RoandE. The motion starts from solid body rotation. Attention is focused on time-dependent effects in the extended core (outside theE1/2Ekman andE1/3Stewartson layers) and their influence on the behaviour of the drag force. When the particle velocity is established by an impulsive start, the flow-field and the drag build up on the spin-up time scale; the particle advances∼2RoE−1/2Hduring the transient stage. Results for the time-dependent geostrophic case, and also for various values of&Vegr;=(12H)1/2E1/4, for both sphere and disk particles, are presented. It is shown that when the particle velocity is established by release under a constant axial (say, buoyant) force, during the spin-up some rapid inertial oscillations may appear which are inconsistent with some of the assumption of the present analysis and require a separate investigation. The present theory is in qualitative agreement with available experiments, but a quantitative comparison requires new experiments. ©1997 American Institute of Physics.