A rigid body \(\mathcal {B}\) moves in an otherwise quiescent viscous liquid filling the whole space outside \(\mathcal B\) , under the action of a time-periodic force \(\varvec{\textsf{f}}\) of period T applied to a given point of \(\mathcal B\) and of fixed direction. We assume that the average of \(\varvec{\textsf{f}}\) over an interval of length T does not vanish, and that the amplitude, \(\delta \) , of \(\varvec{\textsf{f}}\) is sufficiently small. Our goal is to investigate when \(\mathcal {B}\) executes a nonzero net motion; that is, \(\mathcal {B}\) is able to cover any prescribed distance in a finite time. We show that, at the order \(\delta \) , this happens if and only if \(\varvec{\textsf{f}}\) and \(\mathcal B\) satisfy a certain condition. We also show that this is always the case if \(\mathcal B\) is prevented from spinning. Finally, we provide explicit examples where the condition above is satisfied or not. All our analysis is performed in a general class of weak solutions to the coupled system body-liquid problem.
