For the study of drag phenomena in linear hydrodynamics, the method of induced forces has proved to be an efficient tool. Here the authors demonstrate that this method is also a suitable tool to calculate the drag on a submerged body in various linear approximations to the Navier-Stokes equations. It is shown that this method may be derived from a variational principle, and the stationary value of an appropriate functional is the drag. The derivation of the equations' set for the induced force moments is given explicitly for two relevant hydrodynamical problems: a sphere moving slowly along the axis of a rotating viscous fluid, and a sphere in Oseen's flow.