Gyromagnetic experiments, the Barnett effect (magnetization by rotation) and the Einstein-de Haas effect (rotation by magnetization), were initially devised to test the molecular current hypothesis and determine the electron g-factor at the dawn of quantum mechanics. Advances in fabrication techniques and an ever increasing control of materials has led to continued miniaturization of not only integrated circuits but also of mechanical elements, such as cantilevers, giving rise to the field of micro-electro-mechanical devices (MEMS) and eventually nano-electro-mechanical devices (NEMS). In this thesis, we focused on the Barnett effect and the interplay of this effect with its close relative, the Einstein-de Haas effect in mesoscopic or nanoscopic systems. We perform a feasibility study concerning the Barnett effect in magnetic nanostructures and thin films. Furthermore, we study the the interplay of the Barnett and Einstein-de Haas effects in a suspended quasi one-dimensional magnetic wire containing a tail-to-tail domain wall and in a freely rotating grain containing a magnetic moment.