In this paper the problem of a quadrotor that physically interacts with the surrounding environment through a rigid tool is considered. We present a theoretical design that allows to exert an arbitrary 3D force by using a standard near-hovering controller that was originally developed for contact-free flight control. This is achieved by analytically solving the nonlinear system that relates the quadrotor state, the force exerted by the rigid tool on the environment, and the near-hovering controller action at the equilibrium points, during any generic contact. Stability of the equilibria for the most relevant actions (pushing, releasing, lifting, dropping, and left-right shifting) are proven by means of numerical analysis using the indirect Lyapunov method. An experimental platform, including a suitable tool design, has been developed and used to validate the theory with preliminary experiments.