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A new model for the numerical simulation of a rigid body moving in a viscous fluid flow using FEM is presented. One of the most interesting features of this approach is the small computational effort required to solve the motion of the rigid body, comparable to a pure fluid solver. The model is based on the idea of extending the fluid velocity inside the rigid body and solving the flow equations with a penalization term to enforce rigid motion inside the solid. In order to get the velocity field in the fluid domain the Navier-Stokes equations for an incompressible viscous flow are solved using a fractional-step procedure combined with the two-step Taylor-Galerkin for the fractional linear momentum. Once the velocity field in the fluid domain is computed, calculation of the rigid motion is obtained by averaging translation and angular velocities over the solid. One of the main challenges when dealing with the fluid-solid interaction is the proper modelling of the interface which separates the solid moving mass from the viscous fluid. In this work the combination of the level set technique and the two-step Taylor-Galerkin algorithm for tracking the fluid-solid interface is proposed. The good properties exhibited by the two-step Taylor-Galerkin, minimizing oscillations and numerical diffusion, make this method suitable to accurately advect the solid domain avoiding distortions at its boundaries, and thus preserving the initial size and shape of the rigid body. The proposed model has been validated against empirical solutions, experimental data and numerical simulations found in the literature. In all tested cases, the numerical results have shown to be accurate, proving the potential of the proposed model as a valuable tool for the numerical analysis of the fluid-solid interaction.