The vortex induced vibrations (VIVs) of a pivoted cylinder with finite height have been numerically investigated. A mathematical model is introduced and described, and the resulting equations are numerically solved for low Reynolds number Re = 100, 200 and several combinations of the governing parameters. Results on the solid body trajectories, the maximum amplitude of the oscillations, the hydrodynamic force coefficients, the wake structure, and details on the vortex shedding near the cylinder are presented and discussed. The numerical results compare reasonably well with the canonical system of VIV of two-degrees of freedom circular cylinder in the laminar regime. Also, qualitative similarities with closely related VIV systems at larger Re suggest interesting lines of future research. Analytical approximations for limiting cases are done and an excellent agreement with the numerical results is obtained.