Recent theoretical work has shown that radiation pressure effects can in principle cool a mechanical degree of freedom to its ground state. In this paper, we apply this theory to our realization of an opto-mechanical system in which the motion of mechanical oscillator modulates the resonance frequency of a superconducting microwave circuit. We present experimental data demonstrating the large mechanical quality factors possible with metallic, nanomechanical beams at 20 mK. Further measurements also show damping and cooling effects on the mechanical oscillator due to the microwave radiation field. These data motivate the prospects for employing this dynamical backaction technique to cool a mechanical mode entirely to its quantum ground state.