We present an optomechanical cavity which is fabricated by patterning a suspended metallic mirror on the tip of an optical fiber. Optically induced self-excited oscillations of the suspended mirror are experimentally demonstrated. The feasibility of employing on-fiber optomechanical cavities for sensing applications is discussed. A theoretical analysis evaluates the sensitivity of the proposed sensor, which is assumed to operate in the region of self-excited oscillations and the results are compared with the experimental data.