We analyze experimentally and theoretically mechanical oscillation within an optical cavity stimulated by the pressure of circulating optical radiation. The resulting radio frequency cavity vibrations (phonon mode) cause modulation of the incident, continuous-wave (cw) input pump beam. Furthermore, with increasing cw pump power, an evolution from sinusoidal modulation to random oscillations is observed in the pump power coupled from the resonator. The temporal evolution with pump power is studied, and agreement was found with theory. In addition to applications in quantum optomechanics, the present work suggests that radiation-pressure-induced effects can establish a practical limit for the miniaturization of optical silica microcavities.