In this paper, we analyze for stability the problem of planar vibrational motion of a satellite about its centre of mass. The satellite is dynamically symmetric whose center of mass is moving in a circular orbit. The in-plane motion is a simple pendulum-like motion in which the axis of symmetry of the satellite remains in the orbital plane. It is expressed in terms of elliptic functions of time. Using Routh's equations we study the orbital stability of planar vibrations of the satellite, in the sense that the stable in-plane motions remain under perturbation very near to the orbital plane. The linearized equation for the out-of-plane motion takes the form of a Hill's equation. Detailed analysis of stability using Floquet theory is performed analytically and numerically. Zones of stability and instability are illustrated graphically in the plane of the two parameters of the problem: the ratio of moments of inertia and the amplitude of the unperturbed motion.