The planar vibration of a thin elastic rod with circular cross section in viscous medium is discussed. Based on the Kirchhoff's theory the dynamical equations of the rod are established in the Frenet coordinates of the centerline. The torsional vibration is decoupled from the flexural vibration when the centerline is an arbitrary planar curve. The planar torsional vibration of an arbitrary planar rod and the planar flexural vibrations of an axially compressed straight rod and a ring without torsion are discussed when the ends of the rod are fixed. The natural frequencies and the damping coefficients are derived. It is proved that the Lyapunovs and Eulers conditions of stability of an axially compressed straight rod in the space domain are the sufficient and necessary condition of asymptotic stability of the rod in the time domain, or the necessary condition of stability of the rod without damping. The asymptotic stability of a ring in viscous medium is always satisfied.