For a moored semisubmersible platform, the hydrodynamic viscous damping on slender members is essential for the prediction of platform motions in resonant slow-drift oscillations. This study presents an experimental study of hydrodynamic forces acting on a slowly surging vertical circular cylindrical model in still water and regular waves, emphasizing on the viscous force induced by both the wave-frequency motion and low-frequency oscillation of the cylinder. Experimental results were analyzed using both the Independent-Flow form of Morison equation and the Relative-Velocity form of Morison equation, and compared with those for the cylinder model fixed in regular waves and oscillating with single- and dual-frequency in still water. The mutual influences between wave-frequency and slow-surging motion were discussed. Hydrodynamic coefficients based on these two force models were shown against various suitable non-dimensional parameters and the correlations between them were discussed. The applicability of these two forms of Morison equation was investigated and their applicable ranges were given. In addition, experimental results of the corresponding transverse forces were also presented and discussed.