In this paper, the analytical conditions for sliding and passable motions on the periodically time-varying boundary in a friction-induced oscillator are presented first through the relative force product. The force product for the sliding motion in such a friction-induced oscillator is less than zero. However, the force product is greater than zero for the passable motion. Based on such analytical conditions, the criteria for the onset and vanishing of the sliding motion are developed. The sliding fragmentation for such an oscillator is discussed. The effects of system parameters to the sliding motion are discussed by use of the sliding mapping. Finally, the sliding motions for such a friction-induced oscillator are illustrated. From this investigation, the force product criterion is very useful to determine existence of the sliding and passable motions in such discontinuous dynamical systems. Such an investigation may provide a different way for sliding mode controls in dynamical systems.