The friction-induced vibration of a mass-slider with in-plane and transverse springs and dampers in sliding contact with a spinning elastic disc in three different situations of spinning speed, i.e. constant deceleration, constant acceleration and constant speed, is studied. The stick-slip motion in the circumferential direction and separation-re-contact behaviour in the transverse direction are considered, which make the system responses non-smooth. It is observed that the decelerating rotation of the disc can make the in-plane stick-slip motion of the slider more complicated in comparison with constant disc rotation and thereby exerting significant influence on the transverse vibration of the disc, while the accelerating rotation of the disc contributes to the occurrence of separation during the vibration and thus influencing the vibration behaviour of the system. Numerical simulation results show that distinct dynamic behaviours can be observed in the three situations of spinning speed of disc and three kinds of particular characteristics of differences are revealed. The significant effects of decelerating and accelerating disc rotation on the friction-induced dynamics of the system underlie the necessity to consider the time-variant spinning speed of disc in the research of friction-induced vibration and noise.