Dry Friction Camouflaged in Viscous Drag

Here is presented an interesting problem that can be used to introduce students to a variety of physics topics including non-inertial frames and frictional forces, rotational dynamics, and damped oscillations; the normal force also appears, but not in its usual guises. The problem is a generalized version of problem 3.34 presented in the “Oscillations and Waves,” part of the collection of problems in general physics by I. E. Irodov. The formulation of the problem, now extended by the kinetic friction inclusion, is given in what follows. In the setup shown in Fig. 1, a massive sleeve is fixed between two long identical massless springs. The sleeve can slide over a long horizontal bar against the dry friction of kinetic friction coefficient μ, which emerges to oppose to the relative lateral motion of the two solid surfaces in contact. The horizontal bar rotates at a constant angular velocity about a vertical axis passing through the middle. Above what critical value of the angular velocity (ωC* = ?) will there no longer be oscillations of the sleeve if in the absence of the friction that value amounts to ωC? Initially, the sleeve is given a velocity to start its motion from the center along the rotating bar. Gravitational effects are considered negligible here, but we do use N for the normal force of contact that the bar exerts on the sleeve in the x-y plane. The original problem by Irodov illustrates the same example in order to determine the critical value of angular velocity ωC, but specifically disregards friction (μ = 0).