A uniform solid cylinder of mass \(M\) and radius \(R\) is held a negligible distance above a rough horizontal surface. The cylinder is given an initial clockwise angular velocity \(\omega_0\) about its central axis and is then dropped so that it lands gently on the surface without bouncing. The coefficient of kinetic friction between the cylinder and the surface is \(\mu_k\). The cylinder initially slips on the surface, but after a time interval \(\Delta t\), it begins to roll without slipping. The rotational inertia of a uniform solid cylinder is \(I = \dfrac{1}{2}MR^2\).