A ring of mass M, radius R, and rotational inertia MR² is initially sliding on a frictionless surface at constant velocity v₀ to the right, as shown above. At time t=0 it encounters a surface with coefficient of friction µ and begins sliding and rotating. After traveling a distance L, the ring begins rolling without sliding. Express all answers to the following in terms of M, R, v₀ , µ, and fundamental constants, as appropriate.
(a) Derive an expression for the magnitude of the following as the ring is sliding and rotating.
(i) The linear velocity v of the ring as a function of time t
(ii) The angular velocity ω of the ring as a function of time t
