Chapter 10, Problem 10.92CP

When an object is rolling without slipping, the rolling friction force is much less than the friction force when the object is sliding: a silver dollar will roll on its edge much farther than it will slide on its flat side (sec Section 5.3). When an object is rolling without slipping on a horizontal surface, we can approximate the friction force to be zero, so that a_x and α_z are approximately zero and v_x and ω_z. are approximately constant. Rolling without slipping means v_x = rω_z and α_x= rα_z. If an object is set in motion on a surface without these equalities, sliding (kinetic) friction will act on the object as it slips until rolling without slipping is established. A solid cylinder with mass M and radius R, rotating with angular speed ω₀ about an axis through its center, is set on a horizontal surface for which the kinetic friction coefficient is μ_k.(a) Draw a free-body diagram for the cylinder on the surface. Think carefully about the direction of the kinetic friction force on the cylinder. Calculate the accelerations a_x of the center of mass and α_z of rotation about the center of mass, (b) The cylinder is initially slipping completely, so initially ω_z = ω₀ but v_x = 0. Rolling without slipping sets in when v_x = rω_z. Calculate the distance the cylinder rolls before slipping stops, (c) Calculate the work done by the friction force on the cylinder as it moves from where it was set down to where it begins to roll without slipping.