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. When an object is rolling without slipping on a horizontal surface, we can approximate the friction force to be zero, so that ax and z are approximately zero and vx and z are approximately constant. Rolling without slipping means vx $=$ rz and ax $=$ rz $.$ 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 $0$ about an axis through its center, is set on a horizontal surface for which the kinetic friction coefficient is 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 ax of the center of mass and z of rotation about the center of mass.

$($b$)$ The cylinder is initially slipping completely, so initially z $=$ $0$ but vx $=0.$ Rolling without slipping sets in when vx $=$ rz $.$ 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.