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 (see 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 \(a_z\) are
approximately zero and \(v_x\) and \(\omega_z\) are approximately constant.
Rolling without slipping means \(v_x = r\omega_z\) and \(a_x = r\alpha_z\) . 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 \(\omega_0\) about an axis through its center, is
set on a horizontal surface for which the kinetic friction coefficient is
\(\mu_k\). (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 \(a_z\) of rotation
about the center of mass. (b) The cylinder is initially slipping completely,
so initially \(\omega_z = \omega_0\) but \(v_x =\) 0. Rolling without slipping
sets in when \(v_x = r\omega_z\) . 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.