Chapter 10, Problem 80P

(III) A wheel with rotational inertia $I=\frac{1}{2}M{R}^2$ about its central axle is set spinning with initial angular speed ω₀, and is then lowered onto the ground so that it touches the ground with no horizontal speed. Initially it slips, but then begins to move forward and eventually rolls without slipping. (a) In what direction does friction act on the slipping wheel? (b) How long does the wheel slip before it begins to roll without slipping? (c) What is the wheel’s final translational speed? [Hint: Use   $\sum \stackrel{\to }{F}=m\stackrel{\to }{a},\sum {\tau }_{\text{CM}}=I_{\text{CM}}{\alpha }_{\text{CM}},$ and recall that only when there is rolling without slipping is ${\upsilon }_{\text{CM}}=\omega R$.]