# A uniform wheel of radius R is set into rotation about its axis at an angular speed w.This rotating wheel is now placed on a rough horizontal surface with its axis horizontal. Because of friction at the contact , the wheel accelerated forward and its rotation decelerates till the wheel starts pure rolling on the surfaces . Find the linear speed of the wheel after it starts pure rolling.

$\begin {array} {1 1} (a)\;wR & \quad (b)\;\frac{wR}{3} \\ (c)\;\frac{wR}{2} & \quad (d)\;\frac{wR}{4} \end {array}$

Since friction is acting at the lower most point P, angular momentum conservation can be applied about the lower most point.
Angular momentum initial
$L_i= MV_{cm}R+ Iw$
$\qquad = Iw$[since $v_{cm}=0$]
$\qquad= \large\frac{MR^2w}{2}$
Angular momentum final
$L_f= MV_{cm}R+Iw'$
$w'= \large\frac{V_{cm}}{R}$
$\therefore L_f= MV_{cm}R+\large\frac{MR^2}{2} \frac{V_{cm}}{R}$
$\qquad= \large\frac{3}{2}$$MV_{cm}R L_i=L_f \large\frac{3}{2}$$MV_{cm}R= \large\frac{MR^2w}{2}$
$V_{cm}=\large\frac{Rw}{3}$
edited Jun 24, 2014