Question

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

Answer 1

A uniform wheel of radius R is set in to rotation about its axis (case -1) at an angular speed $\omega$. This rotating wheel is now placed on rough horizontal. Because of its friction at contact, the wheel accelerates forward and its rotation decelarates. As the rotation, if we consider the net momentum before pure rolling and after pure rolling remains constant pure. Before rolling the wheel was only rotating around its axis.

Therefore, Angular momentum

$=I_{cm}\omega +m(V\times R)$

$=\frac{1}{2}m{R}^2\left(\frac{V}{R}\right)+mVR$

$=\frac{3}{2}mVR$

Angular momentum,

= $mVR=\frac{1}{2}m{R}^2\omega =\frac{1}{2}mVR$

= $\Rightarrow V=\frac{\omega R}{3}$