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 into rotation about its axis

The wheel iskept on a rough surface rotating at an angular speed $\omega$

As the rotation decelerates the frictional force will act backward.

If we consider the net momentum at $A$ then it is zero.

Therefore Angular momentum

$L=I\omega$

$=\frac{1}{2}M{R}^2\omega$ ...(1)

After pure rolling the velocity of the wheel is considered to be $v$

Therefore angular momentum

$L^{\prime }=I_{cm}\omega +m(v\times R)$

$=\frac{1}{2}m{R}^2\frac{v}{R}+mvR$

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

Because Eq (1) and (2) are equal

$\therefore \frac{3}{2}mvR=\frac{1}{2}m{R}^2\omega$

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