A wheel is rolling without slipping on a horizontal surface. In an inertial frame of reference in which the surface is at rest, is there any point on the wheel that has a velocity that is purely vertical? Is there any point that has a horizontal velocity component opposite to the velocity of the center of mass? Explain. Do your answers change if the wheel is slipping as it rolls? Why or why not?

The velocity on the point of a rim of the wheel is given by,

$\mathit{v}\mathbf{=}{\mathit{V}}_{\mathbf{C}}\mathit{M}\mathbf{+}\mathit{\omega }\mathit{R}\mathbf{(}\mathbf{-}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{\theta }{\mathit{e}}_{\mathbf{x}}\mathbf{+}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{\theta }{\mathit{e}}_{\mathbf{y}}\mathbf{)}$ (1)

Where$\omega$ angular speed of the wheel is,R is its radius and$V_{CM}$ is the velocity of its center of mass.

We have given that A wheel is rolling without slipping on a horizontal surface.

Assuming$V_{CM}=v_{cm}{e}_x$

So, from equation (1) we can write,

$v=(v_{CM}-\omega Rcos\theta )e_x+\omega Rsin\theta e_y$

When there is no slipping we have

$v_{CM}=\omega R$

So, we get

$v=(\omega R-\omega Rcos\theta )e_x+\omega Rsin\theta e_y$

Now, if velocity is purely vertical. Then,

$$\begin{gathered} \omega R\left(1-\cos \theta \right)=0 \\ \cos \theta =1 \\ \theta =0 \end{gathered}$$

However, when$\theta =0$ the vertical component of the velocity is zero.

Since$\omega R\sin \theta =0$

We can conclude that there is no point on the wheel that has a purely vertical velocity.

The horizontal velocity component opposite to the velocity of the center of mass when

$$\begin{gathered} 1-\cos \theta =0 \\ \cos \theta >1 \end{gathered}$$

This can never be satisfied.

So, there is no point that has a horizontal velocity component opposite to the velocity of the center of mass.

When there is slipping the condition$v_{CM}=\omega R$ is not satisfied.

In this case we have,$v_{CM}>\omega R$

In this case the horizontal components of the velocity is never zero which means the velocity is never purely in the vertical direction.