Rotation of center in rotational motion

For a wheel in pure rotational motion; Does the center rotate? Is there a point in wheel which do not rotate? What can we say about angular velocity of the center? From where we can start to use the term "center" for that area?

Thanks.

Answer

The angular velocity is not well defined for the center of a wheel, or for the center of mass of a rigid body, in the same sense that the number $0/0$ is indeterminate. This is a consequence of the fact that the center of mass does not move if the motion is purely rotational. To see this, let us start ab ovo...

The angular velocity is defined as

The angular velocity of a particle is measured around or relative to a point, called the origin.

(Wikipedia)

You see that the angular velocity of any particle or of any point in a rigid body is defined with respect with the origin, which is usually the center of mass, which is at rest in the case of a purely rotational motion. More precisely, one can define the angular momentum of a point of mass $m$ with respect to a chosen origin $O$ as $$ \boldsymbol{\omega}= \frac{\mathbf{r}\times \mathbf{v}}{r^2}= \frac{\mathbf{r}\times\mathbf{p}}{r m^2}= \frac{\mathbf{L}}{r^2 m} $$ where $\mathbf{r}$ is the position measured from the origin $O$. On the other hand $\mathbf{v}$, $\mathbf{p}=m\mathbf{v}$, $\mathbf{L}$ are the velocity, momentum, and angular momentum, which do not depend on the choice of the origin $O$ of the reference frame. Now, what happens if we consider another reference frame, with a different origin $O$? The angular velocity will change. To see this, imagine to calculate the angular velocity of a wheel in uniform and purely rotational motion with respect to its center, and respect to a point $O$ external to the wheel. The former is constant, the latter is not.

We can finally come back to your question. What is the angular velocity of the center $C$ of a wheel in a purely rotational motion? We again have to specify the origin $O$. Well if the origin is different from the center $O\neq C$, the angular velocity is zero, since $\mathbf{v}=0$ and $\mathbf{r}\neq0$. If instead we calculate the angular velocity of the center with respect to the center $O=C$, the angular velocity is simply not defined, since one has $\mathbf{r}= \mathbf{v}=0$, which gives the indeterminate expression $0/0$ in the equation above.

Another observation. The angular velocity is a local quantity: it can be intended as a global properties only in the case of rigid bodies, where the rotation is, by definition, rigid.