Analysis of rolling without slipping

Question

For the following picture that illustrates how we can analyze rolling without slipping by considering the point of contact B to be an instantaneous axis of rotation:

It is said that the angular velocity of this rotation about B is the same as the angular velocity $\omega$ of the rotation about the center of mass. I am wondering why is this true? I understand that every point on the wheel has the same angular velocity about its center of mass C. However, I can't see how this translates to the angular velocity about B.

Answer 1

for circular motion, you know that $v=\omega r$. since we are instantaneously rotating around $B$, the speed of $A$ is $v_A=\omega_1 R$, where $R$ is the distance between A and B, and $\omega_2$ is the angular speed about $B$.

However, since we are rotating without slipping, we know that $v_{\rm CoM} = \omega R$ so $v_A=\omega_2 R$ where $\omega_2$ is the angular speed about $A$.

Thus, we have $\omega_1=\omega_2=\omega$