# Analysis of rolling without slipping

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.

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 $$\omega_1=\omega_2=\omega$$