Rigorous proof about the nature of rolling motion

[quincyboy7]

In Resnick/Halliday, they describe how rolling can be described as the sum of a rotational force centered at the center of mass (for a wheel, say) and translational motion. The next part involves them saying that the motion can also be described as a completely rotational motion centered at the bottom of the wheel, pointing to how the velocities work in this framework for the bottom, center of mass, and top of the wheel as a "proof".

How does one generalize this into a complete proof however, that the velocities obtained by summing rotational motion from the center of the wheel plus translational motion is equal to the velocities by a rotational motion from the bottom of the wheel? It just doesn't seem very intuitive to me and a rigorous proof might clear up some doubts. Thanks as always!

 

[ashishsinghal]

Since in rotational motion the velocity of the bottom most point is zero, there is no change in frame of reference. It is same as ground.
When we take rotational velocity from center of mass and add velocity of center of mass we are basically applying:
V(particle/ground) = V(particle/com) + V(com/ground)---> concept of relative velocity
In case of the bottom most point its velocity is zero, hence normal velocity is considered.