u/FedericoArduino

I have a doubt about the second cardinal equation of the rigid body in the plane with a moving pole. If I choose the IRC (instantaneous center of rotation) as pole P, why does the correction term vp×mvg vanish? I know that instantaneously it has zero velocity, but it is not fixed in time, so it doesn’t logically make sense to me to set vp=0.
Let’s take pure rolling of a disk as an example. Here I don’t know which methodology is correct:
1. The IRC is the material point of the disk that instantaneously touches the ground. That point at that instant has zero velocity by definition of IRC, so vp=0 and the term vanishes.
2. The IRC is the geometric point of contact, i.e., the location in space where contact occurs. This point moves along the ground at the same velocity at which the disk advances, so vp = vg and is not zero. However, vp is parallel to vg (both horizontal), and the cross product is zero due to parallelism.
So the two approaches give the same result (vanishing term) but for different reasons, and I don’t understand the ‘correct’ method that holds in the general case.