Why don't the Newton-Euler equations contradict Newton's second law?

Question

The Newton-Euler equation I'm looking at says that "With respect to a coordinate frame whose origin coincides with the body's center of mass," the total force required to give an object a certain acceleration with a certain rotational acceleration is: F = m(a + ω×v).

My immediate question was how does this not violate Newton's second law? Doesn't F = ma always?

So I started going through the derivation (found here: http://hades.mech.northwestern.edu/images/2/25/MR-v2.pdf), and noticed something. They say if the rigid body is modeled as a collection of point masses, if you take one that has position p, then:

\[\frac{dp}{dt} = v + ω × p\]

\[\frac{d^2p}{dt^2} = a + \frac{dω}{dt}×p + ω×\frac{dp}{dt} = a + \frac{dω}{dt}×p + ω×(v + ω×p)\]My problem/confusion with this was that as far as I can tell, in the equation dp/dt = v + ω×p, the two p variables are actually different, because when the center of mass of the object moves, I don't think it should change the velocity of any of the given points.

Can anyone help me out with this? Explain why it is set up this way and/or why this isn't a contradiction of newton's second law?

Comments

Everything after $a$ in the last equality is separately zero.