We have a disk rotating about an arbitrary axis, and we can supposedly quantify the kinetic energy of such a disk by $K = \frac 12Iw^2$.
Now, it is also true that, as the disk is rotating, each elemental mass, $dm_j$, of the body possesses an instantaneous tangential/linear velocity, $v_j$. Does this not mean that the kinetic energy of the disk is also characterized by some form of $K = \frac 12mv^2$?
I am confused as to several things. Firstly, how do we know that we are not double counting the kinetic energy of the body, by supposing it is given by $K = \frac 12Iw^2$ and $K = \frac 12mv^2$? Second, if this were true, how can we find such a velocity, $v$, given that it varies throughout the body?
I also wish to understand how this could bode well with the angular momentum of the disk. I am certain, in this case, that the body would well need have some linear momentum, given that $L$ depends upon the existence of $p$, by $L = r × p$. So, I suppose that each elemental mass has a certain $dp$, that then "sums up" to the body's overall $p$... Yet somehow, this conclusion seems funny as well...