Proving Rotational K.E. Formula?

[greswd]

The total kinetic energy (as viewed from one inertial frame) of a free, rigid body is the sum of all the infinitesimal kinetic energies of the components that comprise the body.


How do we prove that for a rotating body

$$E_k=\frac{1}{2}\left(M_{T} v_{c}^{2} + I_{c} ω^{2}\right)$$

 

[greswd]

Where M_T stands for the total mass of all the infinitesmal components combined.

 

[mfb]

Integrate $\int \frac{1}{2}v^2 \rho dV$ (in other words, kinetic energy = 1/2m^2 for all infinitesimal m) and split v into components from translation and rotation and you will get the correct result.