# Angular Momentuem & Kinetic Energy

The kinetic Energy of a spinning body is:

$$E_{K_{\rm total}} = \frac{1}{2}mv_{CM}^2 + I\omega^2$$

The question is: do I have to use the moment of Inertia of the center of mass? and if so, why?

If so, you can see how the first part in your expression for the kinetic energy ($$\frac{1}{2}mv_{CM}^2$$, with emphasis on the $$CM$$ in the subscript) is not independent from the second part, containing the moment of inertia $$I$$.
If you want to convince yourself, you can do a simple calculation. take a pendulum of mass $$m$$ on a mass-less rode of length $$l$$ and calculate its kinetic energy in two ways: 1) just by $$\frac{1}{2}mv^2$$ and 2) by considering it as a rigid body rotating about the point where it is fixed to the ceiling.