Total Kinetic Energy of Rigid Systems

A rigid body undergoing general planar motion possesses kinetic energy from both translation and rotation. The total kinetic energy $K_{\text{tot}}$ separates into the translational energy of the center of mass and the rotational energy about the center of mass.

The translational kinetic energy $K_{\text{trans}}$ depends on the total mass $M$ and the speed of the center of mass $v_{\text{CM}}$:

$$K_{\text{trans}} = \frac{1}{2} M v_{\text{CM}}^2$$

The rotational kinetic energy $K_{\text{rot}}$ depends on the moment of inertia about the center of mass $I_{\text{CM}}$ and the angular velocity $\omega$:

$$K_{\text{rot}} = \frac{1}{2} I_{\text{CM}} \omega^2$$

The total kinetic energy is the scalar sum of these two independent components:

$$K_{\text{tot}} = \frac{1}{2} M v_{\text{CM}}^2 + \frac{1}{2} I_{\text{CM}} \omega^2$$

In general motion, the translational velocity $v_{\text{CM}}$ and the angular velocity $\omega$ are independent variables. The total energy represents the combined work required to accelerate the object linearly and spin it up to its rotational speed.