A rigid body can have kinetic energy from two kinds of motion at the same time: translation of its center of mass and rotation about the center of mass.

For a rigid body of total mass $M$, center-of-mass speed $V_{\rm CM}$, angular speed $\omega$, and moment of inertia $I_{\rm CM}$ about the rotation axis through its center of mass, the total kinetic energy is

$$K=\frac12MV_{\rm CM}^2+\frac12I_{\rm CM}\omega^2.$$

The first term is the kinetic energy associated with motion of the body as a whole. The second is the kinetic energy of motion relative to the center of mass.

Why the energies separate

For each mass element, write its velocity as the sum of the center-of-mass velocity and its velocity relative to the center of mass:

$$\mathbf v_i=\mathbf V_{\rm CM}+\mathbf v_i'.$$

Then

$$K=\frac12\sum_i m_i|\mathbf v_i|^2.$$

Expanding the square gives a translational term, a relative-motion term, and a cross term. By the definition of the center of mass,

$$\sum_i m_i\mathbf v_i'=\mathbf0,$$

so the cross term vanishes. For rigid rotation about the center of mass,

$$\frac12\sum_i m_i|\mathbf v_i'|^2=\frac12I_{\rm CM}\omega^2.$$

Thus translation and rotation contribute additively to the body's kinetic energy.

Worked example

A uniform solid cylinder has mass $M=2.0,\mathrm{kg}$, radius $R=0.50,\mathrm m$, center-of-mass speed $V_{\rm CM}=3.0,\mathrm{m/s}$, and angular speed $\omega=4.0,\mathrm{rad/s}$. About its symmetry axis,

$$I_{\rm CM}=\frac12MR^2=0.25,\mathrm{kg,m^2}.$$

Its translational kinetic energy is

$$K_{\rm trans}=\frac12(2.0)(3.0)^2=9.0,\mathrm J,$$

and its rotational kinetic energy is

$$K_{\rm rot}=\frac12(0.25)(4.0)^2=2.0,\mathrm J.$$

Therefore

$$K=11.0,\mathrm J.$$

If the same mass and radius were concentrated in a thin-walled hollow cylinder, $I_{\rm CM}\approx MR^2=0.50,\mathrm{kg,m^2}$. At the same translational and angular speeds its rotational kinetic energy would be $4.0,\mathrm J$, giving $13.0,\mathrm J$ total.

Two bodies can therefore have the same mass, translational speed, and angular speed but different total kinetic energies because their mass distributions give different moments of inertia.

This decomposition applies generally to rigid bodies that translate while rotating; rolling is one important application.