Kinetic energy in rotation

The kinetic energy of an object rotating with angular speed \(\omega\) is

\[E_{\mathrm{K}} = \frac12 I \omega^2.\]

If the motion of the object also has a translational component, its kinetic energy is the sum of its translational and rotational kinetic energy:

\[E_{\mathrm{K}} = E_{\mathrm{K,\ translational}} + E_{\mathrm{K,\ rotational}} = \frac12 m v_{\mathrm{CM}}^2 + \frac12 I_{\mathrm{CM}} \omega^2,\]

where \(v_{\mathrm{CM}}\) is the speed of the centre of mass and \(I_{\mathrm{CM}}\) is the moment of inertia for an axis passing through the centre of mass parallel to the angular velocity vector.

With this definition of the kinetic energy, the work–kinetic energy theorem is also valid for rotating objects: the net external work done on an object equals the change in the kinetic energy of the object, including both translational and rotational kinetic energy,

\[W = \Delta E_{\mathrm{K}} = \Delta E_{\mathrm{K,\ translational}} + \Delta E_{\mathrm{K,\ rotational}}.\]

In addition to the work-energy theorem, we can also apply the principle of the conservation of mechanical energy to rotation without friction if we use the combined translational-rotational kinetic energy:

\[E_{\mathrm{M}} = E_{\mathrm{P}} + E_{\mathrm{K,\ translational}} + E_{\mathrm{K,\ rotational}} = \mathrm{constant\ if\ there\ is\ no\ friction}.\]