Rotational Kinetic Energy Revisited
If a point-mass is located at and is rotating about an axis-of-rotation with angular velocity , then the distance from the rotation axis to the mass is , or, in terms of the vector cross product, . The tangential velocity of the mass is then , so that the kinetic energy can be expressed as (cf. Eq.(B.23))
In a collection of masses having velocities , we of course sum the individual kinetic energies to get the total kinetic energy.
Finally, we may also write the rotational kinetic energy as half the inner product of the angular-velocity vector and the angular-momentum vector:B.27
Principal Axes of Rotation