Rotational Kinetic Energy

The instantaneous rotational kinetic energy of a rotating rigid body is written

$$K = \frac{1}{2}\sum_{i=1,N} m_i\,{\bf v}_i\cdot{\bf v}_i.$$ (1.193)

Making use of Equation (1.181), and some vector identities (see Section A.10), the kinetic energy takes the form

$$K = \frac{1}{2}\sum_{i=1,N} m_i\,( \mbox{\boldmath$\omega$} \times {\bf r}_i)\cdot( \mbox{\boldmath$\omega$} \times {\bf r}_i) = \frac{1}{2}\, \mbox{\boldmath$\omega$} \cdot\!\! \sum_{i=1,N} m_i\,{\bf r}_i\times \mbox{\boldmath$\omega$} \times {\bf r}_i).$$ (1.194)

Hence, it follows from Equation (1.182) that

$$K = \frac{1}{2}\,\, \mbox{\boldmath$\omega$} \cdot {\bf L}.$$ (1.195)

For the special case of an axisymmetric body, making use of Equation (1.190), we obtain

$$K = \frac{1}{2}\,I_\perp\,(\omega_x^{\,2} + \omega_y^{\,2}) + \frac{1}{2}\,I_\parallel\,\omega_z^{\,2}.$$ (1.196)

For the special case of a body rotating about a principal axis of rotation,

$$K = \frac{1}{2}\,I\,\omega^2,$$ (1.197)

where $I$ is the associated principal moment of inertia. More generally,

$$K = \frac{1}{2}\,I_{xx}\,\omega_x^{\,2} + \frac{1}{2}\,I_{yy}\,\o... ...L_x^{\,2}}{2\,I_{xx}}+ \frac{L_y^{\,2}}{2\,I_{yy}}+\frac{L_z^{\,2}}{2\,I_{zz}},$$ (1.198)

assuming that the Cartesian axes are parallel to the principal axes of rotation.