The total kinetic energy of the object is found by summing the kinetic energies of all of the particles: \[\begin{aligned} K_{tot} &= \sum \frac{1}{2}m_i(\vec v_i\cdot \vec v_i) \\ &=\frac{1}{2} \sum_...The total kinetic energy of the object is found by summing the kinetic energies of all of the particles: \[\begin{aligned} K_{tot} &= \sum \frac{1}{2}m_i(\vec v_i\cdot \vec v_i) \\ &=\frac{1}{2} \sum_i m_i (\vec\omega \times \vec r_i + \vec v_{CM}) \cdot (\vec\omega \times \vec r_i + \vec v_{CM})\\ &=\frac{1}{2} \sum_i m_i (\vec\omega \times \vec r_i)\cdot(\vec\omega \times \vec r_i ) + \frac{1}{2} \sum_i m_i (\vec v_{CM}) \cdot (\vec v_{CM}) + \sum_i m_i (\vec\omega \times \vec r_i) \cdot (\ve…
