Euler's Dynamic Equations - Theory - helix

= 1 2 m~vT C ~v C + 1 2 ~!T I C ~! (42)This expression represents the kinetic energy as a sum of translationaland rotational parts about the center-of-mass. It can also beexpressed using linear and angular momentum,dT = 1 2 (~vT C ~ L+ ~! T ~H C ) (43)Inertial power is the time rate of change of kinetic energy,1dt T = d dt2 (~vT~ C L+~! T ~H C )= 1 2 (~aT C ~ L+~v T C( d dt ~ L)+ ~® T ~H C +~! T ( d dt ~ H C ))= 1 2 (~aT C(m~v C )+~v T C( P ~ F)+ ~®T(IC ~!)+ ~! T ( P ~ MC ))= 1 2 (~vT C(m~a C )+~v T C( P ~ F))+12 (~!T (I C ~®)+ ~! T ( P ~ MC ))= ~v T C( P ~ F)+12 (~!T (I C ~®+ ~!£I C ~!)+~! T ( P ~ MC ))= ~v T C (P ~F)+~! T ( P ~ M C ) (44)where the triple product~! T (~!£I C ~!)is introduced but identicallyvanishes. Similar to kinetic energy, power is expressed simply interms of linear and angular parts about the center-of-mass.4