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Rotational Kinetic Energy

The kinetic energy of rotation of a rigid body is obtained by first dividing it up into a collection of smaller masses, and then summing up the kinetic energies due to the tangential velocities of the individual masses making up that rigid body:
KEr = $\displaystyle\sum$($\displaystyle{1\over 2}$mivi2)   
  = $\displaystyle\sum$($\displaystyle{1\over 2}$miri2$\displaystyle\omega^{2}_{}$) = $\displaystyle{1\over 2}$$\displaystyle\omega^{2}_{}$$\displaystyle\sum$(miri2)   
  = $\displaystyle{1\over 2}$I$\displaystyle\omega^{2}_{}$. (12)
Note: The units of rotational kinetic energy are Joules (J).

When considering the total mechanical energy of a rigid body, this kinetic energy must be added to the kinetic energy of translation:

KEt = $\displaystyle{1\over 2}$mtotalvcm2 (13)

where vcm is the velocity of the center of mass.