### 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))

 (B.25)

where

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

where the second form (introduced above in Eq.(B.7)) derives from the vector-dot-product form by using Eq.(B.20) and Eq.(B.22) to establish that .

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